From 6cadd736c4a2e7ed008ede045841713fbabeffc8 Mon Sep 17 00:00:00 2001 From: Abhist17 Date: Mon, 28 Sep 2026 09:44:13 +0530 Subject: [PATCH] fix: evaluate `stats/base/dists/normal/quantile` via `erfcinv` to keep the lower tail `mu + sigma*sqrt(2)*erfinv(2p-1)` loses the low-order digits of small `p` when forming `2p-1`, and for `p < 2^-54` that difference rounds to `-1`, so the function returned `-Infinity` (e.g., `quantile( 1e-20, 0, 1 )`, whose value is about `-9.26`). `lognormal/quantile` inherits the problem and returned `0`. Evaluate `mu - sigma*sqrt(2)*erfcinv(2p)` instead, as `binomial/quantile` and `poisson/quantile` already do, and add fixtures for both tails generated in 2048-bit precision. --- .../base/dists/normal/quantile/lib/factory.js | 4 +- .../base/dists/normal/quantile/lib/main.js | 14 ++- .../base/dists/normal/quantile/manifest.json | 6 +- .../base/dists/normal/quantile/src/main.c | 14 ++- .../quantile/test/fixtures/julia/REQUIRE | 3 + .../test/fixtures/julia/lower_tail.json | 1 + .../quantile/test/fixtures/julia/runner.jl | 87 +++++++++++++++++++ .../test/fixtures/julia/upper_tail.json | 1 + .../normal/quantile/test/test.factory.js | 44 ++++++++++ .../dists/normal/quantile/test/test.native.js | 40 +++++++++ .../normal/quantile/test/test.quantile.js | 40 +++++++++ 11 files changed, 245 insertions(+), 9 deletions(-) create mode 100644 lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/REQUIRE create mode 100644 lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/lower_tail.json create mode 100644 lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/runner.jl create mode 100644 lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/upper_tail.json diff --git a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/lib/factory.js b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/lib/factory.js index 9ff5e0596d98..dda1b22a1d5f 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/lib/factory.js +++ b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/lib/factory.js @@ -22,7 +22,7 @@ var constantFunction = require( '@stdlib/utils/constant-function' ); var degenerate = require( '@stdlib/stats/base/dists/degenerate/quantile' ).factory; -var erfinv = require( '@stdlib/math/base/special/erfinv' ); +var erfcinv = require( '@stdlib/math/base/special/erfcinv' ); var isnan = require( '@stdlib/math/base/assert/is-nan' ); var sqrt = require( '@stdlib/math/base/special/sqrt' ); @@ -72,7 +72,7 @@ function factory( mu, sigma ) { if ( isnan( p ) || p < 0.0 || p > 1.0 ) { return NaN; } - return A + ( B * erfinv( (2.0*p) - 1.0 ) ); + return A - ( B * erfcinv( 2.0*p ) ); } } diff --git a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/lib/main.js b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/lib/main.js index 653eacf86865..fc3002e485df 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/lib/main.js +++ b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/lib/main.js @@ -20,7 +20,7 @@ // MODULES // -var erfinv = require( '@stdlib/math/base/special/erfinv' ); +var erfcinv = require( '@stdlib/math/base/special/erfcinv' ); var isnan = require( '@stdlib/math/base/assert/is-nan' ); var sqrt = require( '@stdlib/math/base/special/sqrt' ); @@ -30,6 +30,16 @@ var sqrt = require( '@stdlib/math/base/special/sqrt' ); /** * Evaluates the quantile function for a normal distribution with mean `mu` and standard deviation `sigma` at a probability `p`. * +* ## Method +* +* The quantile function is +* +* ```tex +* Q(p) = \mu + \sigma \sqrt{2} \operatorname{erf}^{-1}(2p-1) = \mu - \sigma \sqrt{2} \operatorname{erfc}^{-1}(2p) +* ``` +* +* The function evaluates the second form. In the first, \\(2p-1\\) loses the low-order digits of small \\(p\\) and rounds to \\(-1\\) for \\(p < 2^{-54}\\), which returns \\(-\infty\\). +* * @param {Probability} p - input value * @param {number} mu - mean * @param {NonNegativeNumber} sigma - standard deviation @@ -87,7 +97,7 @@ function quantile( p, mu, sigma ) { } A = mu; B = sigma * sqrt( 2.0 ); - return A + (B * erfinv( (2.0*p) - 1.0 )); + return A - ( B * erfcinv( 2.0*p ) ); } diff --git a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/manifest.json