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RWWYU; 9c!j99 G  .1<Q<<5 Jr5r;N)rrrrrFrZrrrnr$rrrrrrr r;r5r3rqrqsC 2  1 . .&HN( EH#Jr5rqc \rSrSrSrSrSrg)TestSkewCauchyi.c[R"SSS5n[[RR USS9[R R U55 [[RRUSS9[R RU55 [[RRUSS9[R RU55 gNrrr]rra) rfrkrr9 skewcauchyrerrnrrs r3 test_cauchyTestSkewCauchy.test_cauchy/s KKAs #!%"2"2"6"6qA"6">"',,"2"21"5 7!%"2"2"6"6qA"6">"',,"2"21"5 7!%"2"2"6"6qA"6">"',,"2"21"5 7r5c[RRS5 [RRS5S-S- n[RRS5S-S- n/SQn/SQn[ [ R RX!5U5 [ [ R RX!5U5 [ [ R RXA5U5 g)Nrr\rcr r) g&45?gQʶ?g-4[u?g a2?g Rn?gѦ}cي?) gN@Q?gPPz@ ?g?g; 7?glg?g!?%?gQҙ?gC"?go?g@Sg?) rfrrrrr9rrernr)rqr1r[rerns r3test_skewcauchy_R TestSkewCauchy.test_skewcauchy_R8s& q IINN2  "Q & IINN2  #a '%$ ((,,Q2C8((,,Q2C8((,,S4a8r5r;N)rrrrrrr r;r5r3rr.s 7 9r5rc\rSrSrSr\R S5r\RRS/SQ5S5r Sr g) TestJFSkewTi[cS=pUS-n/SQn/SQn[R"X5n[R"U5n[UR U5UR U55 [UR U5UR U55 [UR U5UR U55 [URS5URS55 g)Nrrc)rrrfr)rrYrrrrfr)r9 jf_skew_trrrernr)rqr1r2rr[rjfrs r3test_compare_tTestJFSkewT.test_compare_t\s  U ! - __Q " GGBKq 1558,q 1558,q 1558,(!''&/:r5c[R"[[5RS- 5n[R R USS9$)atSample data points computed using the `ST5` distribution from the GAMLSS package in R. The pdf has been calculated for (a,b)=(2,3), (a,b)=(8,4), and (a,b)=(12,13) for x in `np.linspace(-10, 10, 41)`. N.B. the `ST5` distribution in R uses an alternative parameterization in terms of nu and tau, where: - nu = (a - b) / (a * b * (a + b)) ** 0.5 - tau = 2 / (a + b) z"data/jf_skew_t_gamlss_pdf_data.npyz x,pdf,a,bnames)rfloadr__file__parentrec fromarraysr,s r3gamlss_pdf_dataTestJFSkewT.gamlss_pdf_datalsAww N ! !$H H vv  [ 99r5za,b)r)rr)rh cUUSU:HUSU:H-nUSUSpe[U[R"X#5RU5SS9 g)zCompare the pdf with a table of reference values. The table of reference values was produced using R, where the Jones and Faddy skew t distribution is available in the GAMLSS package as `ST5`. r1r2r[rer rN)rr9rre)rqrr1r2rr[res r3test_compare_with_gamlss_r&TestJFSkewT.test_compare_with_gamlss_r|s[  S !Q &?3+?1+D E cDK3U__Q266q9Fr5r;N) rrrrrrr`rrrrr r;r5r3rr[sH;  ^^ : : [[U$>? 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 2..0#EBr5cZ[[RRSS5SS5 g)Nrrcr|rre rws r3test_regression_ticket_1326$TestChi2.test_regression_ticket_1326=sEJJNN32C~~!!s!4**4*: Z%r5r;N) rrrrrrrrrrr r r;r5r3ru ru Bs:   [[URG,&-&r5ru c \rSrSrSrSrSrg) TestGumbelRiacT[[RRS5SSS9 g)NrM?