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fix: avoid cancellation for small sigma in stats/base/dists/lognormal/kurtosis #15776

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fix: avoid cancellation for small sigma in stats/base/dists/lognormal/kurtosis #15776

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db0e2c3
fix: avoid cancellation for small `sigma` in `stats/base/dists/lognor…
Abhist17 Oct 1, 2026
78298fb
docs: move comment into `gen` in `stats/base/dists/lognormal/kurtosis…
Abhist17 Oct 13, 2026
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29 changes: 20 additions & 9 deletions lib/node_modules/@stdlib/stats/base/dists/lognormal/kurtosis/lib/main.js
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Expand Up @@ -21,14 +21,30 @@
// MODULES //

var isnan = require( '@stdlib/math/base/assert/is-nan' );
var exp = require( '@stdlib/math/base/special/exp' );
var expm1 = require( '@stdlib/math/base/special/expm1' );


// MAIN //

/**
* Returns the excess kurtosis for a lognormal distribution with location `mu` and scale `sigma`.
*
* ## Method
*
* The excess kurtosis is
*
* ```tex
* e^{4\sigma^2} + 2e^{3\sigma^2} + 3e^{2\sigma^2} - 6
* ```
*
* which cancels for small \\(\sigma\\). With \\(u = e^{\sigma^2} - 1\\), evaluated by `expm1`, it expands to
*
* ```tex
* u(16 + u(15 + u(6 + u)))
* ```
*
* whose terms are all positive.
*
* @param {number} mu - location parameter
* @param {PositiveNumber} sigma - scale parameter
* @returns {number} excess kurtosis
Expand All @@ -54,21 +70,16 @@ var exp = require( '@stdlib/math/base/special/exp' );
* // returns NaN
*/
function kurtosis( mu, sigma ) {
var out;
var s2;
var u;
if (
isnan( mu ) ||
isnan( sigma ) ||
sigma <= 0.0
) {
return NaN;
}
s2 = sigma * sigma;
out = exp( 4.0*s2 );
out += 2.0 * exp( 3.0*s2 );
out += 3.0 * exp( 2.0*s2 );
out -= 6.0;
return out;
u = expm1( sigma*sigma );
return u * ( 16.0 + ( u * ( 15.0 + ( u * ( 6.0 + u ) ) ) ) );
}


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6 changes: 3 additions & 3 deletions lib/node_modules/@stdlib/stats/base/dists/lognormal/kurtosis/manifest.json
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Expand Up @@ -40,7 +40,7 @@
"dependencies": [
"@stdlib/math/base/napi/binary",
"@stdlib/math/base/assert/is-nan",
"@stdlib/math/base/special/exp"
"@stdlib/math/base/special/expm1"
]
},
{
Expand All @@ -56,7 +56,7 @@
"libpath": [],
"dependencies": [
"@stdlib/math/base/assert/is-nan",
"@stdlib/math/base/special/exp",
"@stdlib/math/base/special/expm1",
"@stdlib/constants/float64/eps"
]
},
Expand All @@ -73,7 +73,7 @@
"libpath": [],
"dependencies": [
"@stdlib/math/base/assert/is-nan",
"@stdlib/math/base/special/exp"
"@stdlib/math/base/special/expm1"
]
}
]
Expand Down
29 changes: 20 additions & 9 deletions lib/node_modules/@stdlib/stats/base/dists/lognormal/kurtosis/src/main.c
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Original file line number Diff line number Diff line change
Expand Up @@ -18,11 +18,27 @@

#include "stdlib/stats/base/dists/lognormal/kurtosis.h"
#include "stdlib/math/base/assert/is_nan.h"
#include "stdlib/math/base/special/exp.h"
#include "stdlib/math/base/special/expm1.h"

