User Manual for STABLE 5.1 Mathematica Version
Contents
1. 15 lt p e bead a ele he i 15 Jare gi aaron E doors di 15 2 6 Simulate discrete stable random variates o e 15 a ES 15 war E OR a b Ede dek 15 17 4 Multivariate Stable Functions 18 4 1 Define multivariate stable distribution AAA 18 4 1 1 Independent components 0 02 00 000200004 18 4 1 2 Isotropic stable AAA 19 GATA 19 E EET An Gu dk GE gia 19 id Ge bee a e 19 NA E e EEE Ee te aes 20 42 Basie TUNCUONS e ice 4 E A Rd A 20 4 2 1 Density TUNCUON exec oras eH SEER Oe e a GE EES 20 o Pab gt gut ZUEK dri i Pab dirura bg 20 4 2 3 Cumulative function Monte Carlo 2 20 0 0 a 20 Re SEK Ne E e e See eee ea gago 21 HN 21 A A ak OA EEN Ta ROS Li E Gte 21 ute 21 LE E E AEE e a 21 er gag 22 A AR A 22 4 4 1 Amplitude cumulative distribution TOGO 22 4 4 2 Amplitude density oo 22 e e Gutar dete i e Guz e E E d e EE ie ga 22 E a ed E E dk Ede Gn 22 a E Ea E GO du a a Sr B 22 Se Sd EG dp a a Ep 23 e Sub Sariek beti Gat HE Eo b E dE 23 ega Ea 23 a GOL islets bugi Be ety ae ae 23 e SEE AEE Egk Ed Bod a 23 4 6 1 Discrete bivariate density 2 2 0 2 0 2 000000000004 24 kag Ea Eur ba d Gu e ert eo br Za 24 4 7 1 Information about a distribution e e e e 24 SE dt Zu labe a E gt 24 La a GE d 24 25 26 27 STABLE User Manual 4 1 Univariate Stable Introduction Stable distributions are a class of probabi2. k 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 STABLE User Manual 13 2 4 1 Series approximation of stable pdf around the origin Mathematica function not implemented in Mathematica 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 Mathematica function not implemented in Mathematica 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 nterms 2 corresponds to a linear term etc 2 4 3 Series approximation of stable pdf at the tail Mathematica function not implemented in Mathematica 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 n
3. method used to estimate the parameters see Section 2 2 I for codes 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 3 4 for the estimate of y 5 for the estimate of at that angle STABLE User Manual 22 4 3 3 Fit an elliptical stable distribution to multivariate data Mathematica function not implemented in Mathematica x contains the data values d vectors 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 This information can be used to define a multivariate stable distribution for simulation 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 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 difficulties these r
4. 1 parameterization STABLE will do a linear interpolation to compute the pdf or cdf at that point The thresholds used in rounding and linear approximation are described on page 12 You can manually reset these values but be careful the algorithms may yield poor values in some cases The remainder of this manual is a description of the functions in the STABLE library 2 1 Basic functions 2 1 1 Test scripts Mathematica function not implemented These will test most of the STABLE routines and can be used as a source of examples on how to use the functions 2 1 2 Stable densities Mathematica function StablePDF x alpha beta gamma delta param This function computes stable density functions pdf y f x f ajla 8 7 6 param i 1 n The algorithm is described in Nolan 1997 STABLE User Manual 6 2 1 3 Stable distribution functions Mathematica function StableCDF x alpha beta gamma delta param This function computes stable cumulative distribution functions cdf y F x F x la P y 9 param 1 n The algorithm is described in Nolan 1997 2 1 4 Stable quantiles Mathematica function StableQuantile x alpha beta gamma delta param 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 Extreme tail quantiles may be hard to find because of subtractive cancelation and the fact that cdf calculatio
5. 2 16 Likelihood ratio test Mathematica function StableLRT x abnd bbnd This function computes the likelihood ratio Ly 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 6 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 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 results 9 estimated value of alpha without assuming HO results 10 estimated value of beta without assuming HO results 11 estimated value of gamma without assuming HO results 12 estimated value of delta without assuming HO 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 sam
6. 7 Derivative of stable densities A 2 1 8 Second derivative of stable densities 2 0 0 ee ee ee Za ay du eat Sache td ee ek b oh ye E a Gee Sr 2 2 Statistical functions o aoao a 2 2 1 Estimating SORET GA 2 2 4 Maximum likelihood estimation with search control O E iaz Gag Gr 2 2 9 Quantile based estimation version 2 a ee es erek dE E Fe A Br EE a bu eit E Sb E E 2 2 13 Log likelihood computation o o e 2 2 14 Chi squared goodness of fit test uuu 2 2 15 Kolmogorov Smirnov goodness of fit test 2 16 Likelihood ratio test Informational utility functions 2 3 1 Version information 2 2 3 2 Modes of stable distributions n oon ooo a e ee 2 3 3 Set internal tolerancel a 2 3 4 Get internal tolerancea 2 4 1 Series approximation of stable pdf around the origin 2 4 3 Series approximation of stable pdf at the tail 2 4 4 Series approximation of stable cdf at the tail 2 6 Discrete stable distributions oa a a NI 2 3 A ME O ME MO 00 00 00 00 00 00 NN NNN AAAA ZG GN ZG LA LO La Y STABLE User Manual 3 2 6 1 Discrete stable density 2 2 0 20 00 00000000000 000 15 2 6 2 Quick discrete stable density 2 2 2 0 0 0 200 000
