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User Manual for STABLE 5.1 STABLE Library Version

1. f xi f xila 3 7 6 param i 1 n 2 1 8 Stable score nonlinear function STABLE function stablenonlinfn n x y alpha beta gamma delta param ierr This function computes the score or nonlinear function for a stable distribution g x f x f x d dx ln f x The routine uses stablepdf to evaluate f x and numerically evaluates the derivative f x Warning this routine will give unpredictable results when 4 1 The problems occur where f a 0 is small in this region calculations of both f x and f x are of limited accuracy and their ratio can be very unreliable 2 2 Statistical functions 2 2 1 Estimating stable parameters STABLE function stablefit n x theta method param ierr Estimate stable parameters from the data in x11 T using method as described in the following table This routine calls one of the functions described below to do the actual estimation method value algorithm notes 1 maximum likelihood a gt 0 4 2 quantile a gt 0 1 3 empirical characteristic function a gt 0 1 4 fractional moment a gt 0 4 8 0 uses power p 0 2 5 log absolute moment B 8 0 6 modified quantile a gt 0 4 7 U statistic method B d Note that the fractional moment and log absolute moment methods do not work when there are zeros in the data set 2 2 2 Maximum likelihood estimation STABLE function stablefitmle n x theta param ierr Estimate the st
2. 4 4 5 Fit amplitude data STABLE function mvstablefitamplitude d nr r alpha gamma0 method ierr Estimate the parameters a and yy for amplitude data r contains the univariate amplitude data values d is the dimension of the underlying distribution that the amplitude data comes from met hod is the method to use for estimating This initial implementation allows only method 5 which uses method of moments on the log of the amplitude data Other methods are planned for the future The function returns the estimated value of a and yo 4 4 6 Amplitude score function STABLE function mvstableamplitudescorefn nr r g alpha gamma0 d ierr Compute the score function of the amplitude g r f r f r where f r fr rla yo d is the amplitude pdf defined above Current implementation works for a 0 8 2 d lt 98 There seems to be a relative error of approximately 3 for large r 4 5 Faster approximations to multivariate routines There are a limited number of functions for quickly calculating multivariate functions in the 2 dimensional isotropic case Such a distribution is specified by the index of stability a the scale yo and the location 01 62 Because the description is simple these functions use those arguments directly and do not use a distribution descriptor 4 5 1 Quick log likelihood for bivariate isotropic case STABLE function mvstableqkloglikisotropic2d n x loglik alpha gamma0 delta ierr Comp
3. Computes the stable cumulative distribution function using a series expansion with nterms in it This function is best used to calculate the cdf near the origin in the 1 parameterization The series is not defined for a 1 Note that nterms 1 corresponds to a constant term nt erms 2 corresponds to a linear term etc STABLE User Manual 13 2 4 3 Series approximation of stable pdf at the tail STABLE function stablepdfseriestail n x y alpha beta gamma delta param nterms ierr Computes the stable probability distribution function using a series approximation with nterms in it This function is best used to calculate points on the tail of a distribution The series is defined only for x gt 0 For x lt 0 replace x by x and 8 by 6 The series is not defined for a 1 2 4 4 Series approximation of stable cdf at the tail STABLE function stablecdfseriestail n x y alpha beta gamma delta param nterms ierr Computes the stable cumulative distribution function using a series approximation with nterms in it This function is best used to calculate points on the tail of a distribution The series is defined only for x gt 0 For x lt 0 replace x by x and 8 by 6 The series is not defined for a 1 2 5 Faster approximations to basic functions The functions described in preceding sections are accurate but can take a long time to compute For evalu ating a single pdf or cdf at a single set of parameter value
4. 1 if it has characteristic function _ fexp 7 ule 1 ib tan E signu i u afl iio in i y ul 1 782 sign u In u i u a 1 2 Note that if 6 0 then these two parameterizations are identical it is only when 8 0 that the asymmetry term the imaginary factor involving tan 4 or 2 becomes relevant More information on parameterizations and about stable distributions in general can be found at http academic2 american edu Jpnolan which has a draft of the first chapter of Nolan 2010 The next section gives a description of the different functions in STABLE 2 Univariate Stable Functions If you are calling the library directly from a C Fortran or Visual Basic program you need to understand the calling sequence and arguments All functions are implemented through a set of internal routines For portability reasons the internal routines pass all arguments including lengths of any vector used error return codes and results of the computations For these users the function prototypes are provided All real variables x y alpha beta etc are double precision numbers e g 8 byte long floating point numbers while all integers n param ierr are 4 bytes long Do not use complex numbers while calling these routines Basic error checking is done but no guarantee is made about the behavior of a function if invalid arguments are used when calling them The descriptions below assume array subscripts start at 1 i
5. A ei ek 2 1 4 Simulate stable random variates 2 es 2 1 5 Stable hazard function 2 1 6 Derivative of stable densities ee 2 1 7 Second derivative of stable densities 2 1 8 Stable score nonlinear function Statistical functioms 20 Qt A ERE ee LA aed Me AA Rie EG AE 2 2 1 Estimating stable parameters e 2 2 2 Maximum likelihood estimation 2 2 3 Maximum likelihood estimation with restricted parameters 2 2 4 Maximum likelihood estimation with search control 2 2 5 Quantile based estimation 0 ee 2 2 6 Empirical characteristic function estimation o 2 2 7 Fractional moment estimation 2 2 8 Log absolute moment estimation o 2 2 9 Quantile based estimation version 2 e 2 2 10 U statistic based estimation 2 2 11 Confidence intervals for ML estimation sooo a 2 2 12 Information matrix for stable parameters ooo a 2 2 13 Log likelihood computation e 2 2 14 Chi squared goodness of fit test o oo a 2 2 15 Kolmogorov Smirnov goodness of fit test 2 ee ee NO 00 00 00 00 00 00 NANN N N YI A DODODODODOQOQDO0ssunr Ur Ur Ur Un Un A 2 2 16 Likelihood ratio test ok a A ee ee eR ea Oe 9 2 27 Stable TESTESSION s 2 5 aie Reeth EA AON Rd ke ee Ee 10 2 2 18 Stable regression profile likelihood parameter confidence intervals 10 Inform
6. all stable distributions any finite spectral measure A can be approximated by a discrete measure see Byczkowski et al 1993 Below is a plot of the density surface of a bivariate stable density with three point masses each of weight 1 at locations cos 7 3 sin 7 3 1 0 and cos 57 3 sin 57 3 STABLE User Manual 18 density surface contours of density surface triangle alpha 1 3 XX 0 4 Multivariate Stable Functions Since the specification of a multivariate stable distribution is somewhat cumbersome a different approach from the univariate case is taken in these routines Two steps are needed to work with a multivariate stable distribution First the distribution is specified by calling a function to define the distribution Second call a separate functions to compute densities cumulatives simulate etc The programs for working with multivariate stable distributions are less well developed and generally limited to 2 dimensions At the current time when dimension is greater than 2 you can a simulate using mvstablernd b calculate the pdf using mvstablepdf if the components are independent OR the spectral mea sure has exactly d point masses c calculate the cdf using mvstablecdf if the components are independent or d calculate the cdf using mvstablecdfMC by Monte Carlo estimation for any type of distribution The accuracy of the pdf and cdf calculations are limited In all cases X is a column vector this is i