~T}%m;r r)rr9rarrws r3rTestGumbelR.test_sfcs# ))"-/E" $r5cT[[RRS5SSS9 g)NrTr7 r r)rr9rarrws r3rTestGumbelR.test_isfls# **513D" $r5r;N)rrrrrrr r;r5r3r r as $$r5r c \rSrSr\R "SS9S5r\R S5r\R S5r\R S5r \RR\RRS\R"S 5\R"S \RRS 9/5\RRS S S/5\RRS/SQ5\RRSSS/5S55555r\RR\RRSSS/5S55rSr\RR%SS9\RRSSS/5\RRSS S/5S 555rS!rS"rS#r\RR\RRS$\R"/S%Q/S&Q/S'Q5\R"/S(Q/S)Q/S*Q\RRS 9\R"/S+Q\R0"S,S-S.5\R0"S/SS05\RRS 9/5S155r\RRS$\R"/S%Q/S&Q/S'Q5\R"/S(Q/S)Q/S*Q\RRS 9\R"/S+Q\R0"S,S-S.5\R0"S/SS05\RRS 9/5S25r\RRS3S4S/5\RRS5S6S7/5S855r\RRS9S:S;/SS?S4\R:\R<\R<4/S@SA//5SB5r\RRSC/SDQ5\RRSE\ RBRDSF\R0"SGS4SH5SI4\ RBRFSF\R0"SGS4SH5SI4\ RBRDSJ\R0"S4SKSH5SF4\ RBRFSJ\R0"S4SKSH5SI4/5SL55r$\RRSM/SNQ5SO5r%\RR$\RRSM/SPQ5SQ55r&SRr'SSr(STr)gU)VTestLevyStableivT)autousecS[RlS[RlS[Rl[R R [Rlg)z2Setup default parameters for levy_stable generatorS1 piecewiseN)r9rparameterizationcdf_default_methodpdf_default_method _levy_stable _QUAD_EPSquad_epsrws r3reset_levy_stable_params'TestLevyStable.reset_levy_stable_paramswsH.2*/:,/:,%*%7%7%A%A"r5c[R"[[5RS- 5n[R R URSS9nU$)a{Sample data points for pdf computed with Nolan's stablec See - http://fs2.american.edu/jpnolan/www/stable/stable.html There's a known limitation of Nolan's executable for alpha < 0.2. The data table loaded below is generated from Nolan's stablec with the following parameter space: alpha = 0.1, 0.2, ..., 2.0 beta = -1.0, -0.9, ..., 1.0 p = 0.01, 0.05, 0.1, 0.25, 0.35, 0.5, and the equivalent for the right tail Typically inputs for stablec: stablec.exe << 1 # pdf 1 # Nolan S equivalent to S0 in scipy .25,2,.25 # alpha -1,-1,0 # beta -10,10,1 # x 1,0 # gamma, delta 2 # output file z.data/levy_stable/stable-Z1-pdf-sample-data.npyx,p,alpha,beta,pctrrfrrrrrrTr,s r3nolan_pdf_sample_data$TestLevyStable.nolan_pdf_sample_datasL6ww N ! ! < = vv  /C D r5c[R"[[5RS- 5n[R R URSS9nU$)aSSample data points for cdf computed with Nolan's stablec See - http://fs2.american.edu/jpnolan/www/stable/stable.html There's a known limitation of Nolan's executable for alpha < 0.2. The data table loaded below is generated from Nolan's stablec with the following parameter space: alpha = 0.1, 0.2, ..., 2.0 beta = -1.0, -0.9, ..., 1.0 p = 0.01, 0.05, 0.1, 0.25, 0.35, 0.5, and the equivalent for the right tail Ideally, Nolan's output for CDF values