7. Mathematica function StablePDFSecondDeriv x alpha beta gamma delta param This function computes the second derivative of stable density functions y f xi f xila B y param t LM 2 1 9 Stable score nonlinear function Mathematica function StableScore x alpha beta gamma delta param This function computes the score or nonlinear function for a stable distribution g x f x f d dx ln f x The routine uses stablepaf to evaluate f x and numerically evaluates the derivative STABLE User Manual 7 f x Warning this routine will give unpredictable results when 6 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 Mathematica function StableFit x method param Estimate stable parameters from the data in 71 using method as described in the following table This routine calls one of the functions described below to do the actual estimation see those sections for references method value algorithm notes 1 maximum likelihood a gt 0 2 2 quantile a gt 0 1 3 empirical characteristic function a gt 0 1 4 fractional moment a gt 0 4 8 d 0 uses p 0 2 5 log absolute moment B 09 6 modified quantile a gt UA 7 U statistic method B 09 Note that the fractional moment log absolute
8. 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 SN le u s gt la 1 k A 1 lt u 0 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 This plot was produced using the test script mvstabletest see Section 2 1 STABLE User Manual 18 density surface contours of density surface triangle alpha 1 3 SOS 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 2 lt d lt 100 you can e simulate
9. 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 The method assumes the data is symmetric and centered at 0 departures from either assumption may generate unreliable estimates 2 2 8 Log absolute moment estimation Mathematica function no direct interface use StableFit with method 5 Estimate stable parameters for the data in x using the log absolute moment method as in Nikias and Shao 1995 Section5 7 and Zolotarev 1986 Section 4 1 This routine only works in the symmetric case it will always return 8 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 x is 0 Remove zero values and possibly values close to 0 from the data set if you want to use this method The method assumes the data is symmetric and centered at 0 departures from either assumption may generate unreliable estimates 2 2 9 Quantile based estimation version 2 Mathematica function no direct interface use StableFit with method 6 Estimate stable parameters for the data in x using a modified quantile method of Nolan 2017 It should work for any values of the parameters but
10. some extreme values may be unreliable 2 2 10 U statistic based estimation Mathematica function no direct interface use StableFit with method 7 STABLE User Manual 9 Estimate stable parameters for the data in x using the method of 2006 The U statistic method does not work when there are zeros in the data set Remove zero values and possibly values close to 0 from the data set if you want to use this method The method assumes the data is symmetric and centered at 0 departures from either assumption may generate unreliable estimates 2 2 11 Confidence intervals for ML estimation Mathematica function StableMLEConfidencelnterval 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 confidence interval for each parameter in theta alpha beta gamma delta These values depend on the confidence level you are seek ing 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 interval is theta 1 sigtheta 1 For 6 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 bounda
11. this function 4 3 Statistical functions 4 3 1 Estimate a discrete spectral measure fit a stable distribution to bivariate data Mathematica function not implemented in Mathematica 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 method2d 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 for some discussion of suggested values and diagnostics Suggest using nspectral 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 6 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 Mathematica function not implemented in Mathematica 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
12. using mvstablernd e fit multivariate data with an elliptical model e calculate the pdf using mvstablepdf if the components are independent OR the spectral measure has exactly d point masses OR the distribution is isotropic or elliptical e calculate the cdf using mvstablecdf if the components are independent e calculate the cdf using mvstablecdfMC by Monte Carlo estimation for any type of distribution The accuracy of the multivariate 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 the multivariate pdf or cdf 4 1 Define multivariate stable distribution STABLE has 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 changed 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 4 1 1 Independent components Mathematica function Not implemented in Mathematica STABLE User Manual 19 Define a multivariate stable distribution with dimension d lt 100 and independent components with char acteristic function 6 beta gamma and delta should be vectors of length d
