User Manual for STABLE 5.1 R Version
Contents
1. 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 4 1 4 Discrete spectral measure R function mvstable discrete spec meas alpha s lambda delta beta param 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 1ambda should be a row vector of length nlambda containing the weights beta should be a row vector of length nlambda specifying the skewness at each point mass delta is the shift as a column vector param is the parameterization must be 0 or 1 The spectral measure is defined by putting mass 1ambdal j 1 betalj 2 at s and mass 1ambda j 1 betalj 2 at s Setting all beta equal to 1 gives the standard definition of a spectral measure with mass lambda j ats j Setting all beta equal to 0 guarantees that the distribution is symmetric putting weight 1 ambdal j 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 bet a by defining antipodal points and weights and values of beta that balance correctly However STABLE does not detect this Some parts of STABLE are signi2. Stochastic Models 13 687 702 Nolan J P and B Rajput 1995 Calculation of multidimensional stable densities Commun Statist Simula 24 551 556 Index S Plus functions amplitude score 22 damplitude 21 damplitude2d quick 22 ddiscstable 14 ddiscstable quick 14 ddiscstable2d 23 discstable find gamma 15 dmvstable 19 dstable 5 dstable deriv 5 dstable log quick 13 dstable quick 13 dstable secondderiv 6 dstable series origin 12 dstable series tail 12 hstable 5 hstable quick 13 mvstable convert 23 mvstable discrete spec meas 18 mvstable discrete spec meas2d 19 mvstable elliptical 18 mvstable find rectangle 20 mvstable fit 20 mvstable fit amplitude 22 mvstable fit elliptical 21 mvstable fit parfn2d 20 mvstable indep 18 mvstable info 23 mvstable isotropic 18 mvstable loglik isotropic2d quick 22 mvstable parfn2d 23 mvstable undefine 19 pamplitude 21 pdiscstable 14 pdiscstable quick 14 pmvstable 19 pmvstable MC 20 pstable 5 pstable quick 13 pstable series origin 12 pstable series tail 12 qamplitude 21 qstable 5 qstable quick 13 ramplitude 21 rdiscstable 15 rdiscstable2 15 rmvstable 20 rstable 5 stable chisq 8 27 stable convert 11 stable fit 6 stable fit dmle 15 stable fit ecf 7 stable fit fractional moment 7 stable fit logabs 7 stable fit mle 6 stable fit mle restricted 6 stable fit quantile 7 stable fit quantile2 7 stable fi
3. 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 important to remember when you specify x for calculating say a pdf 4 1 Define multivariate stable distribution The interfaced versions of STABLE have the ability to work with multiple distributions When a multivariate stable distribution is defined a distribution descriptor is returned That descriptor must be used when computing quantities for that distribution Note The descriptor should not be chan
4. 1 0 1 2 Note that under the standard assumptions 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 bbnd z 0 0 in which case results 2 will have 2 df To test HO data comes from a symmetric stable distribution vs H1 data 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 R function stable regression x y trimprob symmetric Computes regression coefficients b1 55 by for the problem yi bizia boti E bktik bei 1 1 n STABLE User Manual 10 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 a n 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 is the intercept trimprob is a vector of length 2 e g 0 1 0 9 which gives the lower and upper quantile
5. o ee eee 14 2 6 3 Discrete stable cumulative distribution function 0 0 14 2 6 4 Quick discrete stable cumulative distribution function 14 2 6 5 Simulate discrete stable random variates o o a 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 oaoa 15 3 Multivariate Stable Introduction 16 4 Multivariate Stable Functions 17 4 1 Define multivariate stable distribution eee 17 4 1 1 Independent components 0 000000 000 18 4 4 2 Isotropicstable ss pk oe AG RR RR Re ee Re 18 4 1 3 Ellipticalstable oou ome ie Re Se es 18 4 1 4 Discrete spectral measure e 18 4 1 5 Discrete spectral measure in 2 dimensions 0 19 4 1 6 Undefine a stable distribution lees 19 4 2 Basicf ncti Ds xod we Pe Se Pos AAA OR POS Ru EA woe EL Y 19 422 1 Density Tnction 4 343 Se Beek he ed eh oe BPS We dd pea 19 4 22 Cumulative function c os xy eher gro US m bie Rob UM be Sow RP nd 19 4 2 3 Cumulative function Monte Carlo lll 20 4 24 Multivariate simulation 20 4 25 Find a rectangle with probability at least p lees 20 4 3 Statistical functions s re ne ok Rogo o RERO os eo OE Boy oe RATER Rx 20 4 3 1 Est