/**
* Returns the excess kurtosis for a lognormal distribution with location `mu` and scale `sigma`.
*
* ## Method
*
* The excess kurtosis is
*
* ```tex
* e^{4\sigma^2} + 2e^{3\sigma^2} + 3e^{2\sigma^2} - 6
* ```
*
* which cancels for small \\(\sigma\\). With \\(u = e^{\sigma^2} - 1\\), evaluated by `expm1`, it expands to
*
* ```tex
* u(16 + u(15 + u(6 + u)))
* ```
*
* whose terms are all positive.
*
* @param mu location parameter
* @param sigma scale parameter
* @return excess kurtosis
Expand All @@ -32,19 +48,14 @@
* // returns ~110.936
*/
double stdlib_base_dists_lognormal_kurtosis( const double mu, const double sigma ) {
double out;
double s2;
double u;
if (
stdlib_base_is_nan( mu ) ||
stdlib_base_is_nan( sigma ) ||
sigma <= 0.0
) {
return 0.0/0.0; // NaN
}
s2 = sigma * sigma;
out = stdlib_base_exp( 4.0*s2 );
out += 2.0 * stdlib_base_exp( 3.0*s2 );
out += 3.0 * stdlib_base_exp( 2.0*s2 );
out -= 6.0;
return out;
u = stdlib_base_expm1( sigma * sigma );
return u * ( 16.0 + ( u * ( 15.0 + ( u * ( 6.0 + u ) ) ) ) );
}
1 change: 0 additions & 1 deletion lib/node_modules/@stdlib/stats/base/dists/lognormal/kurtosis/test/fixtures/julia/REQUIRE
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@@ -1,3 +1,2 @@
Distributions 0.23.8
julia 1.5
JSON 0.21
2 changes: 1 addition & 1 deletion lib/node_modules/@stdlib/stats/base/dists/lognormal/kurtosis/test/fixtures/julia/data.json
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@@ -1 +1 @@
{"sigma":[0.013606039203587494,1.6770196416730887,4.426416512870041,3.939767112070066,4.419662226502438,0.8434960457795337,1.0026940133952622,1.4512138282428566,2.8623145368917067,3.2153907122177996,2.549051222353713,0.7208063555120703,2.044091167217031,0.9107223308474277,0.32806114406364606,1.7706768795382821,2.7706240687282433,2.290652267933818,4.429448105152064,3.6411858326315416,3.936440891920885,2.6572952706035355,4.846500506506592,3.4914522517183633,4.514825261652415,3.3816883097647055,4.531386430969631,1.5646050466274741,1.1698142455987182,2.3306365033036047,0.25298270392462463,3.432215428336759,2.6393454298572716,4.546023064172649,4.9614981823819315,0.6767380270254053,1.995626862775618,0.9182724132017517,1.6123270518329025,1.176263648034963,4.161705200208267,3.8202396331483337,4.504624824960235,1.9047359026014143,0.03996494779838655,1.0943749037658457,4.294849529418365,1.7338675156060535,2.575391132109978,4.940118927276637,4.6043311839301815,4.766208202403544,0.223040099918973,2.8938559859622193,3.6441421854062006,3.774078854693675,2.281404509767518,1.8760172456551683,3.80300783924309,1.7744213999186365,2.480611887972033,0.9001588606748534,3.007506136357504,2.99031232372142,3.1316138424957716,3.626505161580307,1.7433973679516745,3.391897895536502,1.7749204524124174,4.153575862028073,1.908680439266962,4.6881269264290975,0.9231496485240065,3.5400867587100016,1.1607450221163806,4.806666723939346,2.683479522317742,3.6652189461301674,0.43170185675238004,1.5845150389696294,4.665099694101583,4.157859502858453,3.902770391049705,0.9327170836030874,2.168638486308275,4.997796469821671,3.5833264426009426,2.232959786563533,1.341290922694096,4.860616170905477,2.81668647327327,3.496961574934864,3.592712077690484,4.606904812578824,1.221039634545329,2.0474878204998936,4.775991769508673,3.2005905180987346,1.1102791583885163,4.517173024418933],"expected":[0.0029627772282978526,86904.32612102535,1.0884551812127136e34,9.205927173872686e26,8.570694117978124e33,40.57097462916551,113.02347398129417,5860.96113123167,1.7087123219377416e14,9.125455894930533e17,1.9449352569240494e11,19.975805678561414,1.8701350925232217e7,61.435473542836206,2.0207080750853397,305519.8480698636,2.1657186531854242e13,1.3173712422554023e9,1.211850959251774e34,1.0762506965795024e23,8.290046159529648e26,1.8507069543325483e12,6.365322628572092e40,1.501821208843442e21,2.570525418574072e35,7.346089523598491e19,4.680353750231139e35,21378.184211619016,400.03102625461463,2.753764569169632e9,1.1247575548773288,2.9116631204282466e20,1.2654699173055764e12,7.963149180311166e35,5.796666143283337e42,15.644800914707993,8.603750872707954e6,64.46248406783172,38219.60701320712,421.9616562892313,1.2233683097642807e30,2.2530773927674577e25,1.7790882302630783e35,2.1177298955578553e6,