b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/manifest.json index 709351d4c22f..7a058454c73e 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/manifest.json +++ b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/manifest.json @@ -41,7 +41,7 @@ "@stdlib/math/base/napi/ternary", "@stdlib/math/base/assert/is-nan", "@stdlib/math/base/special/sqrt", - "@stdlib/math/base/special/erfinv", + "@stdlib/math/base/special/erfcinv", "@stdlib/constants/float64/sqrt-two" ] }, @@ -59,7 +59,7 @@ "dependencies": [ "@stdlib/math/base/assert/is-nan", "@stdlib/math/base/special/sqrt", - "@stdlib/math/base/special/erfinv", + "@stdlib/math/base/special/erfcinv", "@stdlib/constants/float64/eps", "@stdlib/constants/float64/sqrt-two" ] @@ -78,7 +78,7 @@ "dependencies": [ "@stdlib/math/base/assert/is-nan", "@stdlib/math/base/special/sqrt", - "@stdlib/math/base/special/erfinv", + "@stdlib/math/base/special/erfcinv", "@stdlib/constants/float64/eps", "@stdlib/constants/float64/sqrt-two" ] diff --git a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/src/main.c b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/src/main.c index 039d7d8b12d3..e1917a9883b5 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/src/main.c +++ b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/src/main.c @@ -19,12 +19,22 @@ #include "stdlib/stats/base/dists/normal/quantile.h" #include "stdlib/math/base/assert/is_nan.h" #include "stdlib/math/base/special/sqrt.h" -#include "stdlib/math/base/special/erfinv.h" +#include "stdlib/math/base/special/erfcinv.h" #include "stdlib/constants/float64/sqrt_two.h" /** * Evaluates the quantile function for a normal distribution with mean `mu` and standard deviation `sigma` at a probability `p`. * +* ## Method +* +* The quantile function is +* +* ```tex +* Q(p) = \mu + \sigma \sqrt{2} \operatorname{erf}^{-1}(2p-1) = \mu - \sigma \sqrt{2} \operatorname{erfc}^{-1}(2p) +* ``` +* +* The function evaluates the second form. In the first, `2p-1` loses the low-order digits of small `p` and rounds to `-1` for `p < 2^{-54}`, which returns `-\infty`. +* * @param p probability * @param mu mean * @param sigma standard deviation @@ -48,5 +58,5 @@ double stdlib_base_dists_normal_quantile( const double p, const double mu, const if ( sigma == 0.0 ) { return mu; } - return mu + ( ( sigma * STDLIB_CONSTANT_FLOAT64_SQRT2 ) * stdlib_base_erfinv( ( 2.0 * p ) - 1.0 ) ); + return mu - ( ( sigma * STDLIB_CONSTANT_FLOAT64_SQRT2 ) * stdlib_base_erfcinv( 2.0 * p ) ); } diff --git a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/REQUIRE b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/REQUIRE new file mode 100644 index 000000000000..9d632823d594 --- /dev/null +++ b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/REQUIRE @@ -0,0 +1,3 @@ +Distributions 0.25 +julia 1.5 +JSON 0.21 diff --git a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/lower_tail.json b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/lower_tail.json new file mode 100644 index 000000000000..fdca946f7843 --- /dev/null +++ b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/lower_tail.json @@ -0,0 +1 @@ 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diff --git a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/runner.jl b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/runner.jl new file mode 100644 index 000000000000..b1bc520e2288 --- /dev/null +++ b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/runner.jl @@ -0,0 +1,87 @@ +#!/usr/bin/env julia +# +# @license Apache-2.0 +# +# Copyright (c) 2026 The Stdlib Authors. +# +# Licensed under the Apache License, Version 2.0 (the "License"); +# you may not use this file except in compliance with the License. +# You may obtain a copy of the License at +# +# http://www.apache.org/licenses/LICENSE-2.0 +# +# Unless required by applicable law or agreed to in writing, software +# distributed under the License is distributed on an "AS IS" BASIS, +# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +# See the License for the specific language governing permissions and +# limitations under the License. + +import Distributions: quantile, Normal +import JSON + +""" + gen( p, mu, sigma, name ) + +Generate fixture data and write to file. + +# Arguments + +* `p`: probabilities +* `mu`: mean +* `sigma`: standard deviation +* `name::AbstractString`: output filename + +# Examples + +``` julia +julia> p = rand( 1000 ); +julia> mu = rand( 1000 ); +julia> sigma = rand( 1000 ) .