7. between 1 and 1 9 and beta is near 0 the approximation is good 2 6 Discrete stable distributions Given a stable distribution X S a 3 7 6 param and a pair of cutoff values a lt b the random variable Y integer part of max a min X b 1s a discrete stable distribution These distribution arise in signal processing where a continuous quantity is quantized digitized and limited accuracy is kept It is assumed that the cutoff values are integers The saturation probability is P X lt a 1 2 P X gt b 1 2 and is a measure of how much of the distribution is lost by truncating at the cutoff values In the routines below the cutoff is specified by a vector of length 2 cutoff a b In this section X will always refer to the continuous stable distribution while Y will always refer to a discrete quantized integer valued distribution In the internal routines the x values are integers The matlab R Mathematica interfaces use double pre cision values 2 6 1 Discrete stable density STABLE function stablepdfdiscrete n ix f alpha beta gamma delta cutoff param ierr Calculates f P Y 2 i 1 n STABLE User Manual 15 2 6 2 Quick discrete stable density STABLE function stablegkpdfdiscrete n ix f alpha beta gamma delta cutoff param ierr Calculates f P Y x i 1 n Faster than above less accurate 2 6 3 Discrete stable cumulative distribution function STABLE func
8. distributions Results are approximately the same as log f x 2 5 4 Quick stable quantile computation STABLE function stablegkinv n p x alpha beta gamma delta param ierr STABLE User Manual 14 Call is identical to Section 2 1 3 but much faster Note the comments in that section about extreme upper quantiles 2 5 5 Quick stable hazard function computation STABLE function stablegkhazard n x h alpha beta gamma delta param ierr Call is identical to Section 2 1 5 2 5 6 Quick stable likelihood computation STABLE function stableqkloglik n x theta param loglik ierr Call is identical to Section 2 2 13 2 5 7 Quick stable score nonlinear function STABLE function stablegknonlinfn n x y alpha beta gamma delta param method ierr This function approximates the score or nonlinear function for a stable distribution g a f 1 f x d dx ln f x The algorithm used depends on the value of method When method 1 stablegkpadf is used to compute f a and in the numerical evaluation of f x When method 2 stablescorefn is used to compute g x on a grid then a spline is fit to those values The resulting spline is used to approxi mate g x If n is large this is noticeably faster than either stablescorefn or method 1 above When method 3 a rational function approximation is used to approximate g x This is the fastest method but the accuracy depends on the values of alpha and beta If alpha is
9. form A r and search for a value of r that makes P X A r close to p That procedure involves bivariate numerical integration will take much longer than this function 4 3 Statistical functions 4 3 1 Estimate a discrete spectral measure fit a stable distribution to bivariate data STABLE function mvstablefit nvect x alpha delta nspectral angle lambda method2d methodld iparam ierr x contains the data values nspectral is the number of points in the estimated spectral measure must be divisible by 4 met hod1d is the method to use for estimating univariate stable parameters internally see Section 2 2 1 for codes only used if method2d 1 method2d is the method to use in estimating bivari ate distribution Use met hod2d 1 for Rachev Xin Cheng method met hod2d 2 for projection method method2d 3 for empirical characteristic function method The methods are described in Nolan et al 2001 see Nolan and Panorska 1997 for some discussion of suggested values and diagnostics Suggest using nspect ral 40 method1d 3 method2d 2 param 1 The function returns a list structure that contains information about the fit which is always done as a discrete spectral measure The fields in the fit are the estimated value of a the estimated shift location vector 0 angle which is a uniform grid from 0 to 27 of length nspect ral and Lambda for the estimated weights at each position 4 3 2 Estimate parameter functions STABLE function mvs
10. parameters were estimated from the data then this tail probability will be an overestimate of the significance level 2 2 16 Likelihood ratio test STABLE function stablelrt n x abnd bbnd results ierr This function computes the likelihood ratio Lo L1 where Lo is the maximum likelihood of the data x under the assumption that x is an i i d sample from a stable distribution with a and restricted to the range abnd 1 lt a lt abnd 2 and bbnd 1 lt lt bbnd 2 and L is the maximum likelihood of the data under an unrestricted stable model The function computes the maximum likelihood using the quick approximation to stable likelihoods so is limited to a in the range 0 4 2 The vector results will contain the results of the computations results ratio of the likelihoods results 2 log ratio of likelihoods results log likelihood of the data for the restricted HO results log likelihood of the data for the unrestricted H1 results 5 estimated value of alpha under HO results estimated value of gamma under HO results estimated value of delta under HO results estimated value of alpha without assuming HO results estimated value of beta without assuming HO results results estimated value of gamma without assuming HO 1 2 3 4 5 results 6 estimated value of beta under HO 7 8 9 1 1 12 estimated value of delta without assuming HO 0 1 2 Note that under the standard assumpti
11. set to 1 and 8 is left unchanged This is to avoid computations involving 3 tan ra 2 which blows up as a gt 1 if 8 40 3 Near Cauchy case if a 1 lt e and 8 lt e then ais set to 1 and Pis set to 0 4 Near L vy case if la 1 2 lt e and G 1 lt e then ais set to 1 2 and is set to 1 if a 1 2 lt e and 8 1 lt e then a is set to 1 2 and is set to 1 STABLE User Manual 27 code type meaning 101 error Invalid input parameter 102 warning Accuracy warning alpha lt 1 103 warning vmax exceeded in mvstablepdf 104 error Too many points in spectral measure 105 error nspectral must be divisible by 4 106 error This parameterization is not allowed in this function 107 error Too few uniform 0 1 input values for simulation 108 error Distribution not defined 109 error mvstablecdf not implemented for nonsymmetric case 110 error Matrix is not positive definite 111 error alpha must be at least 0 8 112 error Definition error 113 error Dimension is greater than the max allowed 115 error Spline error 150 error Not enough memory 151 error Error in a subroutine Table 4 Multivariate error codes References Abdul Hamid H and J P Nolan 1998 Multivariate stable densities as functions of one dimensional projections J Multivar Anal 67 80 89 Byczkowski T J P Nolan and B Rajput 1993 Approximation of multidimensional stable densiti
12. specifying the skewness at each point mass delta is the shift as a column vector param is the parameterization must be 0 or 1 dist contains an identifier used to specify the distribution when calling other functions The spectral measure is defined by putting mass 1ambdal j 1 betal j 2 at sj and mass 1ambdafj 1 betalj 2 at s Setting all beta equal to 1 gives the standard definition of a spectral measure with mass lambda 3 at s 3 Setting all beta equal to 0 guarantees that the distribution is symmetric putting weight 1 ambdalj 2 at s If any element of beta is not 0 the distribution is assumed to be nonsymmet ric It is possible to manually make a spectral measure symmetric with nonzero beta by defining antipodal points and weights and values of beta that balance correctly However STABLE does not detect this Some parts of STABLE are significantly faster and more accurate in the symmetric case e g density calculations and simulations 4 1 5 Discrete spectral measure in 2 dimensions STABLE function mvstablediscspecmeas2d alpha nlambda angle lambda delta beta iparam dist ierr STABLE User Manual 20 Define a bivariate stable distribution with discrete spectral measure This is a special case of the previous function In two dimensions the locations of the point masses can be specified by angles anglel j gives the angle in radians of the location of s cos ang1le j sin angle dist contains a