6. p and 1 are indistinguishable for small p less than approximately107 1 STABLE will correctly return F 1 p F 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 z a 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 4 co 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 random variates R function rstable n alpha beta gamma delta param This function simulates n stable random variates 1 12 n with parameters a B y 9 in parame terization param It is based on Chambers et al 1976 2 1 5 Stable hazard function R function hstable x alpha beta gamma delta param This function computes the hazard function for a stable distribution h f a 1 F a i n 1 prey 2 1 6 Derivative of stable densities R function dst
7. FALSE Compute confidence intervals for regression 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 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 value 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 2 3 Informational utility functions 2 3 1 Version information R function stable version 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 STABLE User Manual 11 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 R function stable mode theta param Retu
8. X2 lt a 1 2 P X2 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 Discrete bivariate density R function ddiscstable2d dist x cutoff eps method 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 X X2 are truncated at the same value The function returns a vector p of length n with pi P Y x PL t1 Y2 a 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 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 R function mvstable info dist Returns information about distribution dist Useful for checking that
9. Y lt xi i 1 n Faster than above less accurate STABLE User Manual 15 2 6 5 Simulate discrete stable random variates R function rdiscstable n alpha beta gamma delta cutoff param Simulates discrete stable random variates with the specified parameters and cutoffs 2 6 6 Simulate discrete stable random variates with specified saturation probability R function rdiscstable2 n alpha beta delta psaturation cutoff param 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 R function discstable find gamma alpha beta delta saturation cutoff param Given a 6 and cutoff a b the scale y is computed to get the requested saturation probability e g psaturation P X lt a 1 2 P X b 1 2 2 6 8 Discrete maximum likelihood estimation R function stable fit dmle x method cutoff param Estimate the stable parameters for the discrete stable data in 7 2 in parameterization param using maximum likelihood estimation The likelihood is numerically evaluated and
10. alpha beta gamma delta param 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 R function hstable quick x alpha beta gamma delta param Call is identical to Section 2 1 5 2 5 6 Quick stable likelihood computation R function stable loglik quick x theta param Call is identical to Section 2 2 13 STABLE User Manual 14 2 5 7 Quick stable score nonlinear function R function stable score quick x alpha beta gamma delta param method This function approximates the score or nonlinear function for a stable distribution g a f z f x d dx In f x The algorithm used depends on the value of method When method 1 stablegqkpdf is used to compute f x 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 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 6 y param and a pair of cutoff values a l
11. 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 jpenolan 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 Interfaced STABLE functions require input variables and return the results of the computations The interface computes the lengths of all arrays specifies default values for some of the variables in some case and handles return codes and results The parameters of the stable distribution must be specified The R interface requires alpha but lets the user leave off other arguments in which case bet a defaults to be 0 gamma defaults to be 1 and delta defaults to be 0 param defaults to O in all cases In some of the utility or statistical routines the 4 stable parameters are passed in a vector theta alpha beta gamma delta In these cases R requires all 4 parameters to be specified The STABLE interface prints an error message when an error occurs If an error occurs execution is aborted if a warning occurs execution continues There is basic help information built into the interfaces In R you can type the name function e g dstable without parenthesis to get the function definition The STABLE library is not reentrant on a single computer only one user sh
12. squared goodness of fit test R function stable chisq data theta param nclass Compute chi squared goodness of fit statistic for the data in z1 using nclass equally probable classes bins 2 2 15 Kolmogorov Smirnov goodness of fit test R function stable ks gof x theta method param This function computes the Kolmogorov Smirnov two sided test statistic D sup F x F a oo r oo STABLE User Manual 9 where F is the stable cdf with parameters a theta 1 8 theta 2 y theta 3 theta 4 and is the sample cdf of the data in x Use method 0 for quick computations stableqkcdf is used to compute cdf use method 1 for slower computations stablecaf 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 tha