should match the percentile from where they have been sampled from. Even more so as we extract percentile x positions from stablec too. However, we note at places Nolan's stablec will produce absolute errors in order of 1e-5. We compare against his calculations here. In future, once we less reliant on Nolan's paper we might switch to comparing directly at percentiles (those x values being produced from some alternative means). Typically inputs for stablec: stablec.exe << 2 # cdf 1 # Nolan S equivalent to S0 in scipy .25,2,.25 # alpha -1,-1,0 # beta -10,10,1 # x 1,0 # gamma, delta 2 # output file z.data/levy_stable/stable-Z1-cdf-sample-data.npyr rr r,s r3nolan_cdf_sample_data$TestLevyStable.nolan_cdf_sample_datasMJww N ! ! < = vv  /C D r5cf[R"[[5RS- 5nU$)zSample data where loc, scale are different from 0, 1 Data extracted in similar way to pdf/cdf above using Nolan's stablec but set to an arbitrary location scale of (2, 3) for various important parameters alpha, beta and for parameterisations S0 and S1. z1data/levy_stable/stable-loc-scale-sample-data.npy)rfrrrrr,s r3nolan_loc_scale_sample_data*TestLevyStable.nolan_loc_scale_sample_datas/ww N ! ! ? @  r5 sample_sizerMr{marksr S0r z alpha,beta))rfr)rfr)rr)rr|z gamma,deltar rrcU[Rl[R"X#XES9n[R"UR USS9UR 5upU S:deg)N)r.rrbrarrr|)r9rr rrrn) rqr r.rrrr lsrrs r3rZTestLevyStable.test_rvss\(.>*   % || FF $F 7 4xxr5rr|r c[RRS5 SnSnSn[RR X!X4SS9n[R "USX!X44S9upgUS :deg ) z3Additional test cases for rvs for alpha equal to 1.h:rfr|rryrarbrrrrN)rfrrr9rrr)rqrr.rarbr[statrs r3test_rvs_alpha1TestLevyStable.test_rvs_alpha1sl y!    ! !%3'+ " -,,q-%*#$=?4xxr5cl/SQn[RRU5up#pE[USSSS9 [ USS5 [ USS 5 [ US S5 U/S Q-n[RRU5upxp[ US5 [ US 5 [ U S S 5 [ U SS 5 g)N)wgr rrrr羼u?r r r r 67',?r r r r tA}˜.?r r r r Gz?rrr{)\(̿rcgۧ1?rg9]c?)r r r r r rgv?r )r9rr#rrr rqr[alpha1beta1loc1scale1r!alpha2beta2rrs r3r(TestLevyStable.test_fits  ',&7&7&A&A!&D#t148E4+FGQ/ '1 9 9&+&7&7&A&A"&E#tVQUBFFA.D&!,r5zUnknown problem with fitstart.rzalpha,beta,delta,gamma)rrrcr)rfrrcrparametrizationc U[Rl[RRXX4SSS9n[RR U5nUupp[ XX4/XX/SS9 g)z7Test that fit agrees with rvs for each parametrization.r)rrrrN)r9rr rr#r) rqr.rrrr rr alpha_obsbeta_obs delta_obs gamma_obss r3 test_fit_rvsTestLevyStable.test_fit_rvsss-<)  $$ Ue$% ))$/471 Y % ' ) 7 r5cB[R"/SQ5n[RR U5up#pE[RR U*5upgp[ US5 US:wde[ Xb5 [ Xs*5 [ X*5 [ X5 g)N r r rrr\r\r\rJrJr]r]r r)rfrr9rr#rr) rqr[r r r r r r rrs r3test_fit_beta_flip!TestLevyStable.test_fit_beta_flip8s HH? @&+&7&7&A&A!