13. symmetric 3 1 if distribution is symmetric 0 otherwise indep 4 1 if the components are independent 0 otherwise nlambda 5 number of point masses in the spectral measure may be 0 param 6 parameterization ns 7 number of values in the s array seqnum 8 internal index of this distribution Table 1 Integer values returned by mvstableinfo where cutoff a b are the upper and lower cutoff values Note that the same cutoff is used for both components of X These distributions arise in signal processing where a bivariate continuous quantity is quantized digitized and limited accuracy is kept It is assumed that the cutoff values are integers The satura tion probability is Psat P X lt a 1 2 P X gt b41 2 P X2 lt a 1 2 P X3 gt b 1 2 and is a measure of how much of the distribution is lost by truncating at the cutoff values In the internal routines the x values are integers The R Mathematica and matlab interfaces store these integer values in double precision numbers 4 6 1 Discrete bivariate density STABLE function mvstablepdfdiscrete2d n x p cutoff eps method ierr Compute the pdf of a discrete bivariate stable distribution x should be a 2 x n matrix of integer values cutoff is a vector of length 2 with upper and lower cutoff values for the truncation The typical value for cutoff is 128 127 both components of X4 X2 are truncated at the same value The function returns a vector p o
14. 00 4 21 4 2 5 Find a rectangle with probability at least p o a 21 43 Statistical functions s re a s e a e A Body oe gas 21 4 3 1 Estimate a discrete spectral measure fit a stable distribution to bivariate data 21 4 3 2 Estimate parameter functions e a 21 4 3 3 Fit an elliptical stable distribution to multivariate data 22 4 4 Amplitude distribution k 0 E aa ae ee 22 4 4 1 Amplitude cumulative distribution function 22 44 2 Amplitudedensity e 00 R E y E a ee ee 22 4 4 3 Amplitude quantiles e 22 4 4 4 Simulate amplitude distribution e e 22 4 43 Fitcamplitude data 490 a e rE a oe E E S 23 4 4 6 Amplitude score function o oaoa a 23 4 5 Faster approximations to multivariate routines o oo 2 2 0 ee ee 23 4 5 1 Quick log likelihood for bivariate isotropic case 2 o ee 23 4 5 2 Quick amplitude density in bivariate case 2 2 2 2 a 23 4 6 Bivariate discrete stable distribution o o ooa e 23 4 6 1 Discrete bivariate density a 24 4 7 Multivariate informational utility functions oaoa e e 24 4 7 1 Information about a distribution oaoa e e o 24 4 7 2 Compute projection parameter functions o 25 4 7 3 Multivariate convert parameterization a 25 5 Error return codes 26 Re
15. 01 Estimation of stable spectral measures Mathematical and Computer Modelling 34 1113 1122 Nolan J P and A K Panorska 1997 Data analysis for heavy tailed multivariate samples Comm in Stat Stochastic Models 13 687 702 Nolan J P and B Rajput 1995 Calculation of multidimensional stable densities Commun Statist Simula 24 551 556 Index STABLE functions 10 21 mvstableamplitudecdf 22 mvstableamplitudepdf 22 mvstableamplitudequant 22 mvstableamplitudernd 22 mvstableamplitudescorefn 23 mvstablecdf 20 mvstableconvert 25 mvstablediscspecmeas 19 mvstablediscspecmeas2d 19 mvstableelliptical 19 mvstablefindrect 21 mvstablefit 21 mvstablefitamplitude 23 mvstablefitelliptical 22 mvstablefitparfn2d 21 mvstableindep 19 mvstableinfo 24 mvstableisotropic 19 mvstableparfn2d 25 mvstablepdf 20 mvstablepdfdiscrete2d 24 mvstableqkamplitudepdf2d 23 mvstableqkloglikisotropic2d 23 mvstablernd 21 mvstableundefine dist ierr 20 stablecdf 5 stablecdfdiscrete 15 stablecdfseriesorigin 12 stablecdfseriestail 13 stablechisq 8 stableconvert 12 stablediscretefindgamma 15 stablediscretemle 15 stablefit 6 stablefitecf 7 stablefitfracmoment 7 stablefitlogabs 7 stablefitmle 6 stablefitmleO 7 stablefitmleci 8 stablefitmleinfomatrix 8 stablefitmlerestricted 7 stablefitquant 7 stablefitquant2 8 stablefituest 8 stablegettol 12 stablehazard 5 stableinv 5 stableksgo
16. 1 Discrete stable density o e 0000000 14 2 6 2 Quick discrete stable density 2 2 o e 15 2 6 3 Discrete stable cumulative distribution function 0 0 15 2 6 4 Quick discrete stable cumulative distribution function 15 2 6 5 Simulate discrete stable random variates o o e 15 2 6 6 Simulate discrete stable random variates with specified saturation probability 15 2 6 7 Find scale y to have a specified saturation probability for a discrete stable distribution 15 2 6 8 Discrete maximum likelihood estimation 0 0 15 3 Multivariate Stable Introduction 17 4 Multivariate Stable Functions 18 4 1 Define multivariate stable distribution e e o 18 4 1 1 Independent components 19 4 1 2 Isotropic Stable vos A ee ke 19 4 1 3 Ellipticalstable cen a ae een Be A ee 19 4 1 4 Discrete spectral measure 2 2 0 0 0000000 E 19 4 1 5 Discrete spectral measure in 2 dimensions 0 19 4 1 6 Undefine a stable distribution e e o 20 4 2 BAasIC TUOCUONS 2er A RA A OR 20 422 17 Density Tnction vaa dl ee aed Re hee BPS SW a 20 4 22 Cumulative functio n os 2s hte tle eee eve OR Ge we Arve ak es ee oe 20 4 2 3 Cumulative function Monte Carlo 0 2 00 00 2 eee eee 21 4 2 4 Multivariate simulation 0 2 0 0 0 00 0 0
17. User Manual for STABLE 5 1 STABLE Library Version Abstract This manual gives information about the STABLE library which computes basic quantities for univari ate stable distributions densities cumulative distribution functions quantiles and simulation Statistical routines are given for fitting stable distributions to data and assessing the fit Utility routines give in formation about the program and perform related calculations Quick spline approximations of the basic functions are provided Densities cumulative distribution functions and simulation for discrete quantized stable distributions are described The multivariate module gives functions to compute bivariate stable densities simulate stable random vectors and fit bivariate stable data In the radially symmetric case the amplitude densities cumulative distribution functions quantiles are computed for dimension up to 100 92002 2009 by Robust Analysis Inc 6618 Allegheny Avenue Takoma Park MD 20912 4616 USA phone and fax 301 891 8484 www RobustAnalysis com Revised 9 July 2009 processed July 11 2009 STABLE User Manual 2 Contents 1 Univariate Stable Introduction 2 Univariate Stable Functions 2 1 2 2 2 3 2 4 2 5 2 6 A BASIC TUDCUODS 4 4 reia A Qo BS A IG Gs EI E ER DARAS E Bek 2 11 Stableidensiti s ot ean a Bg ee Be eS ee ee ee 2 1 2 Stable distribution functions o sooo e ee 213 Stable quantiles dell ess ii Se Bats Ao
18. able parameters for the data in x1 T in parameterization param using maximum likelihood estimation The likelihood is numerically evaluated and maximized using an optimization routine This program and the numerical computation of confidence intervals below are described in Nolan 2001 For speed reasons the quick log likelihood routine is used to approximate the likelihood this is where the restriction a gt 0 4 comes from STABLE User Manual 7 2 2 3 Maximum likelihood estimation with restricted parameters STABLE function stablefitmlerestricted n x alpha beta gamma delta param restriction ierr This is a modified version of maximum likelihood estimation where some parameters can be estimated while the others are restricted to a fixed value The function receives theta alpha beta gamma delta and if restriction i 1 then theta i is fixed 2 2 4 Maximum likelihood estimation with search control STABLE function stablefitmle0 n x method gamma_init delta_init alpha0 beta0 gamma0 delta0 ftol tlower tupper ierr This is maximum likelihood estimation with greater control over the search and ranges for the parameters It is used internally 2 2 5 Quantile based estimation STABLE function stablefitquant nx x theta param ierr Estimate stable parameters for the data in x using the quantile based on the method of McCulloch 1986 It sometimes has problems when a is small say lt 1 2 and the d
19. ata is highly skewed Try the modified version below in such cases 2 2 6 Empirical characteristic function estimation STABLE function stablefitecf nx x gamma0 delta0 theta param ierr Estimate stable parameters for the data in x using the empirical characteristic function based method of Koutrovelis Kogon Williams described in Kogon and Williams 1998 An initial estimate of the scale gamma0 and the location delta0 are needed to get accurate results We recommend using the quantile based estimates of these parameters as input to this routine 2 2 7 Fractional moment estimation STABLE function stablefitfracmoment nx x theta p ierr Estimate stable parameters for the data in x using the fractional moment estimator as in Nikias and Shao 1995 This routine only works in the symmetric case it will always return 3 0 and 6 0 In this case the 0 parameterization coincides with the 1 parameterization so there is no need to specify parameterization p is the fractional moment power used A reasonable default value is p 0 2 take p lt a 2 to get reasonable results This method does not work if there are zeros in the data set negative sample moments do not exist Remove zero values and possibly values close to 0 from the data set if you want to use this method 2 2 8 Log absolute moment estimation STABLE function stablefitlogabs nx x theta ierr Estimate stable parameters for the data in x using the log absolute mo