13. 5 error nspectral must be divisible by 4 106 error This parameterization is not allowed in this function 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 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 1331 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 Birkh user McCulloch J H 1986 Simple consistent estimators of stable distribution parameters Communications in Statistics Simulation and Computation 15 110
14. 9 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 Multivariate elliptically contoured stable distributions theory and estimation Submitted Nolan J P 2017 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 Revah 2013 Linear and nonlinear regression with stable errors J of Economet rics 172 186 194 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 Stochastic Models 13 687 702 STABLE User Manual 27 Nolan J P and B Rajput 1995 Calculation of multidimensional stable densities Commun Statist Simula 24 551 556 Zolotarev V M 1986 One dimensional Stable Distributions Volume 65 of Translations of mathematical monographs American Mathematical Society Tra
15. User Manual for STABLE 5 3 Mathematica 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 2016 by Robust Analysis Inc www RobustAnalysis com info Orobustanalysis com Processed July 18 2017 STABLE User Manual Contents 1 Univariate Stable Introduction 2 Univariate Stable Functions 2 1 Basic f NC ONS A 2 1 1 Testscriptsh ees t ra a gi E dd E Gi E EEE dE de GE ia BUZ Stable densities s i i ec irera ee POR PEER MS Sw HEELERS ES EE Eza 2 1 3 Stable distribution functions 2 a Sad ie Gia Weegee AL A te ce EE E ae Bence Os Ee 2 1 5 Simulate stable random variates e ERT e ec ts Sk een Ea E WGA a EEE EE 2 1
16. al integration 2 relative error for quantile search 3 alpha and beta rounding 4 x tolerance near zeta 5 exponential cutoff 6 peak strim location tolerance 7 stabletrim tolerance 8 minimum alpha 9 minimum xtol 10 threshold for quantile search 11 x tolerance 2 3 4 Get internal tolerance Mathematica function StableGet Tolerance Returns the value of the internal tolerances see the preceding function for the meanings of each variable 2 3 5 Convert between parameterizations Mathematica function StableConvert param theta newparam Convert from the parameters given in theta alpha beta gamma delta given in the param parameterization to the parameters t het anew given in the newparam parameterization Currently param and newparam are restricted to the values 0 1 2 and 3 2 3 6 Omega function Mathematica function not implemented in Mathematica Compute the function w u a 8 k i 1 n where Ju 1 ib tan 2 sign u ult 1 a 1 w ula 6 0 E KO 1 BO 0 signu In ul Gea 1 3 SEE ZEZ e b 1 Gian 22 signu a 1 u 1 82 sign u In ul a 1 These functions are from the characteristic functions of standardized univariate stable distributions if Z S a B 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 w ular
17. at s 3 Setting all beta equal to 0 guarantees that the distribution is symmetric putting weight lambdalj 2 at ts 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 bet a 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 Mathematica function Not implemented in Mathematica 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 ang1el j 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 sy
18. atistical routines the 4 stable parameters are passed in a vector theta alpha beta gamma delta In these cases Mathematica 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 Mathematica type a question mark before the name e g StablePDF to get the function definition The STABLE library is not reentrant 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 There are certain values of the parameters a near 2 a near 1 8 near 1 etc where there are complicated numerical problems with calculations In these cases the STABLE program may approximate values by rounding parameters For example if you try to calculate a stable pdf or cdf for a 1 009 and 6 0 009 the STABLE program will round to a 1 and 8 0 and compute the value for these values of the parameters Likewise when zx is near 0 in the
19. cases e g the one sided distributions when a lt 1 and 8 1 A random variable X is S a 6 y 9 0 if it has characteristic function Fourier transform exp loti 1 i tan 72 sign u yuji29 1 i u a l exp y u 1 482 sign u In ylul i u ei E Eexp iuX STABLE User Manual 5 A random variable X is S a B y 9 1 if it has characteristic function exp 7 u 1 Bltan signu i u aA exp ylul 1 82 sign u In u i u a 1 2 EexpliuX 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 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 2017 The next section gives a description of the basic univariate 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 Mathematica interface requires the pa rameter values alpha beta gamma delta paramare passed individually In some of the utility or st