0.025613923280804585,219.97520809679804,1.1053996735304558e32,184640.86231706975,3.3359483533629706e11,2.485630020639736e42,6.728816398359076e36,2.904159117938304e39,0.8559062403015236,3.532055572328829e14,1.1730830263714398e23,5.543598714594206e24,1.1129010052559166e9,1.3802617072481334e6,1.3322219979787817e25,321799.3124232156,4.914289785487553e10,57.468152808846455,5.164623455702553e15,3.4190282136833995e15,1.0878296725197038e17,7.023703106411123e22,210107.91542297904,9.687038239190974e19,324036.5658544803,9.33529272228243e29,2.2472380548227155e6,1.5157385016621903e38,66.50824522254032,5.897787620089486e21,371.3336164258881,1.3672155967061007e40,3.237237724100069e12,2.1724715507692697e23,3.960706589227197,27171.320180376802,6.4043162912494e37,1.0764083038567397e30,2.8842213527928303e26,70.7403144211999,1.5060746781812477e8,2.4613727341799226e43,2.021918544042339e22,4.652789296591771e8,1879.5675199228124,1.1011657934479049e41,6.061768403997698e13,1.751870105915238e21,2.647069569361544e22,7.398109259386784e36,617.4468441893677,1.9761933212281045e7,4.218885118179443e39,6.241641662908753e17,248.55305197979465,2.7980710753276238e35],"mu":[-3.785188913233297,-2.3558649744770355,-3.3138459690143613,-2.9223611054467513,-2.277843450292301,-2.8313848475061523,-2.122842969889887,-2.779670277997986,-3.4129542144140266,-3.017605174232774,-3.2270755124211306,-3.4270122548111033,-3.9717943231511783,-2.3830370154185094,-2.5550404092066095,-2.8817967495314214,-2.502531286316559,-2.4357741219952023,-2.1733363701090442,-2.728269488988314,-2.710744423754661,-2.527014372496091,-3.6913646196420067,-3.9858664200566967,-3.3972121018385684,-2.6181944319390817,-2.737859280611349,-3.1187275564953567,-2.0777464142254476,-2.8035994714081585,-3.0331760887506274,-2.8664954691854305,-3.385305429742553,-2.3982491604332288,-3.4207936628141016,-2.7738508798259445,-3.3212546352727386,-3.133191803271066,-2.574504826434971,-2.003442385493707,-2.2254394340378583,-3.3018369649841675,-2.3637046976567095,-3.7428617372308106,-2.7093912186654125,-2.785996368097992,-2.0199160183078315,-3.632329689818935,-2.580119206736008,-3.930258571516214,-2.8283672729962146,-2.0052323273061843,-2.5568706518664057,-2.3781838978276753,-3.5859983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{"expected":[374.6641575762713,0.25813962085582276,1.4447858872781317e6,3.399135621174137e39,3.16008176276834e27,2.369170020695188e33,657842.1230198158,1286.7746878508783,25.291400742239826,5.651081865253103e30,2.1533314229154277e36,1.87770415749761e35,3.0569395328710516e31,2.9340381292384405,7.464285602755757,5.097726402778806e36,7.910465082413064e37,1112.4703001340206,7405.078623373308,10121.912798346424,1.3501134898721713e40,46.8475663168776,1.2314968247021147,1.1567587250723511e41,491460.2135600005,3.9950144862962525e35,1.3777808597406285e18,3.0874297763377917e6,1.2662867856459092e18,2.240252734701454e11,1.9016932398723793e30,1.4289463900371857e20,2.3330153578965325e15,3.0467427575565087,217.0173937193514,3.262882309561579e39,7.895346894203445e31,52.18529377123516,111213.8293969303,3.3352885507406935e7,0.1892280289411066,5.510607969202061e6,1.1389079308742155e31,1.4954980837114608e27,21.41441790941474,1.4694570948468705e13,6.477380609282881e8,1.5772208477237264e6,0.7458796942407523,2.8820320208832912e7,6.341242908833529e28,2.6206898016711603e20,2.3772943611695165e18,1.6554378086983018e16,3.38480397605341e12,9.604897778919878e19,2.0155999680017145e40,3.6969204589727385e35,9.540711002242792e10,6.229872436881677e37,5.69016072869923e30,6.245794854083576e27,4.605375944013919,71660.63775658574,815183.1611355108,813.812969922389,0.03675026419630696,15.012769973600365,1.50674693005475e8,7.482434242009564,1.3161011535825863e21,1.0380749509469504e24,10.348589390718793,128.50575677622444,2.4960865372391215e8,0.005668340232668664,467.2678312079923,1.2083031183089783e33,0.7395662277155438,4.1209481483942305e12,1.263142314213425e26,1.014413562774108,1.1406622380615926e21,3.862782535605391e17,1.6774788339723594e38,1.7870505484339687e43,2.4471090631030197e20,3.293382221951729,5.532690538560754e32,2.320912764190833e18,17.500361835151768,1180.4541468836865,18.03791336731804,2.730594947638017e32,4.249306757383831e33,8.261529718234978