20. ational utility functions 2 2 0 0 0 000000222 eee 11 2 3 1 Version information 2 0 000 000 eee eee eee 11 2 3 2 Modes of stable distributions 2 2 20 0 0 2002 02 0000 11 2 3 3 Setintemal tolerance mos RE ee Be ee 11 2 3 4 Get mtermal tolerance gt cos 22 poa aa a ee a RE ee ee ae 12 2 3 5 Convert between parameterizations 0 00000 12 23 0 Omega UNC is eg De ee te Re a eas ee SRA SE ba a ak 12 Series approximations to basic distribution functions o o 12 2 4 1 Series approximation of stable pdf around the origin 12 2 4 2 Series approximation of stable cdf around the origin 12 2 4 3 Series approximation of stable pdf at the tail 13 2 44 Series approximation of stable cdf at the tail 2 2 2 13 Faster approximations to basic functions 2 o 0 000000002 eee 13 2 5 1 Quick stable density computation o o o 13 2 5 2 Quick stable cumulative computation o o e 13 2 5 3 Quick stable log pdf computation e 13 2 5 4 Quick stable quantile computation o ooa 13 2 5 5 Quick stable hazard function computation s oo 14 2 5 6 Quick stable likelihood computation s s oa e 14 2 5 7 Quick stable score nonlinear function a 14 Discrete stable distributions 2 2 2 ee 14 STABLE User Manual 3 2 6
21. cal value from a normal distribution i e use z 1 96 for a 95 confidence interval For example the point estimate of a is theta 1 and the confidence inter val is theta 1 sigtheta 1 For the confidence interval is theta 2 sigtheta 2 for y the confidence interval is theta 3 sigtheta 3 For the confidence interval is theta 4 sigtheta 4 These values do not make sense when a parameter is at the boundary of the parameter space e g a 20r 8 1 These values are numerically approximated using a grid of numerically computed values in Nolan 2001 The values have limited accuracy especially when a lt 1 2 2 12 Information matrix for stable parameters STABLE function stablefitmleinfomatrix theta infomat ierr Returns the 4 x 4 information matrix for maximum likelihood estimation of the stable parameters for parameter values theta This is done in the continuous 0 parameterization These are approximate values interpolated from a grid of numerically computed values in Nolan 2001 for a gt 0 5 The values have limited accuracy especially when a lt 1 2 2 13 Log likelihood computation STABLE function stableloglik n x theta param loglik ierr Compute the log likelihood of the data assuming an underlying stable distribution with the specified parameters 2 2 14 Chi squared goodness of fit test STABLE function stablechisq n x theta param nclass chisq ierr STABLE User Manual 9 Comput
22. ctral measure OR the shape matrix R for an elliptical stable law s d nlambda 1 to d nlambda ns s matrix with d rows and ns d columns beta d nlambda ns 1 to d 2 nlambda ns 8 values associated with point masses Table 2 Double values returned by mvstableinfo 4 7 2 Compute projection parameter functions STABLE function mvstableparfn2d n angle alpha beta gamma delta ierr Compute the exact parameter functions for a bivariate stable distribution For direction t R X t is univariate stable with parameters a 3 t y t 9 t This function computes the parameter functions and 6 at the values t cos angle j sin anglel j Angles in angle are given in radians 4 7 3 Multivariate convert parameterization STABLE function mvstableconvert newparam d shift ierr Converts between multivariate stable parameterizations newparam must be 0 or 1 In the uninterfaced function mvsconvert the parameterization newparam of the current parame terization stored internally from definition of current stable distribution The new shift vector shift is computed and returned Calling this routine changes the internal parameterization to newparam and changes the internal shift parameter to shift STABLE User Manual 26 5 Error return codes An error is unrecoverable and stops execution For example if you ask to compute the density of a stable parameter with a 3 you will get a return c
23. d 3 2 3 6 Omega function STABLE function stableomega n u omegar omegai alpha beta param Compute the function w u a 8 k i 1 n where tula o e do al aula 91 Ll ee These functions are from the characteristic functions of standardized univariate stable distributions if Z S a 6 1 0 k then Fexp tuZ exp w ula 8 k As before k 0 or k 1 correspond to two different parameterization The function returns two vectors containing the real and imaginary parts of co uja B k tan 2 signu lu 9 1 a 1 sign u In ul a tan 3 amp signu a 1 sign u In u a 1 3 AA A 2 4 Series approximations to basic distribution functions These functions use the Bergstrom series for stable densities and cdfs which are only defined for a 1 2 4 1 Series approximation of stable pdf around the origin STABLE function stablepdfseriesorigin n x y alpha beta gamma delta param nterms ierr Computes the stable probability distribution function using a series approximation with nterms in it This function is best used to calculate the density near the origin in the 1 parameterization The series is not defined for a 1 Note that nterms 1 corresponds to a constant term nterms 2 corresponds to a linear term etc 2 4 2 Series approximation of stable cdf around the origin STABLE function stablecdfseriesorigin n x y alpha beta gamma delta param nterms ierr
24. de of a S theta 1 theta 2 y theta 3 6 theta 4 param distribution If 6 4 0 the mode is determined by a numerical search of the pdf 2 3 3 Set internal tolerance STABLE function stablesettol inum value Sets the value of an internal variable that is used during computations You change these values at your own risk computation times can become very long and some choices of the parameters can cause infinite loops inum meaning 1 enable internal warning messages Warning this will not work when used from R Mathematica or matlab relative error for pdf numerical integration relative error for cdf numerical integration relative error for quantile search alpha and beta rounding x tolerance near zeta exponential cutoff peak strim location tolerance stabletrim tolerance minimum alpha minimum xtol threshold for quantile search x tolerance FP OTOoOmAAANANFWNK OO ja STABLE User Manual 12 2 3 4 Get internal tolerance STABLE function stablegettol inum value Returns the value of the internal settings see the preceding function for the meanings of each variable 2 3 5 Convert between parameterizations STABLE function stableconvert param theta newparam thetanew Convert from the parameters given in theta given in the param parameterization to the parameters thetanew given in the newparam parameterization Currently param and newparam are restricted to the values 0 1 2 an
25. e chi squared goodness of fit statistic for the data in 11 T using nclass equally probable classes bins 2 2 15 Kolmogorov Smirnov goodness of fit test STABLE function stableksgof n x theta method d tailprob param ierr This function computes the Kolmogorov Smirnov two sided test statistic D sup F x F 2 oco lt 4 lt oo where F is the stable cdf with parameters a theta 1 8 theta 2 y theta 3 theta 4 and F is the sample cdf of the data in x Use method 0 for quick computations stableqkcdf is used to compute cdf use met hod 1 for slower computations st ablecdf is used to compute cdf The routine returns the observed value of D and an estimate of the tail probability P D gt d i e the significance level of the test This tail probability is calculated using Stephen s approximation to the limiting distribution e g n 0 12 0 11n 2 D is close to the limiting Smirnov distribution This is close to n _D for large n and a better approximation on the tails for small n Note this calculation is not very accurate if the tail probability is large but these cases aren t of much interest in a goodness of fit test If you don t like this approximation the function returns D and you can compute your own tail probability WARNING the computation of the significance level is based on the assumption that the parameter values theta a 3 7 6 were chosen independently of the data If the