13. User Manual for STABLE 5 1 R 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 2002 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 B siC f tictions 4 reia ete dew PRU AIA Vs ete ev ERU DARAS E Bek 2 11 Stable densities x uo a Bg ee Be eS ee ee ee 2 1 2 Stable distribution functions o sooo e ee 2 13 Stable quantiles 44 wy vp uS Se EAS Se ddd w
14. 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 2001 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
15. able deriv x alpha beta gamma delta param This function computes the derivative of stable density functions y f a f xi o 3 7 6 param Quen STABLE User Manual 6 2 1 7 Second derivative of stable densities R function dstable secondderiv x alpha beta gamma delta param This function computes the second derivative of stable density functions y f xi f xila 3 7 6 param t 1 N 2 1 8 Stable score nonlinear function R function stable score x alpha beta gamma delta param This function computes the score or nonlinear function for a stable distribution g x f z 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 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 R function stable fit x method param Estimate stable parameters from the data in x1 n 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 B 0 uses po
16. al 12 2 3 6 Omega function R function stable omega u alpha beta param Compute the function w u a 8 k i 1 n where u 1 iB tan 2 sign u ju 1 afl QUIERO E u 1 182 sign u In u a 3 alii olo Ll ei These functions are from the characteristic functions of standardized univariate stable distributions if Z S o 6 1 0 k then Eexp iuZ 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 w u a B k tanZ signu a 1 sign u In u Q I p M ea DUM 2 4 Series approximations to basic distribution functions These functions use the Bergstrom series for stable densities and cdfs which are only defined for o 1 2 4 1 Series approximation of stable pdf around the origin R function dstable series origin n x alpha beta gamma delta param nterms 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 R function pstable series origin n x alpha beta gamma delta param nterms Computes the stable cumulative distribution function u
17. definition 4 7 2 Compute projection parameter functions R function mvstable parfn2d angle Compute the exact parameter functions for a bivariate stable distribution For direction t R X t is univariate stable with parameters a B t y t 9 t This function computes the parameter functions 6 y and at the values t cos angle j sin ang1e j Angles in angle are given in radians 4 7 5 Multivariate convert parameterization R function mvstable convert dist newparam Converts between multivariate stable parameterizations newparam must be 0 or 1 In the interfaced versions of STABLE the input distribution dist is converted to the new parameterization and a new distri bution descriptor is returned STABLE User Manual 24 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 code 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 o 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
18. e if la 1 2 lt e and 8 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 code type meaning 101 error Invalid input parameter 102 warning Accuracy warning alpha 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 2 Multivariate error codes 25 STABLE User Manual 26 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 densities 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
19. e 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 Eexp 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 0j k then the charac teristic function is d Eexp i lt u X gt exp M ujla Bj ky i lt u d gt 6 j l 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 A 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 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 17 density surface contours of density surface triangle alpha 1 3 XX 0
20. ensity function R function dmvstable dist x Computes the density f x for stable distribution dist at each value in x Note this routine as sumes 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 point 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 R function pmvstable dist a b 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 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 t