&D#t&+&7&7&A&A1"&E#tUAqyyF+E6*D%(F+r5c(Sn[R"/SQ5n[RR U*5up4pV[RR U*U-5upxp[ Xs5 [ X5 [ XU-5 [ X5 g)Nr r )rfrr9rr#r) rqSHIFTr[r r r r r r rrs r3test_fit_delta_shift#TestLevyStable.test_fit_delta_shiftDsx HH? @&+&7&7&A&A1"&E#t&+&7&7&A&A1"u*&M#tF+E)D,/F+r5cp/SQn[RRU5up#pEUS:d SU35eU[U5:dS[U5SU35e/SQn[RRU5upxpUS:d SU35eU [ U5:dS[ U5SU 35eg) N) r r rrr\r\r\rJrJr r zExpected alpha < 1, got zExpected loc < z, got ) r r rrr\r\r\rJrJr zExpected alpha > 1, got zExpected loc > )r9rr#r!r1r s r3test_fit_loc_extrap"TestLevyStable.test_fit_loc_extrapOs 6&+&7&7&A&A!&D#tz>5fX>>zc!f}DAxvdVDD} 7&+&7&7&A&A"&E#tz>5fX>>zc"g~FR vFF~r5z pct_range,alpha_range,beta_range)rr|r)rYr rc)rrr)rr|r|rr)rYr|r rrc)grrrrr ) rr|rYrffffff?r|?rrrrrYrcrDrrct^^^^Un[R"5nURS:H=(a URS:HmSR URURUR /5nSSUUU4Sj/SSUUU4S j/SSUUUU4S j/SS UUU4S j/S SUUU4Sj/S SUUU4Sj//n[ U5GHSun upn U [Rl U b X\"U5OUn [5nUR[S5 [RRU SU SU SSSS9n[R"SS9 [!U /SQU[R""XS- 5[R""XS- 5[R""U S5- /5nSSS5 WUSU :[R$"U5-nSU S U S!US"UR&R(S#U3 n[+UU SU US$S%9 SSS5 GMV g!,(df  Ns=f!,(df  GMz=f)&z2Test pdf values against Nolan's stablec.exe outputLinuxi686/dnir]c >[R"UST5[R"UST5-[R"UST5-USS:HUSS:H-USS:USS:-USS:H--USS:*[R"USS S /5--USS :*[R"USS S /5--USS:*[R"USSS/5--USS:H[R"USSS/5-[R"[R"US5/SQ5--USS:H[R"USS/5-[R"[R"US5S/5--USS:H[R"USSS/5-[R"[R"US5/SQ5--USS:HUSS:H-USS:H--USS:HUSS:H-USS:H--USS:HUSS :H-USS:H--USS:H[R"USS/5-[R"[R"US5/SQ5--USS:H[R"USSS/5-[R"[R"US5/SQ5--USS:H[R"USS S /5-[R"[R"US5SS/5--USS:-)-$)Npctr.rrr|rr& rrrrr|rrrYrr)r|rrr r )gٿ333333ӿrrr|r rfrYrrr)rrrfrrE)rfisinrr alpha_range beta_range pct_ranges r3rI7TestLevyStable.test_pdf_nolan_samples..sGGAeHi0GGAgJ 45GGAfIz23 vY!^uX_.vY#-wZ3.0uX_. wZ3.GGAeHsCj9: wZ3.GGAeHsCj9:$wZ3.GGAeHr2h78%,wZ3.GGAeHsCj9:GGBFF1V9$5|DE-6wZ3.GGAeHrd34GGBFF1V9$5t<=7@wZ3.GGAeHsCj9:GGBFF1V9$57MNOA#JwZ3.vY#-/uX-/K(TwZ3.vY$.0uX-/U-^wZ3.vY#-/uX-/_2hwZ2-GGAeHrd34GGBFF1V9$57GHIi7rwZ2-GGAeHsCj9:GGBFF1V9$5|DEs<|wZ2-GGAeHsCj9:GGBFF1V9$5RyAB}AF7s*GC,EHr5r rc>[R"UST5[R"UST5-[R"UST5-USS:-USS:g-$)Nr r.rrrfrfr r s r3rIr siGGAeHi0GGAgJ 45GGAfIz23wZ#%'wZ2% 'r5c>USS:HT(+-[R"UST5-ST;-[R"UST5-$)Nr.rfr rr )rr r is_linux_32r s r3rIr sXwZ2%$_&GGAeHi01;&(GGAfIz2 3r5g& .