26. es J Multivar Anal 46 13 31 Chambers J C Mallows and B Stuck 1976 A method for simulating stable random variables Journal of the American Statistical Association 71 354 340 344 Fan Z 2006 Parameter estimation of stable distributions Communications in Statistics Theory and Methods 35 245 256 Kogon S and D Williams 1998 Characteristic function based estimation of stable parameters In R Adler R Feldman and M Taqqu Eds A Practical Guide to Heavy Tailed Data pp 311 338 Boston MA Birkhauser McCulloch J H 1986 Simple consistent estimators of stable distribution parameters Communications in Statistics Simulation and Computation 15 1109 1136 Nikias C L and M Shao 1995 Signal Processing with Alpha Stable Distributions and Applications New York Wiley Nolan J P 1997 Numerical calculation of stable densities and distribution functions Commun Statist Stochastic Models 13 759 774 Nolan J P 2001 Maximum likelihood estimation of stable parameters In O E Barndorff Nielsen T Mikosch and S I Resnick Eds L vy Processes Theory and Applications Boston Birkhauser Nolan J P 2010 Stable Distributions Models for Heavy Tailed Data Boston Birkhauser In progress Chapter 1 online at academic2 american edu jpnolan Nolan J P and D Ojeda 2006 Linear regression with general stable errors Preprint Nolan J P A Panorska and J H McCulloch 20
27. es P a lt X lt b by simulating n indepedent random vectors with the same distribution as X and counting how many are in the interval a b It works for any distribution and dimension that can be simulated 4 2 4 Multivariate simulation STABLE function mvstablernd dist n d x nu u ierr Simulate n stable random vectors from the stable distribution dist This works for any distribution that can be defined in dimensions d gt 2 dis the dimension x will be filled with the results nu is the number of elements of u and u is a vector of user supplied 1 1 d Uniform 0 1 rv used by the simulation routine nu should be at least nx nunif where the latter can be gotten from function mvstableinfo 4 2 5 Find a rectangle with probability at least p STABLE function mvstablefindrect r p ierr Find a number r so that the rectangle A A r r r x r r has P X A gt p where X isa bivariate stable distribution defined by dist This is used for technical calculations e g in approximating the probability of unbounded regions The method uses univariate projections and will generally give an overestimate of r The method is less accurate for small p or if the distribution is not centered or highly skewed it gets more accurate if p is close to 1 and the distribution is centered and symmetric If p is not too close to 1 one can get a better value of r by making repeated calls to the multivariate cdf function with rectangles of the
28. f 9 stableloglik 8 28 stablelrt 9 stablemode 11 stablenonlinfn 6 stableomega 12 stablepdf 5 stablepdfderiv 6 stablepdfdiscrete 14 stablepdfsecondderiv 6 stablepdfseriesorigin 12 stablepdfseriestail 13 stableqkcdf 13 stableqkcdfdiscrete 15 stableqkhazard 14 stableqkinv 13 stableqkloglik 14 stableqklogpdf 13 stableqknonlinfn 14 stableqkpdf 13 stableqkpdfdiscrete 15 stableregression 10 stablernd 5 stablernddiscrete 15 stablernddiscrete2 15 stablesettol 11 stableversion 11
29. f length n with pi P Y xi PY tu Yo 04 Note that eps is the attempted accuracy for each probability p not for the total error The probabilities are computed using the bivariate cdf function above and thus only works for symmetric stable two dimen sional distributions It s accuracy is limited it is likely that when all possible values of x are used gt gt p will be slightly different from 1 The current implementation is slow The method variable is unused at the current time it will be used for faster approximations in future implementations 4 7 Multivariate informational utility functions 4 7 1 Information about a distribution STABLE function mvstableinfo dist ni info nr rinfo Returns information about distribution dist Useful for checking that definition Values are returned in two arrays info and rinfo The user is responsible for specifying ni and nr and allocating an integer array info and a double array rinfo to hold returned values ni should be at least 9 nr must be big enough to hold all the values needed for this distribution Tables 1 and 2 specify how the fields are packed into the two arrays Note that subscripts are 0 origin in C if you call this from some other language e g Fortran R etc you must shift subscripts up by 1 STABLE User Manual 25 field subscript range meaning alpha 0 index of stablity a delta ltod lambda d 1 to d nlambda point masses in a discrete spe
30. f you are using C you must correct for the 0 origin The parameters of the stable distribution must be specified In the basic routines the parameter values alpha beta gamma delta paramare passed individually The base routines return a status error code in the integer variable ierr A zero is a normal return other values are documented in Section 5 The STABLE library is not reentrant on a single computer only one user should be using the library at once The user should be aware that these routines attempt to calculate quantities related to stable distributions with high accuracy Nevertheless there are times when the accuracy is limited If is small the pdf and cdf have very abrupt changes and are hard to calculate When some quantity is small e g the cdf of the light tail of a totally skewed stable distribution the routines may only be accurate to approximately ten decimal places The remainder of this section is a description of the functions in the STABLE library STABLE User Manual 5 2 1 Basic functions 2 1 1 Stable densities STABLE function stablepdf n x y alpha beta gamma delta param ierr This function computes stable density functions pdf y f x f x la 8 y param i 1 n The algorithm is described in Nolan 1997 2 1 2 Stable distribution functions STABLE function stablecdf n x y alpha beta gamma delta param ierr This function computes stable cumulative distribution
31. ferences 27 Index 27 STABLE User Manual 4 1 Univariate Stable Introduction Stable distributions are a class of probability distributions that generalize the normal distribution Stable distributions are a four parameter family a is the tail index or index of stability and is in the range 0 lt a lt 2 is a skewness parameter and is in the range 1 lt lt 1 y is a scale parameter and must be positive and 6 is a location parameter an arbitrary real number There are numerous meanings for these parameters We will focus on two here which we call the 0 parameterization and the 1 parameterization The STABLE programs use a variable param to specify which of these parameterizations to use If you are only concerned with symmetric stable distributions the two parameterizations are identical For non symmetric stable distributions we recommend using the 0 parameterization for most statistical problems and only using the 1 parameterization in special cases e g the one sided distributions when a lt 1 and 8 1 Since there are no formulas for the density and distribution function of a general stable law they are described in terms of their characteristic function Fourier transform A random variable X is S a 8 y 9 0 if it has characteristic function exp y u 1 ib tan 2 sign u yu 2 1 idu a Al Ee X 2 1 OS oe y ul 1 722 sign u In y u idu a 1 A random variable X is S a B y 9
32. functions cdf y F x F x la 6 y 9 param i 1 n The algorithm is described in Nolan 1997 2 1 3 Stable quantiles STABLE function stableinv n p x alpha beta gamma delta param ierr This function computes stable quantiles the inverse of the cdf x F 1 p i 1 n The quantiles are found by numerically inverting the cdf Note that extreme upper tail quantiles may be hard to find because of subtractive cancelation in double precision arithmetic 1 p and 1 are indistinguishable for small p less than approximately10 1 STABLE will correctly return F 1 p F7 1 00 for most values of a and 3 You can get better accuracy on the lower tails where there is no subtractive cancelation use the reflection property F x a 3 1 F zla 6 Also note that the accuracy of the inversion is determined by two internal tolerances See Section 2 3 3 1 tolerance 10 is used to limit how low a quantile can be searched for The default value is p 107 0 quantiles below p will be set to the left endpoint of the support of the distribution which may be oo Likewise quantiles above 1 p will be set to the right endpoint of the support of the distribution which may be 00 2 tolerance 2 is the relative error used when searching for the quantile The search tries to get full precision but if it can t it will stop when the relative error is less than tolerance 2 2 1 4 Simulate stable