e12,0.1503146484066514,3.785067638556271e23,3.2195431394602635e22,19.83892535987055],"mu":[-2.936268483547231,-3.5456407430467287,-2.49801917636995,-2.618252946656159,-2.0466537251006485,-2.7680898517432775,-2.9218119166556513,-2.821305338466147,-2.3507857664865957,-3.440566061106624,-2.8211611958768374,-2.5292492957681727,-2.81215375538012,-2.301579576871421,-2.173445458840998,-2.616630081475671,-2.1430009664551837,-2.9334977377800673,-3.7551044841401326,-3.7618165998806954,-3.0593527055437315,-2.8637232814741362,-3.7970649665660465,-2.609489430698722,-3.520524939250132,-2.3936346100030477,-3.042570407059161,-3.966624207952349,-3.4057618962111387,-3.9898463963056385,-2.0000782726334965,-3.132221671535051,-3.8614716073892144,-3.7908310465377824,-2.1867612028947487,-2.350121414206386,-2.5665092788463353,-3.3336272422938924,-3.6121654291144507,-3.380150333769962,-3.6265414365221185,-3.733532041305666,-2.4456840102716177,-3.2449275225812926,-2.2519890363360857,-2.0945682191109993,-3.5559321694485453,-2.3387300879848723,-3.624880091381444,-2.524104370950912,-2.2783986308237054,-3.7527950766074465,-2.55077892913095,-3.7451982692097214,-2.5199965852831587,-3.201726475991415,-2.99099805724927,-3.9310181413823435,-3.584874534679627,-2.3051876964255826,-2.2116237464290998,-2.8296393607238377,-2.1662507243309403,-2.7505864931201165,-3.057944902385742,-3.8233822698753093,-2.868291630727116,-3.8406269406378533,-2.4312801547487735,-3.454746741655492,-2.812093044037038,-3.8175354740820002,-3.856091813762546,-2.09349523867552,-3.5846319561800746,-2.6297910032704905,-2.4197244231179122,-2.2276301700842636,-3.2998239954204727,-2.530702475024433,-3.867061218893978,-3.8494945095720987,-3.720420788813494,-3.545913663191005,-3.4356969089054554,-2.0507918305066726,-3.5747491472880517,-3.630209442521034,-2.027017436425869,-3.1447549503763716,-3.4757146700719166,-3.1263576823505765,-2.046818479599792,-2.655142149008398,-2.590511030838605,-2.8411850763808406,-3.795422194864786,-3.0509237286700817,-3.197963455853589,-3.854673862143601],"sigma":[1.1618374601125603,0.12558000466621422,1.8791055624062947,4.770333840433493,3.9787056672254124,4.383145210670981,1.825108513213295,1.301966752506325,0.762629870349017,4.207416064409523,4.573294203527411,4.506121616465517,4.257274718606362,0.38339416813371496,0.5419837065220379,4.596788498055146,4.670755694128237,1.286444146606811,1.4725239840884263,1.5004074597003831,4.806339365230706,0.8672214727291544,0.26362663330627645,4.861882626724208,1.8046341425907286,4.527016828375898,3.2313676449874,1.9296382835441221,3.228101643320839,2.5559789424289487,4.1749340534957105,3.4061936582354715,2.974291939103117,0.38928900017469514,1.092595655599145,4.76926172418997,4.285044090986942,0.8847277585025631,1.6958719978806895,2.0794226102337547,0.10784234701417761,1.9672356919130174,4.228185417418687,3.9551313665586214,0.7332180174844305,2.753065436519475,2.2514651911420818,1.8850182036507757,0.20913881135843415,2.0705630442915535,4.071837953229406,3.428378760686874,3.2524007961963655,3.0555380321271812,2.6855558790565723,3.391584059026906,4.816749957739735,4.5248750926663766,2.5138523157846997,4.664359663904484,4.207620801717572,4.000053302764205,0.4570734989970082,1.6621011631078197,1.8400027149086489,1.252261937610346,0.047847044672857386,0.6692465709884676,2.1686643267221877,0.5424211325378964,3.4867230369915077,3.71817859440839,0.601332644704618,1.0210540334233258,2.1976901054565134,0.01881730492160061,1.1884672066630708,4.363900922679868,0.20830531716359846,2.69470312281179,3.876235749338262,0.24129157347034913,3.4815903361228906,3.1817940749478146,4.690829497684816,4.989782698706603,3.425879201850168,0.401626387335699,4.341468564753905,3.251478162812904,0.6970270064484563,1.2927989588822208,0.7024824953249514,4.3210890547188,4.399774537254491,2.7267861350327776,0.09628058468193423,3.6841049912399897,3.59951756460769,0.7195756908041023]}
13 changes: 9 additions & 4 deletions lib/node_modules/@stdlib/stats/base/dists/lognormal/kurtosis/test/fixtures/julia/runner.jl
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Expand Up @@ -16,7 +16,6 @@
# See the License for the specific language governing permissions and
# limitations under the License.