20. d When method 1 stableqkpdf 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 a 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 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 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 STABLE User Manual 15 In the internal routines the x values are integers The matlab R Mathema
21. e regression coefficients The variable symmetric can be used to force the fitting program to assume symmetry in the error terms e param is the parameterization used and must be 0 1 or 2 the default is parameterization 2 This function returns a structure with different fields e theta is the vector of regression coefficients found by maximum likelihood e theta_ols is the initial vector of coefficients from the OLS regression e theta_trimis the initial vector of coefficients from the trimmed regression e psi 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 regression coefficients Note that all parameters are in the param parameterization e param the parameterization used e symmetric whether regression was restricted to the symmetric case Note that in the non Gaussian stable case some of the traditional assumptions in regression are no longer true It is not generally the case that Fe 0 so the estimates may not be unbiased First if a lt 1 the tails of the stable distribution are very heavy and Fe is undefined Second in the non symmetric case i e 8 4 0 we do not require He 0 Also it is not generally the case that the regression line goes through the center of the data This is not an error it is a consequence of regression with non symmetric residuals and how the parameterization centers the distribution What happe
22. emented in Mathematica STABLE User Manual 21 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 Mathematica function not implemented in Mathematica 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 Find a 2 dimensional rectangle with probability at least p Mathematica function not implemented in Mathematica Find a number r so that the 2 dimensional rectangle A A r r r x r r has P X A gt p where X is a 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 gener ally 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 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
23. est Mathematica function StableKolmogorovSmirnov x theta method param This function computes the Kolmogorov Smirnov two sided test statistic D sup F F 2 o0 lt x lt oo STABLE User Manual 10 where F is the stable cdf with parameters a theta 1 8 theta 2 y theta 3 6 theta 4 and F is the sample cdf of the data in z Use method 0 for quick computations the fast approximation is used to compute cdf use met hod 1 for slower computations the slow method 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 parameters were estimated from the data then this tail probability will be an overestimate of the significance level 2
24. fied saturation probability for a discrete stable distribution Mathematica function not implemented in Mathematica 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 gt b 1 2 2 6 8 Discrete maximum likelihood estimation Mathematica function not implemented in Mathematica Estimate the stable parameters for the discrete stable data in 71 2 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 WARNING The Mathematica interface for multivariate func tions is not yet implemented 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 EI 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 s where w ula B k is defined in 3 As in one dimension the 1 parameterizatio
25. hese integer values in double precision numbers STABLE User Manual 24 4 6 1 Discrete bivariate density Mathematica function not implemented in Mathematica 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 X1 X2 are truncated at the same value The function returns a vector p of length n with p EIN xj P Y 214 Yo Zal 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 pi 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 Mathematica function not implemented in Mathematica Returns information about distribution dist Useful for checking that definition 4 7 2 Compute projection parameter functions Mathematica function not implemented in Mathematica Compute the exact parameter functions for a bivariate stable distr
26. his case it is much faster than trying to compute the right hand side above using the bivariate pdf routine in Section 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 Mathematica function not implemented in Mathematica 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 KI has components Y integer part of max a min X b 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 EUA lt a 1 2 P X gt b 1 2 P 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 t
27. ibution For direction t R X t is univariate stable with parameters a 3 t y t 9 t This function computes the parameter functions GLL y and 6 at the values t cos angle j sin angle j Angles in angle are given in radians 4 7 3 Multivariate convert parameterization Mathematica function not implemented in Mathematica 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 25 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 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 retur