33. int masses is equal the dimension of the problem Otherwise only 2 dimensional computations can be done The symmetric case uses the method in Abdul Hamid and Nolan 1998 the nonsymmetric case uses the method in Nolan and Rajput 1995 The symmetric case is faster and more accurate than the nonsymmetric case Both routines are accurate near the center of the distribution and have limited accuracy near the tails 4 2 2 Cumulative function STABLE function mvstablecdf dist d a b epsabs cdf ierr This function approximates P a lt X lt b If the components are independent it computes this by taking the product of the corresponding univariate probabilities d is the dimension In the symmetric two dimensional case the probability is evaluated by numerically integrating the nu merically computed 2 dimensional density f x Due to the limited precision in the numerical calculation of the density and the approximate nature of the integration of this density this routine gives only a few digits of accuracy To find the probability of an unbounded regions it is best to truncate the region using the routine in Section 4 2 5 to find a bounded rectangle containing most of the probability Use the function in Section 4 2 3 to approximate in 2 dimensional nonsymmetric case or in higher di mensions STABLE User Manual 21 4 2 3 Cumulative function Monte Carlo STABLE function not implemented in STABLE library This function approximat
34. is function 12 error sinc error in sfitfracmoment 13 error Internal error in sfitlogabs 14 error Data value near zero in sfitfracmoment or sfitlogabs 15 error Error in subroutine 16 error Internal error while computing derivatives 17 error f a and f b have the same signs 18 error Too many function evaluations 19 error Not enough memory 20 error X zero value 21 error Internal error in quickstable 30 error Two parameterization is required in skewed case Table 3 Univariate error codes Warning code 7 can arise in several ways The purpose of this warning is to avoid numerical problems in internal calculations that can occur near the boundary in the parameter space or to use special cases to increase speed but to let the user know that something nonstandard is being done In the following discussion let e the value of tolerance 4 The default value is e 0 01 You can change the value of tolerance 4 by using the function stablesettolerance above and query it s value by using function stablegettolerance The default value was picked in an ad hoc way you can make it smaller even 0 if you wish to calculate certain quantities in one of the cases below But be aware that numerical errors may arise Special cases where warning code 7 occur are 1 a near 2 if a 2 e 2 then a is set to 2 and is set to 0 2 Near a 1 but not Cauchy if a 1 lt e and 6 gt e then a is
35. ment method as in Nikias and Shao 1995 This routine only works in the symmetric case it will always return 6 0 and 0 In this case the 0 parameterization coincides with the 1 parameterization so there is no need to specify parameterization STABLE User Manual 8 The log absolute moment method does not work when there are zeros in the data set because log x is undefined when z is 0 Remove zero values and possibly values close to 0 from the data set if you want to use this method 2 2 9 Quantile based estimation version 2 STABLE function stablefitquant2 nx x theta param ierr Estimate stable parameters for the data in x using a modified quantile method of Nolan 2010 It should work for any values of the parameters but some extreme values are inaccurate 2 2 10 U statistic based estimation STABLE function stablefituest nx x theta param ierr Estimate stable parameters for the data in x using the method of Fan 2006 It only works for the symmetric case 2 2 11 Confidence intervals for ML estimation STABLE function stablefitmleci theta z n sigtheta ierr This routine finds confidence intervals for maximum likelihood estimators of all four stable parame ters The routine returns a vector sigtheta of half widths of the confidence interval for each param eter These values depend on the confidence level you are seeking specified by z and the size of the sample n The z value is the standard criti
36. mportant to remember when you specify x for calculating say a pdf 4 1 Define multivariate stable distribution When matrices are used in the multivariate routines care must be taken when specifying the allocated di mensions of a matrix and how much is used For example a d dimensional elliptically contoured stable distribution use a d x d shape matrix R A general program to work with elliptically contoured distributions of different dimensions may allocate a matrix R of size dinax X dmax and then use the upper left 2 x 2 matrix when working with a 2 dimensional distribution the upper left 3 x 3 matrix when working with a 3 dimensional distribution etc If you call a STABLE routine in such a case it needs to know the allocated size as well as the current size to correctly access the elements This is done by passing an integer argument giving the leading dimension of the matrix For a matrix R the variable would be called ldr using the STABLE User Manual 19 old programming convention of the first letters from leading dimension of R for a matrix A the variable would be called Ida etc There are different functions used to define each of the different types of distributions that STABLE can work with They are described below 4 1 1 Independent components STABLE function mvstableindep alpha d beta gamma delta iparam dist ierr Define a multivariate stable distribution with independent components with characteris
37. n identifier used to specify the distribution when calling other functions There are several special cases that are handled differently internally e When angle and lambda are of length 2 densities can be calculated in terms of univariate densities e The special case of the previous one is when angle 0 7 2 This corresponds to a distribution with independent components Both density and cdf are calculated in terms of products of univariate density and cdf respectively e If all elements of beta are 0 the distribution is symmetric Cumulative distribution function calcula tions only work in the symmetric case though Monte Carlo based cdf estimation works for any case you can simulate including skewed 4 1 6 Undefine a stable distribution STABLE function mvstableundefine dist ierr Clears the definition of the stable distribution dist 4 2 Basic functions 4 2 1 Density function STABLE function mvstablepdf dist d n x ierr Computes the density f x for stable distribution dist at each value in x d is the number of rows of x Note this routine assumes that the density exists The density will not exist in the discrete spectral measure case if the mass is concentrated on a proper subspace of the domain In the independent case the program computes the pdf as a product of univariate stable pdfs There is one other case that can be evaluated in terms of univariate pdfs if the spectral measure is discrete AND the number of po
38. ode of 1 and your function will stop In contrast a warning is informational and is usually not serious It alerts you to the fact that the results of a calculation may have some inaccuracy For example stable densities have radical changes of the tail behavior when a 2 or 1 and the computations have small inaccuracies in them In practical terms this usually means little as the difference between an a 1 99 stable distribution and an a 2 stable distribution in an statistical problem is likely to be unobservable in practice Return codes for STABLE program are given in the tables below Univariate routines return error codes in the range 1 99 multivariate routines return error codes in the range 100 199 code type meaning 0 No error 1 error Invalid input parameter 2 error alpha parameter outside of tabulated values in QKSTABLE 3 error Too many data points for internal array 4 error Error computing the likelihood e g pdf 0 5 warning Possible approx error while using QKSTABLE for alpha or beta near boundary 6 warning Possible error in confidence intervals because parameter is near boundary 7 warning alpha and or beta rounded to a special value adjust tol 4 8 warning alpha is at lower bound for search may not have found best value for alpha 9 error Too many bins distinct possible values in sdiscretemle 10 error beta must be 0 to use this function 11 error beta near 1 or 1 does not work in th
