stdlib-js · Girish-Garg · Mar 17, 2025 · Mar 17, 2025 · Apr 29, 2025 · Apr 29, 2025
diff --git a/lib/node_modules/@stdlib/stats/incr/nankurtosis/README.md b/lib/node_modules/@stdlib/stats/incr/nankurtosis/README.md
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+<!--
+
+@license Apache-2.0
+
+Copyright (c) 2025 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.
+
+-->
+
+# incrnankurtosis
+
+> Compute a [corrected sample excess kurtosis][sample-excess-kurtosis] incrementally, ignoring `NaN` values.
+
+<section class="intro">
+
+The [kurtosis][sample-excess-kurtosis] for a random variable `X` is defined as
+
+<!-- <equation class="equation" label="eq:kurtosis" align="center" raw="\operatorname{Kurtosis}[X] = \mathrm{E}\biggl[ \biggl( \frac{X - \mu}{\sigma} \biggr)^4 \biggr]" alt="Equation for the kurtosis."> -->
+
+```math
+\mathop{\mathrm{Kurtosis}}[X] = \mathrm{E}\biggl[ \biggl( \frac{X - \mu}{\sigma} \biggr)^4 \biggr]
+```
+
+<!-- <div class="equation" align="center" data-raw-text="\operatorname{Kurtosis}[X] = \mathrm{E}\biggl[ \biggl( \frac{X - \mu}{\sigma} \biggr)^4 \biggr]" data-equation="eq:kurtosis">
+    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@49d8cabda84033d55d7b8069f19ee3dd8b8d1496/lib/node_modules/@stdlib/stats/incr/nankurtosis/docs/img/equation_kurtosis.svg" alt="Equation for the kurtosis.">
+    <br>
+</div> -->
+
+<!-- </equation> -->
+
+Using a univariate normal distribution as the standard of comparison, the [excess kurtosis][sample-excess-kurtosis] is the kurtosis minus `3`.
+
+For a sample of `n` values, the [sample excess kurtosis][sample-excess-kurtosis] is
+
+<!-- <equation class="equation" label="eq:sample_excess_kurtosis" align="center" raw="g_2 = \frac{m_4}{m_2^2} - 3 = \frac{\frac{1}{n} \displaystyle\sum_{i=0}^{n-1} (x_i - \bar{x})^4}{\biggl(\frac{1}{n} \displaystyle\sum_{i=0}^{n-1} (x_i - \bar{x})^2\biggr)^2}" alt="Equation for the sample excess kurtosis."> -->
+
+```math
+g_2 = \frac{m_4}{m_2^2} - 3 = \frac{\frac{1}{n} \displaystyle\sum_{i=0}^{n-1} (x_i - \bar{x})^4}{\biggl(\frac{1}{n} \displaystyle\sum_{i=0}^{n-1} (x_i - \bar{x})^2\biggr)^2}
+```
+
+<!-- <div class="equation" align="center" data-raw-text="g_2 = \frac{m_4}{m_2^2} - 3 = \frac{\frac{1}{n} \displaystyle\sum_{i=0}^{n-1} (x_i - \bar{x})^4}{\biggl(\frac{1}{n} \displaystyle\sum_{i=0}^{n-1} (x_i - \bar{x})^2\biggr)^2}" data-equation="eq:sample_excess_kurtosis">
+    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@49d8cabda84033d55d7b8069f19ee3dd8b8d1496/lib/node_modules/@stdlib/stats/incr/nankurtosis/docs/img/equation_sample_excess_kurtosis.svg" alt="Equation for the sample excess kurtosis.">
+    <br>
+</div> -->
+
+<!-- </equation> -->
+
+where `m_4` is the sample fourth central moment and `m_2` is the sample second central moment.