+ 0.1; +julia> gen( p, mu, sigma, "data.json" ); +``` +""" +function gen( p, mu, sigma, name ) + z = Array{Float64}( undef, length( p ) ); + + # Evaluate the quantile function in extended precision, and round the result to double precision: + setprecision( BigFloat, 2048 ) do + for i in eachindex( p ) + z[ i ] = Float64( quantile( Normal( BigFloat( mu[ i ] ), BigFloat( sigma[ i ] ) ), BigFloat( p[ i ] ) ) ); + end + end + + # Store data to be written to file as a collection: + data = Dict([ + ("p", p), + ("mu", mu), + ("sigma", sigma), + ("expected", z) + ]); + + # Based on the script directory, create an output filepath: + filepath = joinpath( dir, name ); + + # Write the data to the output filepath as JSON: + outfile = open( filepath, "w" ); + write( outfile, JSON.json(data) ); + write( outfile, "\n" ); + close( outfile ); +end + +# Get the filename: +file = @__FILE__; + +# Extract the directory in which this file resides: +dir = dirname( file ); + +# Probabilities in the lower tail, down to `1e-250`: +p = exp10.( rand( 500 ) .* -247.0 .- 3.0 ); +mu = ( rand( 500 ) .* 20.0 ) .- 10.0; +sigma = ( rand( 500 ) .* 10.0 ) .+ 0.1; +gen( p, mu, sigma, "lower_tail.json" ); + +# Probabilities in the upper tail, up to the largest double less than `1`: +p = 1.0 .- exp10.( rand( 500 ) .* -13.0 .- 3.0 ); +mu = ( rand( 500 ) .* 20.0 ) .- 10.0; +sigma = ( rand( 500 ) .* 10.0 ) .+ 0.1; +gen( p, mu, sigma, "upper_tail.json" ); diff --git a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/upper_tail.json b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/upper_tail.json new file mode 100644 index 000000000000..01b4a4fc3742 --- /dev/null +++ b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/fixtures/julia/upper_tail.json @@ -0,0 +1 @@ +{"expected":[37.26871708043865,38.65494567162124,14.44378273985925,20.32855012908123,7.256329258047192,34.677625434289354,36.45166353476877,8.49850645418259,49.69066049145515,1.2403841857845848,20.628492288518903,44.7776687527682,11.821466556311828,19.136557568028252,28.802216438245853,71.6784987630563,40.43770147625403,67.08194431430563,20.337541044370003,25.145358613794432,14.295844909781898,6.9448330463554,16.242157506199554,38.36846898215975,63.1974249490286,34.94550354347651,45.2338909857928,23.465020382725783,15.478341264516263,28.26864902866128,73.3642882467251,46.263817653303256,33.990043426724775,50.61648892509158,39.47778642207126,26.559658242994253,46.0121833365533,37.00092376496334,69.4583189793613,21.467592634782466,44.012484837395235,77.75236138297123,79.07218906517005,14.191381636598166,1.1105865265273525,27.071657984317184,35.90860883074639,32.57041668272115,21.790691323540816,9.851743382922376,11.8009054179651,30.7238443507584,39.026871279406066,52.78741156225492,63.464504138040446,73.77289929126468,14.23130272584018,39.80221742472008,48.41136020837753,14.368735927163039,38.631126174734206,52.69745492382477,60.44541343508607,11.741128146528679,65.75275478881309,28.029286243450013,64.71626334420644,32.962858571054234,5.164771758193122,52.59279584920504,9.21162688573674,35.08867463225813,14.499627179087225,4.180394393339259,54.25577266165005,13.252317452312056,47.17789017442809,31.788368297630957,61.87735736394981,11.266961554685224,77.60185740174592,16.777806539932595,44.41624199651777,20.411886384809026,26.671451288196813,44.51328633687367,14.854670172725454,62.77710771590304,0.023328677618117986,24.921781821240216,-1.9128608481142755,50.17200096273613,41.81431811710524,17.53516872431548,-0.8264010667229734,44.159937812336636,15.66249681625617,24.396277679743015,61.61212602626665,12.275760310065174,44.43375116975956,51.48960046105429,10.841668077232656,11.269367521741364,11.340560681946119,32.709196251059716,19.022933933272824,41.71509091919408,50.439912761414504,68.59370199673013,38.52343071423505,-8.396743148368873,42.46484272365947,22