=c>[R"UST5[R"UST5-[R"UST5-USS:*-$)Nr r.rrr r s r3rIr sVGGAeHi0GGAgJ 45GGAfIz23wZ3&(r5 fft-simpsonrc>USS:[R"UST5-[R"UST5-[R"UST5-$)Nr.rr rr r s r3rIr sVwZ3&GGAeHi01GGAgJ 45GGAfIz23r5rc>[R"UST5[R"UST5-[R"UST5-USS:-USS:-$)Nr r.rr rr r s r3rIr shGGAeHi0GGAgJ 45GGAfIz23wZ!^%wZ#% 'r5Nz2Density calculations experimental for FFT method.*r[r.rr rrbrarrcalcabserrrelerrrr z pdf test  failed with method 'z ' [platform: z]  Ferr_msgverbose)platformunamesystemmachinejoin processorrSr9rr r recordrrerfrrrisnanrrr)rqr r r r rr platform_desctestsixdefault_methodr filter_funcsubdatasuprsubdata2failuresmessager s ``` @r3test_pdf_nolan_samples%TestLevyStable.test_pdf_nolan_samples[sl4% llg-I%--62I  \\5==%// :< tJL ^U% U% W' t& t& kC L"+5!1 B3AE   0 + 7;t,=A "$ "H%%))CLG$FO *[[h/04FF1s|#34FF1s|#34rvvgcl7KK H0$H%-((1+  t#88HI""/HNN4H4H3IH:W CL#! ?%$ "2"0/%$s'5AH' AH)AH' H$ H'' H7 c^^^UnSSUUU4Sj/SSUUU4Sj/SSUUU4Sj/SS UUU4S j/SS UUU4S j/SS UUU4Sj//n[U5GHPunupn U[RlU b XZ"U5OUn [ 5n U R [ S5 [RRU SU SU SSSS9n [R"SS9 [U /SQU [R"XS- 5[R"XS- 5[R"U S5- /5nSSS5 WUSU :[R"U 5-nSUSUSURRSU3n[U U SU US S!9 SSS5 GMS g!,(df  Np=f!,(df  GMw=f)"z3Test cdf values against Nolan's stablec.exe output.r r cb>[R"UST5[R"UST5-[R"UST5-USS:H[R"US/SQ5-USS:H-USS:H[R"US/SQ5-USS:H--)-$ Nr r.rrf)r rrrrYrrrr r s r3rI7TestLevyStable.test_cdf_nolan_samples..PsGGAeHi0GGAgJ 45GGAfIz23 wZ2-GGAfI/ABCuX-/ wZ2-GGAfI?@uX-/  r5r|c`>[R"UST5[R"UST5-[R"UST5-USS:H[R"US/SQ5-USS:H--USS:H[R"US/SQ5-USS:H--$r r r s r3rIr esGGAeHi0GGAgJ 45GGAfIz237r)& +=>?5T)+ 7r)& ?;<5T)+ r5r rc>[R"UST5[R"UST5-[R"UST5-USS:-$)Nr r.r333333?r r s r3rIr wsVGGAeHi0GGAgJ 45GGAfIz23wZ#%'r5rc>[R"UST5[R"UST5-[R"UST5-USS:-USS:*-$)Nr r.rrr r r s r3rIr iGGAeHi0GGAgJ 45GGAfIz23wZ#%'wZ3& (r5rJc>[R"UST5[R"UST5-[R"UST5-USS:-USS:*-$)Nr r.rrX rr r s r3rIr r r5rc>[R"UST5[R"UST5-[R"UST5-USS:-USS:*-$)Nr r.rrfrX r r s r3rIr r r5Nz[Cumulative density calculations experimental for FFT method. Use piecewise method instead.