+
+The previous equation is, however, a biased estimator of the population excess kurtosis. An alternative estimator which is unbiased under normality is
+
+<!-- <equation class="equation" label="eq:corrected_sample_excess_kurtosis" align="center" raw="G_2 = \frac{(n+1)n}{(n-1)(n-2)(n-3)} \frac{\displaystyle\sum_{i=0}^{n-1} (x_i - \bar{x})^4}{\biggl(\displaystyle\sum_{i=0}^{n-1} (x_i - \bar{x})^2\biggr)^2} - 3 \frac{(n-1)^2}{(n-2)(n-3)}" alt="Equation for the corrected sample excess kurtosis."> -->
+
+```math
+G_2 = \frac{(n+1)n}{(n-1)(n-2)(n-3)} \frac{\displaystyle\sum_{i=0}^{n-1} (x_i - \bar{x})^4}{\biggl(\displaystyle\sum_{i=0}^{n-1} (x_i - \bar{x})^2\biggr)^2} - 3 \frac{(n-1)^2}{(n-2)(n-3)}
+```
+
+<!-- <div class="equation" align="center" data-raw-text="G_2 = \frac{(n+1)n}{(n-1)(n-2)(n-3)} \frac{\displaystyle\sum_{i=0}^{n-1} (x_i - \bar{x})^4}{\biggl(\displaystyle\sum_{i=0}^{n-1} (x_i - \bar{x})^2\biggr)^2} - 3 \frac{(n-1)^2}{(n-2)(n-3)}" data-equation="eq:corrected_sample_excess_kurtosis">
+    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@49d8cabda84033d55d7b8069f19ee3dd8b8d1496/lib/node_modules/@stdlib/stats/incr/nankurtosis/docs/img/equation_corrected_sample_excess_kurtosis.svg" alt="Equation for the corrected sample excess kurtosis.">
+    <br>
+</div> -->
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
+<section class="usage">
+
+## Usage
+
+```javascript
+var incrnankurtosis = require( '@stdlib/stats/incr/nankurtosis' );
+```
+
+#### incrnankurtosis()
+
+Returns an accumulator function which incrementally computes a [corrected sample excess kurtosis][sample-excess-kurtosis], ignoring `NaN` values.
+
+```javascript
+var accumulator = incrnankurtosis();
+```
+
+#### accumulator( \[x] )
+
+If provided an input value `x`, the accumulator function returns an updated [corrected sample excess kurtosis][sample-excess-kurtosis]. If not provided an input value `x`, the accumulator function returns the current [corrected sample excess kurtosis][sample-excess-kurtosis].
+
+```javascript
+var accumulator = incrnankurtosis();
+
+var kurtosis = accumulator( 2.0 );
+// returns null
+
+kurtosis = accumulator( 2.0 );
+// returns null
+
+kurtosis = accumulator( -4.0 );
+// returns null
+
+kurtosis = accumulator( -4.0 );
+// returns -6.0
+
+kurtosis = accumulator( NaN );
+// returns -6.0
+```
+
+</section>
+
+<!-- /.usage -->
+
+<section class="notes">
+
+## Notes
+
+-   Input values are type checked. If non-numaric input are possible, you are advised to type check and handle accordingly **before** passing the value to the accumulator function.
+
+</section>
+
+<!-- /.notes -->
+
+<section class="examples">
+
+## Examples
+
+<!-- eslint no-undef: "error" -->
+
+```javascript
+var randu = require( '@stdlib/random/base/randu' );
+var incrnankurtosis = require( '@stdlib/stats/incr/nankurtosis' );
+
+var accumulator;
+var kurtosis;
+var v;
+var i;
+
+// Initialize an accumulator:
+accumulator = incrnankurtosis();
+
+// For each simulated datum, update the corrected sample excess kurtosis...
+for ( i = 0; i < 100; i++ ) {
+    v = (randu() < 0.2 ) ? NaN : randu() * 100.0; // 20% chance of NaN
+    kurtosis = accumulator( v );
+}
+console.log( accumulator() );
+```
+
+</section>
+
+<!-- /.examples -->
+
+* * *
+
+<section class="references">
+
+## References
+
+-   Joanes, D. N., and C. A. Gill. 1998. "Comparing measures of sample skewness and kurtosis." _Journal of the Royal Statistical Society: Series D (The Statistician)_ 47 (1). Blackwell Publishers Ltd: 183–89. doi:[10.1111/1467-9884.00122][@joanes:1998].