39. on xie X which is equivalent to A being a symmetric measure on S i e A A A A for any Borel subset A C S As in the univariate case in the symmetric case the O parameterization and the 1 parameterization coincide The general case is beyond current computational capabilities but several special cases isotropic radially symmetric elliptical independent components and discrete spectral measure are computationally accessible isotropic The spectral measure is continuous and uniform leading to isotropic radial symmetry for the dis tribution The characteristic function is Eexp i lt u X gt exp 9g Jul i lt u gt 4 elliptical The characteristic function is E exp i lt u X gt exp ryan i lt u d gt 5 where R is a positive definite matrix R 7 J is equivalent to the isotropic case above independent components If components are independent with X S a Bj Yj 05 k then the charac teristic function is d Eexp i lt u X gt exp X o ujla Bj k i lt u gt 6 j 1 This is a special case of the discrete spectral measure below the spectral mass is concentrated on the points where the coordinates axes intersect the unit sphere discrete When the spectral measure is discrete with mass ats S j 1 m the characteristic function is Eexp i lt u X gt exp Y w lt u s gt a 1 k Aj i lt u gt 7 j l This discrete class is dense in the class of
40. ons results 2 converges to a chi squared distribution with d f free parameters in H1 parameter space free parameters in HO parameter space as the sample size tends to oo For example to compute the likelihood ratio test for the null hypothesis HO data comes from a normal distribution vs H1 data comes from stable distribution use abnd 2 2 and bbna 0 0 in which case results 2 will have 2 df To test HO data comes from a symmetric stable distribution vs H1 data STABLE User Manual 10 comes from a general stable distribution use abnd 0 4 2 and bbnd 0 0 in which case results 2 will have 3 d f 2 2 17 Stable regression STABLE function stableregression ndata nb x y trimprob binit b alpha beta gamma delta symmetric ierr Computes regression coefficients b1 b2 by for the problem Yi bizi 1 boti bktik tein 1 1 n where the error term e has a stable distribution In matrix form the equation is y Xb e The algorithm is described in Nolan and Ojeda 2006 y is a vector of length n of observed responses x is an x k matrix with the columns of x representing the variables and the rows representing the different observations NOTE if you want an intercept term you must include a column of ones in the x matrix Typically one sets the first column of x to ones and then b 1s the intercept trimprob is a vector of length 2 e g 0 1 0 9 which gives the lower and upper quantiles f
41. or the trimmed regression Trimmed regression trims off extreme values and then performs ordinary least squares regression The resulting coefficients are used to get an initial estimate of the stable regression coefficients symmetric can be used to force the fitting program to assume symmetry in the error terms ej The STABLE library function returns the different fields separately e bis the vector of coefficients e binit is the initial vector of coefficients from the trimmed regression e alpha beta gamma delta are the stable parameters estimated from the residuals They can be regarded as nuisance parameters if you only care about the coefficients Note that all parameters are in the O parameterization You can convert to another representation using the convert parameterization function in Section 2 3 5 Note that in the non Gaussian stable case some of the traditional assumptions in regression are no longer true In particular it is NOT always the case that Fe 0 First if a lt 1 the heavy tails will mean that Fe is undefined Second in the non symmetric case 3 4 0 even if gt 1 we do not require Fe 0 Instead we set delta so that the mode of e is zero The reason for this is to make the regression line go through the center of the data points 2 2 18 Stable regression profile likelihood parameter confidence intervals STABLE function not implemented in STABLE library Compute confidence intervals for regression
42. parameters This function uses profile likelihood for the specific data set to compute confidence intervals for each parameter including the stable parameters a 6 and y as well as the regression coefficients b1 bx It is assumed that the user has already called the regression routine fit lt stable regression x y trimprob stable regression profile likelihood fit x y There are two optional arguments p value to specify the significance level default p val ue 0 05 gives 95 confidence intervals and show plots is a Boolean used to determine if plots of the profile likeli hoods are shown for each parameter STABLE User Manual 11 2 3 Informational utility functions 2 3 1 Version information STABLE function stableversion nv vinfo This functions returns information of the version of STABLE that is being used The values are vinfo 1 major version number vinfo 2 minor version number vinfo 3 modification number vinfo 4 year of release vinfo 5 month of release vinfo 6 day of release vinfo 7 internal serial number For example the values 4 0 2 2005 9 15 123 mean that you are using version 4 0 2 of STABLE which was released on 2005 9 15 with serial number 123 nv is the length of the integer array vinfo If nv is more than 7 other information may be filled into the vinfo array 2 3 2 Modes of stable distributions STABLE function stablemode theta param x ierr Returns the mo
43. random variates STABLE function stablernd n x alpha beta gamma delta n2 unif param ierr This function simulates n stable random variates 1 2 n With parameters a 8 y 6 in param eterization param It is based on Chambers et al 1976 For the Stable library call you need to supply n2 2n Uniform 0 1 random variables in the array unif Different languages have different pseudo random number generators In C stdlib h defines functions srand to set random seed rand to return a ran dom integer and a constant RAND_MAX u rand RAND_MAX 1 0 will give a Uniform 0 1 value Note the default random number generator in C and other languages are generally not very good better ones are available on the web e g www netlib org 2 1 5 Stable hazard function STABLE function stablehazard n x h alpha beta gamma delta param ierr This function computes the hazard function for a stable distribution h f x 1 F x i n 1 gereg STABLE User Manual 6 2 1 6 Derivative of stable densities STABLE function stablepdfderiv n x y alpha beta gamma delta param ierr This function computes the derivative of stable density functions y f a f xila B y param t 1 N 2 1 7 Second derivative of stable densities STABLE function stablepdfsecondderiv n x y alpha beta gamma delta param ierr This function computes the second derivative of stable density functions y
44. ropic stable random vectors Since these are univariate quantities and it is required that the distribution is isotropic one does NOT have to define the isotropic distribution separately Because of computational delicacy these routines are limited to dimension d lt 100 4 4 1 Amplitude cumulative distribution function STABLE function mvstableamplitudecdf nr r f alpha gamma0 d ierr Compute the cdf of the amplitude distribution Fr r P R lt r for R X where X is an d di mensional isotropic stable random vector with characteristic function Fexp i lt u X gt exp y Jul Current implementation works for a 0 8 2 There seems to be a relative error of approximately 3 for large r 4 4 2 Amplitude density STABLE function mvstableamplitudepdf nr r f alpha gamma0 d ierr Compute the density f r where R is described above Current implementation works for a 0 8 2 There seems to be a relative error of approximately 3 for large r 4 4 3 Amplitude quantiles STABLE function mvstableamplitudequant nr p r alpha gamma0 d ierr Compute the quantiles of the amplitude R described above Current implementation works for a 0 8 2 4 4 4 Simulate amplitude distribution STABLE function mvstableamplitudernd nr r alpha gamma0 d nu unif ierr Simulate n i i d values of the amplitude distribution R as described above Current implementation works for a 0 2 2 STABLE User Manual 23