*r[r.rr rr rrr rr z cdf test r z' r Fr )rSr9rr r r rrnrfrrrr rrr)rqr r r r rr r r rr r r rr r r s ``` r3test_cdf_nolan_samples%TestLevyStable.test_cdf_nolan_samples3s2%U% ,T$ &t& t& t& t& GL \"+5!1 B3AE   0 + 7;t,=A "$ "@ %%))CLG$FO *[[h/04FF1s|#34FF1s|#34rvvgcl7KK H0$H%-((1+'rd*??Os&nn2232hZACL#! ;%$ "2"0/%$s'AF6AF%;AF6% F3 /F66 G r*rrRrernc:[R"5nURS:H=(a S[R"5S;nU(aUS:Xa[R "S5 UnS[ RlS[ Rl XfSU:HnSU3[ Rl US ;deUS:Xa[ RRO[ RRnU"US US US S SS9n [XUS5 g)zGTests for pdf and cdf where loc, scale are different from 0, 1 r 32bitrrez4Test unstable on some platforms; see gh-17839, 17859r r*S)rernr[r.rrcrr rN)r r r architecturerrr9rr r r rernr) rqr r*rRr r rr rQr-s r3test_location_scale"TestLevyStable.test_location_scales llg-W'X=R=R=TUV=W2W  45= KKN O*0;,/:,G}-./0[*~%%%%)U]E   ! !8I8I8M8M  CL''*GFO1!  DM40r5zmethod,decimal_placesr rr c [R"/SQ5n[R"/SQ5n[R"/SQ5n[R"SS9 [5nUR [ SS9 U[ Rl[ RRUSUSS S 9n[XtX!5 S S S 5 S S S 5 g !,(df  N=f!,(df  g =f) zwsample points extracted from Tables and Graphs of Stable Probability Density Functions - Donald R Holt - 1973 - p 187. )rrrrr r r r rcrcrcrcrrrrrrrr)gtF_?gܵ|?gQ?g@?gec]?gHP?gPkw?g!rh?gTN?g ?gDioɴ?g r?gH?g+?gHPs?gX?g&?g 0*?gv/?ga2U0*?)rrr|r rrr|r rrr|r rrr|r rrr|r rrzDensity calculation unstable.*)categoryr r rr N) rfrrr rMrr9rr rer)rqrdecimal_placesxsdensitybetasr res r3'test_pdf_alpha_equals_one_beta_non_zero6TestLevyStable.test_pdf_alpha_equals_one_beta_non_zerosXX H ((     [[X &(9(;s JJ'8  4:E   0##''AuA1'EC n )< & &(;(; & &s% C!(AC?C! C C!! C/zparams,expected)r r rr )rcrr\r)r\rrrct[RRUSUSUSUSSS9n[X25 g)Nrr rcrrra)r9rr)rqrrSobserveds r3r TestLevyStable.test_stats sE$$** 1IvayfQivay+  H/r5r.)rr|rzfunction,beta,points,expectedrfrT r\rrrcSUs=:aS:de e[U"XAUS9[R"[U5U55 g)a4Ensure the pdf/cdf routines do not return nan outside support. This distribution's support becomes truncated in a few special cases: support is [mu, infty) if alpha < 1 and beta = 1 support is (-infty, mu] if alpha < 1 and beta = -1 Otherwise, the support is all reals. Here, mu is zero by default. rr r.rN)rrfrr )rqr.rQrrbrSs r3!test_distribution_outside_support0TestLevyStable.test_distribution_outside_supports>P5}1}}} Vt 4 GGCK * r5zx,alpha,beta,expected))r/iZ?s0?gz5F?)r~H?|/mg>m?)r.d5 ?L ,ٿg깃\?)r>?ЀjoؿgA]4?)rpa2?