.83384657930702,4.55568929038452,14.886581112658698,10.703709474397485,5.511252671393019,74.04449119287051,2.0090133561048207,72.51918036361609,62.4434959419195,57.602472642156364,47.31761489738941,-2.0062584805869585,33.03693652308326,2.5072137969202353,17.97887535080751,-0.5317752698923198,40.939530102555736,41.817649058162814,22.76970542246658,34.575574981356745,29.985391733816655,-4.385268110273592,20.697620079743917,31.003001764880555,51.72570052266011,3.968482158502703,10.711080376464444,46.537078510346426,25.844359193907362,26.92055890348751,65.04615045686556,19.086414493425785,23.98160536901051,48.10076219937759,55.422382857949,11.71065155214465,47.713459919005764,66.46299223740064,38.026952755530914,37.292048555737495,51.90379922753748,4.328805823332949,23.071498611040997,15.552295179977955,18.625204313346014,25.749048153917258,22.80905059836309,-8.309158222576698,56.18264449020362,31.308610835597793,39.56576955448574,31.41749110571452,26.70220000420807,52.239577939806615,23.984582330621958,65.04836405338659,12.171658637970042,20.498106260743047,15.776339537293508,36.44781453815461,16.041062259010165,59.40086421777156,51.067595494863276,47.96851944406354,53.590395477831066,48.68256037055249,14.611040927112017,89.67566239425335,77.67816155167858,22.99417320586,10.379027413915258,40.99176097606826,38.28330670188079,63.3538515262861,27.380861169925794,43.24971090282059,55.26537339754464,18.712878555072688,33.70160059675055,18.62714172574306,6.642519434754541,0.9726050191013709,68.62115219285522,54.93181284760247,17.518345913698663,-0.16349304584667965,29.71455990794207,46.89509288587674,11.838663123784013,28.675359595615667,75.60660475564401,24.08945508068969,13.734918011929814,13.842003055579378,15.201183655679989,55.563114928258486,40.01215604076816,15.386586984812084,10.747122209358173,31.917841868924036,1.1974901761513568,6.878776301918836,-0.5350505132240202,2.082159247466988,75.17395725757787,53.34698580099889,55.981792115753294,44.65558817657992,19.393577618100387,5.821048326814777,70.9522147226323,49.809721794025435,19.910365486329848,46.69886244568006,26.061367524362826,43.397996617901484,10.926565732564757,49.09195941887342,14.701656648261444,45.492021069527254,41.9872155355801,15.74605395033022,13.994609259770376,38.769751300316905,33.17672069660595,49.973074689833666,37.71375632868518,3.0617881839560037,24.40223224800617,47.97327146105323,43.54402295503275,28.524952411588945,57.708860262186306,29.98590667064682,61.40350938541719,40.79859230825801,10.050910966328553,18.774522322530526,31.83813428305844,36.74776528958488,42.65084244137101,24.740856673372193,42.260697047251035,43.356319882122705,0.08944584349634621,17.089885872915392,55.014441326992156,9.076428576088547,16.226310906722162,57.37796522789634,65.61637113080778,64.74370014115294,15.537317157102233,0.06745315183296165,39.943344245085186,55.91285803844361,53.7894488679643,37.184570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diff --git a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/test.factory.js b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/test.factory.js index 2f7d1c9f36cc..80ebb338d3c2 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/test.factory.js +++ b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/test.factory.js @@ -31,6 +31,8 @@ var factory = require( './../lib/factory.js' ); // FIXTURES // var data = require( './fixtures/r/data.json' ); +var lowerTail = require( './fixtures/julia/lower_tail.json' ); +var upperTail = require( './fixtures/julia/upper_tail.json' ); var i; for ( i = 0; i < data.expected.length; i++ ) { if ( data.expected[ i ] === 'Inf' ) { @@ -169,3 +171,45 @@ tape( 'the created function evaluates the quantile for `x` given parameters `mu` } t.end(); }); + +tape( 'the created function evaluates the quantile for small probabilities `p` (lower tail)', function test( t ) { + var expected; + var quantile; + var sigma; + var mu; + var p; + var y; + var i; + + expected = lowerTail.expected; + p = lowerTail.p; + mu = lowerTail.mu; + sigma = lowerTail.sigma; + for ( i = 0; i < p.length; i++ ) { + quantile = factory( mu[i], sigma[i] ); + y = quantile( p[i] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 16 ), true, 'p: '+p[i]+'. y: '+y+'. E: '+expected[i] ); + } + t.end(); +}); + +tape( 'the created function evaluates the quantile for probabilities `p` close to one (upper tail)', function test( t ) { + var expected; + var quantile; + var sigma; + var mu; + var p; + var y; + var i; + + expected = upperTail.expected; + p = upperTail.p; + mu = upperTail.mu; + sigma = upperTail.sigma; + for ( i = 0; i < p.length; i++ ) { + quantile = factory( mu[i], sigma[i] ); + y = quantile( p[i] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 96 ), true, 'p: '+p[i]+'. y: '+y+'. E: '+expected[i] ); + } + t.end(); +}); diff --git a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/test.native.js b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/test.native.js index dcfdfeffe82e..f253a18457ad 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/test.native.js +++ b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/test.native.js @@ -40,6 +40,8 @@ var opts = { // FIXTURES // var data = require( './fixtures/r/data.json' ); +var lowerTail = require( './fixtures/julia/lower_tail.json' ); +var upperTail = require( './fixtures/julia/upper_tail.json' ); var i; for ( i = 0; i < data.expected.length; i++ ) { if ( data.expected[ i ] === 'Inf' ) { @@ -137,3 +139,41 @@ tape( 'the function evaluates the quantile for `x` given parameters `mu` and `si } t.end(); }); + +tape( 'the function evaluates the quantile for small probabilities `p` (lower tail)', opts, function test( t ) { + var expected; + var sigma; + var mu; + var p; + var y; + var i; + + expected = lowerTail.expected; + p = lowerTail.p; + mu = lowerTail.mu; + sigma = lowerTail.sigma; + for ( i = 0; i < p.length; i++ ) { + y = quantile( p[i], mu[i], sigma[i] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 16 ), true, 'p: '+p[i]+'. y: '+y+'. E: '+expected[i] ); + } + t.end(); +}); + +tape( 'the function evaluates the quantile for probabilities `p` close to one (upper tail)', opts, function test( t ) { + var expected; + var sigma; + var mu; + var p; + var y; + var i; + + expected = upperTail.expected; + p = upperTail.p; + mu = upperTail.mu; + sigma = upperTail.sigma; + for ( i = 0; i < p.length; i++ ) { + y = quantile( p[i], mu[i], sigma[i] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 96 ), true, 'p: '+p[i]+'. y: '+y+'. E: '+expected[i] ); + } + t.end(); +}); diff --git a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/test.quantile.js b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/test.quantile.js index 28eac26d5a9f..25ce1b0737ab 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/test.quantile.js +++ b/lib/node_modules/@stdlib/stats/base/dists/normal/quantile/test/test.quantile.js @@ -31,6 +31,8 @@ var quantile = require( './../lib' ); // FIXTURES // var data = require( './fixtures/r/data.json' ); +var lowerTail = require( './fixtures/julia/lower_tail.json' ); +var upperTail = require( './fixtures/julia/upper_tail.json' ); var i; for ( i = 0; i < data.expected.length; i++ ) { if ( data.expected[ i ] === 'Inf' ) { @@ -128,3 +130,41 @@ tape( 'the function evaluates the quantile for `x` given parameters `mu` and `si } t.end(); }); + +tape( 'the function evaluates the quantile for small probabilities `p` (lower tail)', function test( t ) { + var expected; + var sigma; + var mu; + var p; + var y; + var i; + + expected = lowerTail.expected; + p = lowerTail.p; + mu = lowerTail.mu; + sigma = lowerTail.sigma; + for ( i = 0; i < p.length; i++ ) { + y = quantile( p[i], mu[i], sigma[i] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 16 ), true, 'p: '+p[i]+'. y: '+y+'. E: '+expected[i] ); + } + t.end(); +}); + +tape( 'the function evaluates the quantile for probabilities `p` close to one (upper tail)', function test( t ) { + var expected; + var sigma; + var mu; + var p; + var y; + var i; + + expected = upperTail.expected; + p = upperTail.p; + mu = upperTail.mu; + sigma = upperTail.sigma; + for ( i = 0; i < p.length; i++ ) { + y = quantile( p[i], mu[i], sigma[i] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 96 ), true, 'p: '+p[i]+'. y: '+y+'. E: '+expected[i] ); + } + t.end(); +});