45. s they are fine However when the functions must be evaluated many times the previous routines are slow For example when estimating stable parameters by maximum likelihood estimation the likelihood is evaluated at each data point for a large number of parameter values during the numerical search for the point where the likelihood is maximized In these cases speed is more desirable than great accuracy The functions described below are approximations to the functions above and are based on pre computed values using those basic functions They are designed to evaluate the quantity of interest at many x values for fixed values of a and 5 Each routine has a setup time and if you change a or 6 that setup code must be rerun It can be slower to run these routines than the basic routines above if you only want to calculate the quantity at a few x values These routines work for 0 2 lt a lt 2 and all 1 lt 6 lt 1 2 5 1 Quick stable density computation STABLE function stableqkpdf n x p alpha beta gamma delta param Call is identical to Section 2 1 1 results are approximately the same 2 5 2 Quick stable cumulative computation STABLE function stableqkcdf n x f alpha beta gamma delta param ierr Call is identical to Section 2 1 2 results are approximately the same 2 5 3 Quick stable log pdf computation STABLE function stableqklogpdf n x logf alpha gamma delta param ierr Approximates log pdf for stable
46. tablefitparfn2d nvect x nproj angle parfn methodld param ierr STABLE User Manual 22 Estimate the parameter functions for the bivariate data in x The data is projected in each direction given by angle and the parameters are estimated in the param parameterization met hod1d is the univariate method used to estimate the parameters see Section 2 2 1 for codes parfn will contain the result The result is a matrix of dimension length x x5 The columns of the result are 1 for the angle 2 for the estimate of a 3 for the estimate of 5 4 for the estimate of y 5 for the estimate of 6 at that angle 4 3 3 Fit an elliptical stable distribution to multivariate data STABLE function mvstablefitelliptical d nvect x alpha delta ldr R methodld ierr x contains the data values met hod1d is the method to use for estimating univariate stable parameters internally see Section 2 2 1 for codes The function returns a list structure that contains information about the fit The fields in the fit are the estimated value of a the estimated shift location vector 6 and R for the estimated shape matrix 4 4 Amplitude distribution For d dimensional random vector X the univariate quantity R X is called the amplitude of X When X is isotropic the radial symmetry allows one to reduce the dimension of the problem to a univariate problem The following routines compute the cdf pdf quantiles simulate and estimate for amplitudes of isot
47. tic function 6 beta gamma and delta should be vectors of length d the dimension of the distribution dist contains an identifier used to specify the distribution when calling other functions 4 1 2 Isotropic stable STABLE function mvstableisotropic alpha d gamma0 delta dist ierr Define a multivariate isotropic stable distribution with characteristic function 4 d is the dimension of the distribution gamma0 is the scale parameter delta is the location vector dist contains an identifier used to specify the distribution when calling other functions 4 1 3 Elliptical stable STABLE function mvstableelliptical alpha 1dr d R delta dist ierr Define a multivariate elliptically contoured sub Gaussian stable distribution with characteristic function 5 The dimension of the distribution is determined from the size of R a positive definite d x d shape matrix and delta is the location vector dist contains an identifier used to specify the distribution when calling other functions 4 1 4 Discrete spectral measure STABLE function mvstablediscspecmeas alpha 1ds d nlambda s lambda delta beta iparam dist ierr Define a multivariate stable distribution with discrete spectral measure having characteristic function 7 s should be a d x nlambda matrix specifying the location of the point masses as columns Lambda should be a row vector of length nlambda containing the weights beta should be a row vector of length nlambda
48. tion stablecdfdiscrete n ix F alpha beta gamma delta cutoff param ierr Calculates F P Y lt 2 1 1 n 2 6 4 Quick discrete stable cumulative distribution function STABLE function stableqkcdfdiscrete n ix p alpha gamma delta cutoff iparam ierr Calculates F P Y lt x i 1 n Faster than above less accurate 2 6 5 Simulate discrete stable random variates STABLE function stablernddiscrete n ix alpha beta gamma delta cutoff n2 unif param ierr Simulates discrete stable random variates with the specified parameters and cutoffs 2 6 6 Simulate discrete stable random variates with specified saturation probability STABLE function stablernddiscrete2 n ix alpha beta delta psaturation cutoff n2 unif param ierr Simulates discrete stable random variates where the scale is computed internally to make the saturation probability psaturation Note that in cases where the stable parameters are passed individually gamma is NOT used In the cases where the vector theta is used the value of y theta 3 is ignored The following function is used to compute y then the previous function is called to generate the values 2 6 7 Find scale y to have a specified saturation probability for a discrete stable distribution STABLE function stablediscretefindgamma alpha beta delta psaturation cutoff param gamma ierr Given a 6 and cutoff a b the scale y is computed to get the requested sat
49. uration probability e g psaturation P X lt a 1 2 P X gt b 1 2 2 6 8 Discrete maximum likelihood estimation STABLE function stablediscretemle n ix theta method cutoff param ierr Estimate the stable parameters for the discrete stable data in 71 in parameterization param using maximum likelihood estimation The likelihood is numerically evaluated and maximized using an STABLE User Manual 16 optimization routine When method 1 stablepdfdiscrete is used to calculate likelihood when method 2 symmetry is assumed 0 and a faster method is used to compute the likelihood STABLE User Manual 17 3 Multivariate Stable Introduction To specify a multivariate stable distribution X X1 X2 Xq in d dimensions requires an index of stability a 0 2 a finite Borel measure A on the unit sphere S s R s 1 and a shift vector R The measure A is called the spectral measure of the distribution The joint characteristic function of X S a A 6 k is given by Eexp i lt u X gt exp a u s gt a 1 k A ds i lt u gt j s where w ula 8 k is defined in 3 As in one dimension the 1 parameterization is more common in theo retical research while the 0 parameterization is better suited to computation and statistical problems Here and below lt u X gt uX uX ugXq is the inner product Symmetric stable distribu tions are defined by the conditi
50. ute the log likelihood of the bivariate isotropic stable data in x with stable index alpha scale gamma 0 and location vector delta An internal approximation is used to compute the single value a Y0 6x1 aae Xn log II fx xila Y0 9 i 1 This function is designed to compute the log likelihood for a fixed many times In this case it is much faster than trying to compute the right hand side above using the bivariate pdf routine in Section 4 2 1 It is also more accurate than that routine especially on the tails The program initially computes an approximation that depends on a if a changes the approximation must be recomputed and it will be slower 4 5 2 Quick amplitude density in bivariate case STABLE function mvstableqkamplitudepdf2d n r y alpha gamma0 ierr Compute the amplitude function fr r a yo d 2 for a 2 dimensional isotropic stable vector For large n it is much faster than the function in Section 4 4 2 4 6 Bivariate discrete stable distribution A bivariate discrete stable distribution is defined by digitizing and truncating a continuous bivariate stable distribution X X1 X2 discrete Y Y1 Y2 7 has components Y integer part of max a min X b STABLE User Manual 24 field subscript meaning type 0 type of distribution one of MVS_TYPE_ISOTROPIC MVS TYPE ELLIPTICAL or MVS_TYPE_DISCRETE nunif 1 number of uniform 0 1 r v needed to simulate d 2 dimension of this distribution

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