21. er 2 1 4 Simulate stable random variates len 2 4 5 Stable hazard function 2 1 6 Derivative of stable densities 2 ee 2 1 7 Second derivative of stable densities 2 1 8 Stable score nonlinear function Statistical functioms zm sow Exe oe ee RICOE rari CIV ER RE Ge tes 2 241 Estimatingstableparameters 2e 2 22 Maximumlikelihoodestimation 2 2 2 3 Maximum likelihood estimation with restricted parameters 2 24 Maximum likelihood estimation with search control 2 2 5 Quantile based estimation 22 2 2 6 Empirical characteristic function estimation llle 2 2 7 Fractional moment estimation 2 28 Log absolute moment estimation e 2 29 Quantile based estimation version 2 2 2222 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 2 2 ee ee 2 2 15 Kolmogorov Smirnov goodness of fit test 22s 2 2 16 Tsikelihood ratio test 41 55 suos eoo c RH Bo RU XS N 00 00 00 090 00 00 NN NNN DDODODODDOQOO0ssUnr Ur Ur Ur Ur Un Un amp 224017 Stable TESTOSSION s 9 aie Reet E AAA da de ee ER 9 2 2 18 Stable regression profile likelihood parameter confidence intervals 10 Informational utilit
22. f stability and is in the range 0 lt a lt 2 B is a skewness parameter and is in the range 1 lt lt 1 y is a scale parameter and must be positive and 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 o 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 5 u 1 iB tan 3 sign u yu 1 i u oa Al E X 2 1 PRO oe y ul 1 722 sign u In y u i u a 1 Y A random variable X is S a B y 0 1 if it has characteristic function _ f exp v u 1 iB tan amp signu i u afl EPUA i y u 1 i82 sign u In lul i u a 1 2 Note that if 6 0 then these two parameterizations are identical it is only when 8 Z 0 that the
23. ficantly faster and more accurate in the symmetric case e g density calculations and simulations STABLE User Manual 19 4 1 5 Discrete spectral measure in 2 dimensions R function mvstable discrete spec meas2d alpha angle lambda beta delta param degrees FALSE 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 angle j gives the angle in radians of the location of s cos angle j sin angle j In R the optional argument degrees can be set TRUE to use degrees instead of radians 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 R function mvstable undefine dist Clears the definition of the stable distribution dist 4 2 Basic functions 4 2 1 D
24. from the data set if you want to use this method 2 2 8 Log absolute moment estimation R function stable fit logabs x Estimate stable parameters for the data in x using the log absolute moment method as in Nikias and Shao 1995 This routine only works in the symmetric case it will always return 6 0 and 6 0 In this case the 0 parameterization coincides with the 1 parameterization so there is no need to specify parameterization 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 0 Quantile based estimation version 2 R function stable fit quantile2 x param 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 STABLE User Manual 8 2 2 10 U statistic based estimation Rfunction stable fit u estno direct interface use StableFit with method 7 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 R function stable mle conf int theta z n 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 confide
25. g 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 1 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 f is set to 0 2 Near a 1 but not Cauchy if o 1 lt e and 8 gt e then a is set to 1 and 8 is left unchanged This is to avoid computations involving 8 tan ra 2 which blows up as a 1 if 8 4 0 3 Near Cauchy case if a 1 lt e and 8 lt e then ois set to 1 and Pis set to 0 4 Near L vy cas
26. ged by a user The descriptor may change between calls and contents may vary in future versions of STABLE There are different functions used to define each of the different types of distributions that STABLE can work with They are described below A simple R example that defines two distributions and works with them is Define two bivariate stable distributions one isotropic and STABLE User Manual 18 one with indep components dist lt mvstable isotropic alpha 1 3 d 2 gamma0 1 delta c 0 0 dist2 lt mvstable indep alpha 1 3 beta c 0 0 gamma c 1 1 delta c 0 0 0 compute the pdf for both distributions x matrix c 0 0 0 1 1 0 1 1 2 4 yl dmvstable dist1 x y2 lt dmvstable dist2 x simulate from both distributions zl rmvstable dist1 10000 z2 lt rmvstable dist2 10000 4 1 1 Independent components R function mvstable indep alpha beta gamma delta param Define a multivariate stable distribution with independent components with characteristic function 6 beta gamma and delta should be vectors of length d the dimension of the distribution 4 1 2 Isotropic stable R function mvstable isotropic alpha d gamma0 delta 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 4 1 3 Elliptical stable R function mvstable elliptical alpha R delta