+
+</section>
+
+<!-- /.references -->
+
+<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->
+
+<section class="related">
+
+* * *
+
+## See Also
+
+-   <span class="package-name">[`@stdlib/stats/incr/mean`][@stdlib/stats/incr/mean]</span><span class="delimiter">: </span><span class="description">compute an arithmetic mean incrementally.</span>
+-   <span class="package-name">[`@stdlib/stats/incr/skewness`][@stdlib/stats/incr/skewness]</span><span class="delimiter">: </span><span class="description">compute a corrected sample skewness incrementally.</span>
+-   <span class="package-name">[`@stdlib/stats/incr/stdev`][@stdlib/stats/incr/stdev]</span><span class="delimiter">: </span><span class="description">compute a corrected sample standard deviation incrementally.</span>
+-   <span class="package-name">[`@stdlib/stats/incr/summary`][@stdlib/stats/incr/summary]</span><span class="delimiter">: </span><span class="description">compute a statistical summary incrementally.</span>
+-   <span class="package-name">[`@stdlib/stats/incr/variance`][@stdlib/stats/incr/variance]</span><span class="delimiter">: </span><span class="description">compute an unbiased sample variance incrementally.</span>
+
+</section>
+
+<!-- /.related -->
+
+<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
+
+<section class="links">
+
+[sample-excess-kurtosis]: https://en.wikipedia.org/wiki/Kurtosis
+
+[@joanes:1998]: http://onlinelibrary.wiley.com/doi/10.1111/1467-9884.00122/
+
+<!-- <related-links> -->
+
+[@stdlib/stats/incr/mean]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/stats/incr/mean
+
+[@stdlib/stats/incr/skewness]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/stats/incr/skewness
+
+[@stdlib/stats/incr/stdev]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/stats/incr/stdev
+
+[@stdlib/stats/incr/summary]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/stats/incr/summary
+
+[@stdlib/stats/incr/variance]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/stats/incr/variance
+
+<!-- </related-links> -->
+
+</section>
+
+<!-- /.links -->
diff --git a/lib/node_modules/@stdlib/stats/incr/nankurtosis/benchmark/benchmark.js b/lib/node_modules/@stdlib/stats/incr/nankurtosis/benchmark/benchmark.js
@@ -0,0 +1,68 @@
+/**
+* @license Apache-2.0
+*
+* Copyright (c) 2025 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.
+*/
+
+'use strict';
+
+// MODULES //
+
+var bench = require( '@stdlib/bench' );
+var pkg = require( './../package.json' ).name;
+var incrnankurtosis = require( './../lib' );
+
+
+// MAIN //
+
+bench( pkg, function benchmark( b ) {
+	var f;
+	var i;
+	b.tic();
+	for ( i = 0; i < b.iterations; i++ ) {
+		f = incrnankurtosis();
+		if ( typeof f !== 'function' ) {
+			b.fail( 'should return a function' );
+		}
+	}
+	b.toc();
+	if ( typeof f !== 'function' ) {
+		b.fail( 'should return a function' );
+	}
+	b.pass( 'benchmark finished' );
+	b.end();
+});
+
+bench( pkg+'::accumulator', function benchmark( b ) {
+	var acc;
+	var v;
+	var i;
+
+	acc = incrnankurtosis();
+
+	b.tic();
+	for ( i = 0; i < b.iterations; i++ ) {
+		v = acc( i );
+		if ( v !== v ) {
+			b.fail( 'should not return NaN' );
+		}
+	}
+	b.toc();
+	if ( v !== v ) {
+		b.fail( 'should not return NaN' );
+	}
+	b.pass( 'benchmark finished' );
+	b.end();
+});