Յ пgqQ?c~S[Rl[[RR XUS9U5 g)aTest pdf for x equal to zeta. With S1 parametrization: x0 = x + zeta if alpha != 1 So, for x = 0, x0 will be close to zeta. When case "x equal zeta" is not handled properly and quad_eps is not low enough: - pdf may be less than 0 - logpdf is nan The points from the parametrize block are found randomly so that PDF is less than 0. Reference values taken from MATLAB https://www.mathworks.com/help/stats/stable-distribution.html Afc=r. Nr9rr rrerqr[r.rrSs r3test_x_equal_zeta TestLevyStable.test_x_equal_zetaIs7Z&-"    ! !!t ! <  r5) )rr1 r2 gz¶?)rr3 r4 gV?)rr5 r6 gRE)3\?)rr7 r8 gB?)rr9 r: g/^?)-C6r1 r2 gH?)rA r3 r4 gn?)rA r5 r6 gx\?)rA r7 r8 gVM%?)rA r9 r: gm?c~S[Rl[[RR XUS9U5 g)aTest pdf for x near zeta. With S1 parametrization: x0 = x + zeta if alpha != 1 So, for x = 0, x0 will be close to zeta. When case "x near zeta" is not handled properly and quad_eps is not low enough: - pdf may be less than 0 - logpdf is nan The points from the parametrize block are found randomly so that PDF is less than 0. Reference values taken from MATLAB https://www.mathworks.com/help/stats/stable-distribution.html r< r. 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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)) )rf)r|rrr)r)gUUUUUU?r g^cQg333333rrN)r9rr)rr1 exact_mvskrs r3test_powerlaw_statsr4 }!s>>, = ?E ~~##Av#6!$3r5c\[RRSS5n[US5 g)Nrr r)r9rrrrs r3test_powerlaw_edger7 !s" a#ACr5cv[RRSS5n[US5 [RR S/SQ5n[U[ R SS/5 [RRS/SQ5n[U[ R S[ R */5 g)Nrr r)rrfrrf)r9rrrrerfr(r6 s r3test_exponpow_edger9 !s a#AC 1./ARVVS#&' a!12ARVVS266'*+r5c^[RRSSS5n[US5 g)Nrr rf)r9gengammarerr6 s r3test_gengamma_edger< !s$ 1a#ACr5za, c, ref, tol))g`6Ar g2j!@r)r r ggA@r)rr rC r)rZr g#@r)g`FAr gI~mr)gl$W}Lrg.I#UrcR[[RRX5X#S9 gr)rr9r; rA)r1rrrs r3test_gengamma_extreme_entropyr> !s"ENN**10#@r5c[RRSSS5nUS:Xde[RRSSS5nU[R *:Xdeg)Nrr rr)r9r; rerrfr()rrs r3!test_gengamma_endpoint_with_neg_cr@ !sO 1a$A 8O8 >> Ar *D BFF7??r5c[RRSSS5n[US5 [RRSSS5n[US5 g)Nr rrfg>r\gqq?)r9r; rKrr6 s r3test_gengamma_munprB !sF Rb)A:& RR(A6"r5cn[R"/SQ5n[R"SS9 [5nUR [ S5 UR [ S5 [RRU5 SSS5 SSS5 g!,(df  N=f!,(df  g=f)N)0g>X*ȿgPix#?g=?g&^?goϿg뵏˿gO_ZQ?g\?g~?g)/?gRܶ?g핚οgH߃[g?g S"gm_ пg7k?g#BF?g KU= ?gT?gZg[Qs?g2?g6Hwƭ8?g*- ?ga]?g'?g޻?g?J_?g~sw?g0FfͿgg"@ӿg?%:˿g&zM?g?Gg F?g~,g HgW2ҿgTzy?gwm ǿg@gW=ֿg 't?g".v1glg.Hꮿg,?g*˿rrz:The maximum number of subdivisions .50. has been achieved.z-floating point number truncated to an integer) rfrrr rMrrr9r+r)rP r s r3test_ksone_fit_freezerD !s  / 0A X &  C JJ)# $ JJ~F H KKOOA  ! 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