27. he 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 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 R function mvstable fit x nspectral methodld method2d param x contains the data values nspect ral 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
28. his 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 STABLE User Manual 20 Use the function in Section 4 2 3 to approximate in 2 dimensional nonsymmetric case or in higher di mensions 4 203 Cumulative function Monte Carlo R function pmvstable MC dist a b n This function approximates 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 R function zmvstable dist n Simulate n stable random vectors from the stable distribution dist This works for any distribution that can be defined in dimensions d gt 2 4 2 5 Finda rectangle with probability at least p R function mvstable find rectangle dist p 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 t
29. imate a discrete spectral measure fit a stable distribution to bivariate data 20 4 3 20 Estimate parameter functions e 20 4 3 3 Fit an elliptical stable distribution to multivariatedata a 21 44 Amplitude distribution s c sers oora lere 21 4 4 1 Amplitude cumulative distribution function lll 21 44 2 Amplitudedensity 2e 21 4 43 Amplitude quantiles e 21 4 4 4 Simulate amplitude distribution lees 21 44 5 Eitamplitude data sae aa e ce be RE ag ea god ee 22 4 4 6 Amplitude score function e 22 4 5 Faster approximations to multivariate routines o o ee eee 22 4 5 1 Quick log likelihood for bivariate isotropic case 2 a 22 4 5 2 Quick amplitude density in bivariate case 2 2 2 a 22 4 6 Bivariate discrete stable distribution e 22 4 6 1 Discrete bivariate density es 23 4 7 Multivariate informational utility functions eee 23 4 7 1 Information about a distribution lees 23 4 7 2 Compute projection parameter functions lees 23 4 7 3 Multivariate convert parameterization e 23 5 Error return codes 24 References 26 Index 26 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 o is the tail index or index o
30. in practice In the S Plus R version of STABLE error messages are controled by the variable err level in the command load stable err level 0 means no messages are printed err level l means no only error messages are printed err leve1 2 the default means both error and warning messages are printed 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 this 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 computin
31. late and estimate for amplitudes of isotropic 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 4 Amplitude cumulative distribution function R function pamplitude r alpha gamma0 d 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 E exp i lt u X gt exp 7 Jul 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 density R function damplitude r alpha gamma0 d 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 396 for large r 4 4 5 Amplitude quantiles R function qamplitude p alpha gamma0 d Compute the quantiles of the amplitude R described above Current implementation works for o 0 8 2 4 4 4 Simulate amplitude distribution R function ramplitude nr alpha gamma0 d Simulate n i i d values of the amplitude distribution R as described above Current implementation works for o 0 2 2 STABLE User Manual 22 4 4 5 Fit amplitude data R function mvstable fit amplitude r d meth
32. maximized using an optimization routine When method 1 stablepdfdiscrete is used to calculate likelihood when method 2 symmetry is assumed 8 0 and a faster method is used to compute the likelihood STABLE User Manual 16 3 Multivariate Stable Introduction To specify a multivariate stable distribution X X4 X5 Xq in d dimensions requires an index of stability a 0 2 a finite Borel measure A on the unit sphere S s IR s 1 and a shift vector IR 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 4 i lt u gt 3 S where w u o 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 u4 Xi ugXq is the inner product Symmetric stable distribu tions are defined by the condition xie X which is equivalent to A being a symmetric measure on S ie 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 measur
33. nce 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 critical 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 6 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 20r PB 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 R function stable mle info matrix theta 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 2 213 Log likelihood computation R function stable loglik x theta param Compute the log likelihood of the data assuming an underlying stable distribution with the specified parameters 2 2 14 Chi
