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Repeat following with beta = -1, -.5, 0, .5 and 1 stablec.exe << 1 # pdf 1 # Nolan S equivalent to S0 in scipy .25,2,.25 # alpha -1,-1,0 # beta -10,10,1 # x 1,0 # gamma, delta 2 # output file sdata/stable-pdf-sample-data.npytnamessx,p,alpha,betatisintbestit quadraturec`s.|ddk|ddddddg@S( NR#g?Riiiii ((R(tnpisin(sClib/python2.7/site-packages/scipy/stats/tests/test_distributions.pytst zolotarevcS`s|ddkS(NR#i((R((sClib/python2.7/site-packages/scipy/stats/tests/test_distributions.pyRscS`s|ddk|ddk@S(NR#iRi((R((sClib/python2.7/site-packages/scipy/stats/tests/test_distributions.pyRsicS`s|ddk|ddk@S(NR#iRi((R((sClib/python2.7/site-packages/scipy/stats/tests/test_distributions.pyRstfftiicS`s|ddkS(NR#i((R((sClib/python2.7/site-packages/scipy/stats/tests/test_distributions.pyRsicS`s|ddk|ddk@S(NR#ig?((R((sClib/python2.7/site-packages/scipy/stats/tests/test_distributions.pyRscS`s|ddk|ddk@S(NR#g?Ri((R((sClib/python2.7/site-packages/scipy/stats/tests/test_distributions.pyRss6Density calculation unstable for alpha=1 and beta!=0.*s2Density calculations experimental for FFT method.*RR#RRRtcalcRg?g$@s&pdf test %s failed with method '%s' %stverboseN(Rttloadtostpathtabspathtjointdirnamet__file__tcoretrecordst fromarraystTR]Rtin1dR\t enumerateRcRtpdf_default_methodtpdf_fft_min_points_thresholdR trecordRRRRtisnanRR]( Rtdatatteststixtdefault_methodtfft_min_pointstdecimal_placest filter_functsubdataRpRtsubdata2tfailures((RsClib/python2.7/site-packages/scipy/stats/tests/test_distributions.pyttest_pdf_nolan_sampless.'!%  " 07c C`stjtjjtjjtjjtd}tjj j |j dd}ddddgdddd gg}x-t |D]\}\}}}}|tj_|tj_|dk r|||n|}t} | jtd d d tjj|d |d|ddddd} t|d| } | tj| |ddd| ktj| B} t| |d|d||| fdtWdQXqWdS(s! 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This unit test is also a regression test for ticket 1548. The exact values are: mean: mu = a / (a + 1) variance: sigma**2 = a / ((a + 2) * (a + 1) ** 2) skewness: One formula (see https://en.wikipedia.org/wiki/Skewness) is gamma_1 = (E[X**3] - 3*mu*E[X**2] + 2*mu**3) / sigma**3 A short calculation shows that E[X**k] is a / (a + k), so gamma_1 can be implemented as n = a/(a+3) - 3*(a/(a+1))*a/(a+2) + 2*(a/(a+1))**3 d = sqrt(a/((a+2)*(a+1)**2)) ** 3 gamma_1 = n/d Either by simplifying, or by a direct calculation of mu_3 / sigma**3, one gets the more concise formula: gamma_1 = -2.0 * ((a - 1) / (a + 3)) * sqrt((a + 2) / a) kurtosis: (See https://en.wikipedia.org/wiki/Kurtosis) The excess kurtosis is gamma_2 = mu_4 / sigma**4 - 3 A bit of calculus and algebra (sympy helps) shows that mu_4 = 3*a*(3*a**2 - a + 2) / ((a+1)**4 * (a+2) * (a+3) * (a+4)) so gamma_2 = 3*(3*a**2 - a + 2) * (a+2) / (a*(a+3)*(a+4)) - 3 which can be rearranged to gamma_2 = 6 * (a**3 - a**2 - 6*a + 2) / (a*(a+3)*(a+4)) g?g?i 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