28. 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 8 gt e then a is set to 1 and is left unchanged This is to avoid computations involving 8 tan ra 2 which blows up as a gt 1 if 6 4 0 3 Near Cauchy case if a 1 lt e and 8 lt e then ais set to 1 and is set to 0 4 Near L vy case if a 1 2 lt e and 8 1 lt 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 26 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 10
29. lity 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 3 is a skewness parameter and is in the range 1 lt 6 lt 1 y is a scale parameter and must be positive and 6 is a location parameter an arbitrary real number 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 see below The main purpose of the STABLE program 1s to make these distributions accessible in practical problems The package enables the calculation of stable densities cumulative distribution functions quantiles etc It also can fit data by several different estimation procedures stable densities 0 25 0 15 Figure 1 Symmetric stable densities 8 0 with a 2 Gaussian in black a 1 5 red and a 1 Cauchy green 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
30. mmetric case though Monte Carlo based cdf estimation works for any case you can simulate including skewed STABLE User Manual 20 4 1 6 Undefine a stable distribution Mathematica function not implemented in Mathematica Clears the definition of the stable distribution dist When a multivariate stable distribution is defined by one of the above functions the STABLE library allocates memory for a distribution descriptor The amount of memory used is not large and this should not normally be a problem But if your program loops and defines many multivariate stable distributions you could have a memory problem as the descriptors accumulate Calling this function when you are done with a stable distribution will deallocate the memory A technical detail distribution descriptors are allocated internally in the STABLE library not in the workspace of the user session If you exit these distribution descriptors are lost EVEN IF YOUR SESSION INFORMATION IS SAVED You must redefine all multivariate distributions when you restart the STABLE library 4 2 Basic functions 4 2 1 Density function Mathematica function not implemented in Mathematica Computes the density f x for stable distribution dist at each value in x Note this routine assumes that the density exists The density will not exist if the spectral measure is supported on a proper linear subspace of the domain When dimension d gt 2 the pdf can be calculated in special cases e
31. moment and U statistic methods do not work when there are zeros in the data set They also assume that the distribution is symmetric and centered at 0 if either of these assumptions are not valid the estimators are unreliable 2 2 2 Maximum likelihood estimation Mathematica function StableMLFit x param Estimate the stable parameters for the data in x1 n 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 gt 0 2 comes from 2 2 3 Maximum likelihood estimation with restricted parameters Mathematica function not implemented in Mathematica 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 takes an input value theta alpha beta gamma delta and if restriction i 1 then theta i is fixed to allow a parameter to vary set restriction i 0 The function then searches over the unrestricted parameters to maximize the likelihood 2 2 4 Maximum likelihood estimation with search control Mathematica function not implemented in Mathematica This is maximum likelihood estimation with greater contr
32. mplementation allows only met hod 5 which uses method of moments on STABLE User Manual 23 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 nonlinear function Mathematica function not implemented in Mathematica Compute the nonlinear function score function for the location 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 Mathematica function not implemented in Mathematica Compute the log likelihood of the bivariate isotropic stable data in x with stable index alpha scale gamma0 and location vector delta An internal approximation is used to compute the single value L a Yo diz SEO b log fx xila 70 1 This function is designed to compute the log likelihood for a fixed a many times In t
33. n 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 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 22 error More Uniform 0 1 values required for simulation 23 error 2 parameterization required Table 1 Univariate error codes Warning code 7 can arise in several ways The purpose of this warning is to avoid numerical problems
34. n 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 u1 Xi ugXq is the inner product Symmetric stable distribu tions are defined by the condition x4 X which is equivalent to A being a symmetric measure on S ie ALAI 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 y u i lt u d 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 I is equivalent to the isotropic case above More information on this accessible class of distributions is given in Nolan 2010 independent components If components are independent with X S a 6 y 0 k then the charac teristic function is d Eexp i lt u X gt exp SN a ujla Bj KZ 1 lt u gt 6 j 1 This is a special case of the discrete spectral measure below the spectral