import Distributions: kurtosis, LogNormal
import JSON

"""
Expand All @@ -39,9 +38,10 @@ julia> gen( mu, sigma, "data.json" );
```
"""
function gen( mu, sigma, name )
z = Array{Float64}( undef, length(mu) );
for i in eachindex(mu)
z[ i ] = kurtosis( LogNormal( mu[i], sigma[i] ) );
# The excess kurtosis is evaluated in extended precision using the textbook formula `exp(4*sigma^2) + 2*exp(3*sigma^2) + 3*exp(2*sigma^2) - 6`, as subtracting `6` in double precision cancels for small `sigma`.
z = setprecision( BigFloat, 2048 ) do
s2 = BigFloat.( sigma ) .^ 2;
Float64.( exp.( 4 .* s2 ) .+ ( 2 .* exp.( 3 .* s2 ) ) .+ ( 3 .* exp.( 2 .* s2 ) ) .- 6 );
end

# Store data to be written to file as a collection:
Expand Down Expand Up @@ -71,3 +71,8 @@ dir = dirname( file );
mu = rand( 100 ) .* 2.0 .- 4.0;
sigma = rand( 100 ) .* 5.0;
gen( mu, sigma, "data.json" );

# Small `sigma` (above `1e-154`, where `sigma^2` underflows):
mu = ( rand( 1000 ) .* 4.0 ) .- 2.0;
sigma = exp10.( range( -150.0, stop = -1.0, length = 1000 ) );
gen( mu, sigma, "small_sigma.json" );
1 change: 1 addition & 0 deletions ..._modules/@stdlib/stats/base/dists/lognormal/kurtosis/test/fixtures/julia/small_sigma.json
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33 changes: 22 additions & 11 deletions lib/node_modules/@stdlib/stats/base/dists/lognormal/kurtosis/test/test.js
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Expand Up @@ -22,16 +22,16 @@

var tape = require( 'tape' );
var isnan = require( '@stdlib/math/base/assert/is-nan' );
var abs = require( '@stdlib/math/base/special/abs' );
var isAlmostSameValue = require( '@stdlib/assert/is-almost-same-value' );
var PINF = require( '@stdlib/constants/float64/pinf' );
var NINF = require( '@stdlib/constants/float64/ninf' );
var EPS = require( '@stdlib/constants/float64/eps' );
var kurtosis = require( './../lib' );


// FIXTURES //

var data = require( './fixtures/julia/data.json' );
var smallSigma = require( './fixtures/julia/small_sigma.json' );


// TESTS //
Expand Down Expand Up @@ -76,9 +76,7 @@ tape( 'if provided a nonpositive `sigma`, the function returns `NaN`', function

tape( 'the function returns the excess kurtosis of a lognormal distribution', function test( t ) {
var expected;
var delta;
var sigma;
var tol;
var mu;
var y;
var i;
Expand All @@ -89,13 +87,26 @@ tape( 'the function returns the excess kurtosis of a lognormal distribution', fu
for ( i = 0; i < mu.length; i++ ) {
y = kurtosis( mu[i], sigma[i] );
if ( expected[i] !== null ) {
if ( y === expected[i] ) {
t.strictEqual( y, expected[i], 'mu:'+mu[i]+', sigma: '+sigma[i]+', y: '+y+', expected: '+expected[i] );
} else {
delta = abs( y - expected[ i ] );
tol = 10.0 * EPS * abs( expected[ i ] );
t.ok( delta <= tol, 'within tolerance. mu: '+mu[i]+'. sigma: '+sigma[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
}
t.strictEqual( isAlmostSameValue( y, expected[ i ], 61 ), true, 'returns expected value' );
}
}
t.end();
});

tape( 'the function returns the excess kurtosis of a lognormal distribution with a small `sigma`', function test( t ) {
var expected;
var sigma;
var mu;
var y;
var i;

expected = smallSigma.expected;
mu = smallSigma.mu;
sigma = smallSigma.sigma;
for ( i = 0; i < mu.length; i++ ) {
y = kurtosis( mu[i], sigma[i] );
if ( expected[i] !== null ) {
t.strictEqual( isAlmostSameValue( y, expected[ i ], 1 ), true, 'returns expected value' );
}
}
t.end();
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