34. od Estimate the parameters a and yo 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 o and o 4 4 6 Amplitude score function R function amplitude score r alpha gamma0 d 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 o 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 91 02 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 R function mvstable loglik isotropic2d quick x alpha gamma0 delta Compute the log likelihood of the bivariate isotropic stable data in x with stable index alpha scale gammao0 and location vector delta An internal a
35. of length nspect ral and Lambda for the estimated weights at each position 4 3 2 Estimate parameter functions R function mvstable fit parfn2d x angle methodld param STABLE User Manual 21 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 The result is a matrix of dimension length x 5 The columns of the result are 1 for the angle 2 for the estimate of a 3 for the estimate of 6 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 R function mvstable fit elliptical x methodld 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 simu
36. ould 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 o 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 Stable densities R function dstable x alpha beta gamma delta param This function computes stable density functions pdf y f x f xi lo 8 y param i 1 n The algorithm is described in Nolan 1997 2 1 2 Stable distribution functions R function pstable x alpha beta gamma delta param This function computes stable cumulative distribution functions cdf y F v F r o 0 y 9 param i 1 n The algorithm is described in Nolan 1997 2 1 3 Stable quantiles R function qstable p alpha beta gamma delta param This function computes stable quantiles the inverse of the cdf x F pi 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
37. pproximation is used to compute the single value n a Y0 xi itus Xn log II fx xila Yo 9 i 1 This function is designed to compute the log likelihood for a fixed a 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 R function damplitude2d quick r alpha gamma0 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 0 STABLE User Manual 23 where cutoffz 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 X1 lt a 1 2 P X gt b4 1 2 P
38. re 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 o and 5 Each routine has a setup time and if you change o 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 o lt 2 and all 1 lt 8 lt 1 2 5 1 Quick stable density computation R function dstable quick x 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 R function pstable quick x alpha beta gamma delta param Call is identical to Section 2 1 2 results are approximately the same 2 5 3 Quick stable log pdf computation R function dstable log quick x alpha gamma delta param Approximates log pdf for stable distributions Results are approximately the same as log f 2 5 4 Quick stable quantile computation R function qstable quick p
39. rns the mode of a S theta 1 8 theta 2 y theta 3 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 R function stable set tolerance 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 DTOoOAmAAIANANFWNK OO Re mA 2 3 4 Get internal tolerance R function stable get tolerance inum Returns the value of the internal settings see the preceding function for the meanings of each variable 2 3 5 Convert between parameterizations R function stable convert param theta newparam Convert from the parameters given in theta given in the param parameterization to the parameters thet anew given in the newparam parameterization Currently param and newparam are restricted to the values 0 1 2 and 3 STABLE User Manu
40. s for 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 e The interfaced versions of this function returns a structure with different fields 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 Ee 0 First if lt 1 the heavy tails will mean that Ee is undefined Second in the non symmetric case 3 Z 0 even if gt 1 we do not require Ee 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 8 Stable regression profile likelihood parameter confidence intervals R function stable regression profile likelihood fit x y p value 0 05 show plots
41. sing 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 nterms 2 corresponds to a linear term etc 2 4 3 Series approximation of stable pdf at the tail R function dstable series tail n x alpha beta gamma delta param nterms 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 8 The series is not defined for a 1 2 4 4 Series approximation of stable cdf at the tail R function pstable series tail n x alpha beta gamma delta param nterms 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 8 The series is not defined for a 1 STABLE User Manual 13 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 values they are fine However when the functions must be evaluated many times the previous routines a