35. ns in the non symmetric case depends on the parameterization used Using param 0 will give a well conditioned problem but the regression line will not go through the center of the data Using param 1 will guarantee that Fe 0 when a gt 1 and the expectation exists but has two consequences the numerics are poorly conditioned if is near 1 so the parameter estimates are very sensitive and the line can be arbitrarily far away from the center of the data Using param 2 will guarantee that the mode of e is zero and the regression line will go through the center of the data points 2 3 Informational utility functions 2 3 1 Version information Mathematica function StableVersion stableversion returns a string with version information 2 3 2 Modes of stable distributions Mathematica function StableMode theta param Returns the mode of a S a theta 1 6 theta 2 y theta 3 6 theta 4 param distribution If 8 4 0 the mode is determined by a numerical search of the pdf STABLE User Manual 12 2 3 3 Set internal tolerance Mathematica function StableSetTolerance inum value Sets the value of internal tolerances that are used during computations You change these values at your own risk computation times can become very long inaccuracy can accumulate and some choices of the parameters can cause infinite loops inum meaning O relative error for pdf numerical integration 1 relative error for cdf numeric
36. ns may only be accurate to 10 decimal places see the notes below 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 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 5 Simulate stable random variates Mathematica function StableRandom n alpha beta gamma delta param This function simulates n stable random variates 1 2 a with parameters a B y 9 in parame terization param It is based on Chambers et al 1976 2 1 6 Stable hazard function Mathematica function StableHazard x alpha beta gamma delta param This function computes the hazard function for a stable distribution h f x 1 F x i egea A 2 1 7 Derivative of stable densities Mathematica function StablePDFDeriv x alpha beta gamma delta param This function computes the derivative of stable density functions y f a f xila 8 y param bg bazi 2 1 8 Second derivative of stable densities
37. nslation from the original 1983 Russian edition Index Mathematica functions StableCDF 8 StableCFFit StableConvert StableFit StableGetTolerance StableHazard 6 StableKolmogorovSmirnov 9 StableLogLikelihood p StableLRT StableMLEConfidenceInterval 9 StableMLFit StableMode StablePDF StablePDFDeriv 6 StablePDFSecondDeriv 6 StableQFit StableQkCDF StableQkHazard StableQkLogLikelihood StableQkPDF StableQkQuantile StableQuantile 6 StableRandom 6 StableScore 6 StableSetTolerance StableVersion
38. ol over the search and ranges for the parameters It is used internally and always uses the 0 parameterization STABLE User Manual 8 2 2 5 Quantile based estimation Mathematica function StableQFit 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 a is small e g a lt 1 2 and the data is highly skewed Try the modified version below in such cases 2 2 6 Empirical characteristic function estimation Mathematica function StableCFFit x 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 Mathematica function no direct interface use StableFit with method 4 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 6 O 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 it is required that p lt 1 Take
39. ot defined for a 1 2 4 4 Series approximation of stable cdf at the tail Mathematica function not implemented in Mathematica 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 values 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 8 Each routine has a setup time and if you change a or p that setup code must be rerun It can be slower to run these routines than the basic routines above if you only want
40. outines are limited to dimension d lt 100 4 4 1 Amplitude cumulative distribution function Mathematica function not implemented in Mathematica 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 y u 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 Mathematica function not implemented in Mathematica 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 Mathematica function not implemented in Mathematica Compute the quantiles of the amplitude R described above Current implementation works for a 0 8 2 4 4 4 Simulate amplitude distribution Mathematica function not implemented in Mathematica Simulate n i i d values of the amplitude distribution R as described above Current implementation works for a 0 2 2 4 4 5 Fit amplitude data Mathematica function not implemented in Mathematica 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 i
41. ple 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 0 0 in which case results 2 will have 2 d f 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 Mathematica function not implemented in Mathematica Computes linear regression coefficients 01 02 0 for the problem Yi 01 1 02 2 kerru 1 1 n where the error term e has a stable distribution The algorithm uses maximum likelihood and is described in Nolan and Ojeda Revah 2013 In matrix form the equation is y X0 e STABLE User Manual 11 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 6 1s the intercept trimprob is a vector of length 2 e g 0 1 0 9 which gives the lower and upper quantiles for the trimming 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 stabl
42. ry of the parameter space e g a 2 or 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 Mathematica function not implemented in Mathematica 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 O 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 Mathematica function StableLogLikelihood x theta param 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 Mathematica function not implemented in Mathematica Compute chi squared goodness of fit statistic for the data in 71 n using nclass equally probable classes bins This test only looks at proportion of the data in each class not how it is spread within that bin This is particularly a problem with the end classes which are infinite regions This test does not consider the tail decay There is also an issue with significance values when parameters are estimated from the data 2 2 15 Kolmogorov Smirnov goodness of fit t
43. the components are independent components e the spectral measure is discrete AND the number of point masses is equal the dimension of the problem e the isotropic case e the elliptical case 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 Mathematica function not implemented in Mathematica This function approximates P a lt X lt b If the components are independent this works in dimensions up to 100 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 4 2 3 Cumulative function Monte Carlo Mathematica function not impl
44. the dimension of the distribution 4 1 2 Isotropic stable Mathematica function Not implemented in Mathematica Define a multivariate isotropic stable distribution with with dimension d lt 100 and characteristic function 4 dis the dimension of the distribution gamma0 is the scale parameter delta is the location vector 4 1 3 Elliptical stable Mathematica function Not implemented in Mathematica Define a multivariate elliptically contoured sub Gaussian stable distribution with dimension d lt 100 and 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 Mathematica function Not implemented in Mathematica Define a multivariate stable distribution with discrete spectral measure with dimension d lt 100 and char acteristic 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 specifying the skewness at each point mass delta is the shift as a column vector paramis the parameterization must be 0 or 1 The spectral measure is defined by putting mass 1ambda j 1 betalj 2 ats and mass lambda j 1 betalj 2 at s Setting all beta equal to 1 gives the standard definition of a spectral measure with mass lambda 3
45. tica interfaces use double pre cision values 2 6 1 Discrete stable density Mathematica function not implemented in Mathematica Calculates f P Y 2 i 1 n 2 6 2 Quick discrete stable density Mathematica function not implemented in Mathematica Calculates f P Y x i 1 n Faster than above less accurate 2 6 3 Discrete stable cumulative distribution function Mathematica function not implemented in Mathematica Calculates F P Y lt 2 1 1 n 2 6 4 Quick discrete stable cumulative distribution function Mathematica function not implemented in Mathematica Calculates F P Y lt x i 1 n Faster than above less accurate 2 6 5 Simulate discrete stable random variates Mathematica function not implemented in Mathematica Simulates discrete stable random variates with the specified parameters and cutoffs 2 6 6 Simulate discrete stable random variates with specified saturation probability Mathematica function not implemented in Mathematica 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 speci
46. to calculate the quantity at a few x values These routines work for 0 2 lt a lt 2 and all 1 lt 8 lt 1 2 5 1 Quick stable density computation Mathematica function StableQkPDF x alpha beta gamma delta param Call is identical to Section results are approximately the same STABLE User Manual 14 2 5 2 Quick stable cumulative computation Mathematica function StableQkCDF x alpha beta gamma delta param Call is identical to Section results are approximately the same 2 5 3 Quick stable log pdf computation Mathematica function not implemented in Mathematica Approximates log f x for stable distributions 2 5 4 Quick stable quantile computation Mathematica function StableQkQuantile p alpha beta gamma delta param Call is identical to Section but much faster Note the comments in that section about extreme upper quantiles 2 5 5 Quick stable hazard function computation Mathematica function StableQkHazard x alpha beta gamma delta param Call is identical to Section 2 1 6 2 5 6 Quick stable likelihood computation Mathematica function StableQkLogLikelihood x theta param Call is identical to Section 2 2 13 2 5 7 Quick stable score nonlinear function Mathematica function not implemented in Mathematica This function approximates the score or nonlinear function for a stable distribution g a f x f d dx ln f x The algorithm used depends on the value of metho
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