42. t b the random variable Y integer part of max a min X b is 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 R function ddiscstable x alpha beta gamma delta cutoff param Calculates f P Y 2 i 1 n 2 6 2 Quick discrete stable density R function ddiscstable quick x alpha beta gamma delta cutoff param Calculates f P Y z i 1 n Faster than above less accurate 2 6 3 Discrete stable cumulative distribution function R function pdiscstable x alpha beta gamma delta cutoff param Calculates F P Y z i 1 n 2 6 4 Quick discrete stable cumulative distribution function R function pdiscstable quick x alpha beta gamma delta cutoff param Calculates F P
43. t the parameter values theta a 3 7 6 were chosen independently of the data If the parameters were estimated from the data then this tail probability will be an overestimate of the significance level 2 2 16 Likelihood ratio test R function stable lrt x abnd bbnd This function computes the likelihood ratio Lo L1 where Ly 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 a abnd 2 and bbnd 1 6 bbnd 2 and L4 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 o in the range 0 4 2 The vector results will contain the results of the computations results results results results estimated value of alpha without assuming HO estimated value of beta without assuming HO estimated value of gamma without assuming HO estimated value of delta without assuming HO results 1 ratio of the likelihoods results 2 2 log ratio of likelihoods results 3 log likelihood of the data for the restricted HO results 4 log likelihood of the data for the unrestricted H1 results 5 estimated value of alpha under HO results 6 estimated value of beta under HO results 7 estimated value of gamma under HO results 8 estimated value of delta under HO 9 1 1
44. t u est 8 stable get tolerance 11 stable ks gof 8 stable loglik 8 stable loglik quick 13 stable lrt 9 stable mle conf int 8 stable mle info matrix 8 stable mleO fit 7 stable mode 11 stable omega 12 stable regression 9 stable regression profile likelihood 10 stable score 6 stable score quick 14 stable set tolerance 11 stable version 10
45. uantile based estimation R function stable fit quantile x param Estimate stable parameters for the data in x using the quantile based on the method of McCulloch 1986 It sometimes has problems when o is small say lt 1 2 and the data is highly skewed Try the modified version below in such cases 2 2 6 Empirical characteristic function estimation R function stable fit ecf x gamma0 delta0 param 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 R function stable fit fractional moment x p Estimate stable parameters for the data in z using the fractional moment estimator as in Nikias and Shao 1995 This routine only works in the symmetric case it will always return 9 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
46. wer p 0 2 5 log absolute moment B 6 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 R function stable fit mle x param Estimate the stable 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 o gt 0 4 comes from 2 2 3 Maximum likelihood estimation with restricted parameters R function stable fit mle restricteddata theta param restriction 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 STABLE User Manual 7 2 2 4 Maximum likelihood estimation with search control R function stable mle0 fit data method gamma init delta init ftol t0 tlower tupper nfunc param This is maximum likelihood estimation with greater control over the search and ranges for the parameters It is used internally 2 2 5 Q
47. y functions e 10 2 3 1 Version information llle 10 2 3 2 Modes of stable distributions e 11 2 3 3 Set intemal tolerance cmo er RR RE erm ee 11 2 3 4 Get mtermal tolerance cos 58g eo a es RUE RATER RU 11 2 3 5 Convert between parameterizations e 11 2 3 0 Omega f nctUonD is exe dore Re a eed ee SERRA bar a dak 12 Series approximations to basic distribution functions 0 12 2 4 1 Series approximation of stable pdf around the origin Ln 12 2 4 2 Senes approximation of stable cdf around the origin sn 12 2 4 3 Senes approximation of stable pdf at the tail 12 2 4 4 Series approximation of stable cdf at the tail 12 Faster approximations to basic functions o 000000000004 13 2 5 1 Quick stable density computation e 13 2 5 2 Quick stable cumulative computation le 13 2 5 3 Quick stable log pdf computation llle 13 2 5 4 Quick stable quantile computation 0 2 02 0000 13 2 5 5 Quick stable hazard function computation llle 13 2 5 6 Quick stable likelihood computation o 13 2 5 7 Quick stable score nonlinear function 0 2 0 0 000 0a ee 14 Discrete stabledistributions es 14 STABLE User Manual 3 2 6 1 Discrete stable density 2e 14 2 6 2 Quick discrete stable density 2 o
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