feat: add stats/incr/mkurtosis

lib/node_modules/@stdlib/stats/incr/mkurtosis/README.md

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+<!--
+
+@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.
+
+-->
+
+# mkurtosis
+
+> Compute a moving [corrected sample excess kurtosis][sample-excess-kurtosis] incrementally.
+
+<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]
+```
+
+<!-- </equation> -->
+
+Using a univariate normal distribution as the standard of comparison, the [excess kurtosis][sample-excess-kurtosis] is the kurtosis minus `3`.
+
+For a window of `W` 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}{W} \displaystyle\sum_{i=0}^{W-1} (x_i - \bar{x})^4}{\biggl(\frac{1}{W} \displaystyle\sum_{i=0}^{W-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}{W} \displaystyle\sum_{i=0}^{W-1} (x_i - \bar{x})^4}{\biggl(\frac{1}{W} \displaystyle\sum_{i=0}^{W-1} (x_i - \bar{x})^2\biggr)^2}
+```
+
+<!-- </equation> -->
+
+The corrected (unbiased under normality) sample excess kurtosis 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)}
+```
+
+<!-- </equation> -->
+
+where `n` is the number of values currently in the window (grows from 1 to `W` during the fill phase, then stays at `W`).
+
+</section>
+
+<!-- /.intro -->
+
+<section class="usage">
+
+## Usage
+
+```javascript
+var mkurtosis = require( '@stdlib/stats/incr/mkurtosis' );
+```
+
+#### mkurtosis( W )
+
+Returns an accumulator `function` which incrementally computes a moving [corrected sample excess kurtosis][sample-excess-kurtosis] over a sliding window of size `W`.
+
+```javascript
+var accumulator = mkurtosis( 4 );
+```
+
+#### accumulator( \[x] )
+
+If provided an input value `x`, the accumulator function returns an updated moving [corrected sample excess kurtosis][sample-excess-kurtosis]. If not provided an input value `x`, the accumulator function returns the current moving [corrected sample excess kurtosis][sample-excess-kurtosis].
+
+```javascript
+var accumulator = mkurtosis( 4 );
+
+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( 3.0 );
+// returns -5.652200677131425
+
+kurtosis = accumulator();
+// returns -5.652200677131425
+```
+
+</section>
+
+<!-- /.usage -->
+
+<section class="notes">
+
+## Notes
+
+-   The `W` parameter defines the **window size**: only the `W` most recently provided values are used when computing the kurtosis. During the initial fill phase (fewer than `W` values have been provided), the kurtosis is computed over however many values are available.
+
+-   The corrected sample excess kurtosis is only defined for **4 or more** values. If fewer than 4 values have been provided, the accumulator returns `null`.
+
+-   **Algorithm** — The implementation avoids a full O(W) recomputation at each step by maintaining running central moment accumulators `M2`, `M3`, and `M4` (à la Welford). When the window is full, adding a new value `x` and discarding the oldest value `xo` is handled in two O(1) operations:
+
+    1.  **Downdate** — algebraically reverse the Welford update that originally incorporated `xo`, recovering the moments as though `xo` were never in the window.
+    2.  **Update** — apply the standard Welford update for `x`.
+
+    The downdate formulas, derived by inverting the Welford recurrence, are:
+
+    ```text
+    mean_new = (W·mean - xo) / (W - 1)
+    delta    = xo - mean_new
+    deltaN   = delta / W
+    term1    = delta · deltaN · (W - 1)
+    M2_new   = M2 - term1
+    M3_new   = M3 - term1·deltaN·(W-2) + 3·deltaN·M2_new
+    M4_new   = M4 - term1·deltaN²·(W²-3W+3) - 6·deltaN²·M2_new + 4·deltaN·M3_new
+    ```
+
+-   **NaN handling** — Input values are **not** type-checked. If provided `NaN` or a value which, when used in computations, results in `NaN`, the accumulated value is `NaN` for, at most, `W` future invocations. Once the `NaN` value leaves the sliding window, the moments are **recomputed from scratch** from the circular buffer and the accumulator recovers. This differs from `@stdlib/stats/incr/kurtosis`, where a `NaN` permanently corrupts all future results.
+
+</section>
+
+<!-- /.notes -->
+
+<section class="examples">
+
+## Examples
+
+<!-- eslint no-undef: "error" -->
+
+```javascript
+var randu = require( '@stdlib/random/base/randu' );
+var mkurtosis = require( '@stdlib/stats/incr/mkurtosis' );
+
+var accumulator;
+var v;
+var i;
+
+// Initialize a moving kurtosis accumulator with a window size of 20:
+accumulator = mkurtosis( 20 );
+
+// For each simulated datum, update the moving corrected sample excess kurtosis:
+for ( i = 0; i < 100; i++ ) {
+    v = randu() * 100.0;
+    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].
+-   Pébay, Philippe. 2008. "Formulas for Robust, One-Pass Parallel Computation of Covariances and Arbitrary-Order Statistical Moments." _Technical Report_ SAND2008-6212. Sandia National Laboratories. <https://prod-ng.sandia.gov/techlib-noauth/access-control.cgi/2008/086212.pdf>.
+-   Welford, B. P. 1962. "Note on a Method for Calculating Corrected Sums of Squares and Products." _Technometrics_ 4 (3): 419–20. doi:[10.1080/00401706.1962.10490022][@welford:1962].
+
+[@joanes:1998]: http://dx.doi.org/10.1111/1467-9884.00122
+[@welford:1962]: https://doi.org/10.1080/00401706.1962.10490022
+
+</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/kurtosis`][@stdlib/stats/incr/kurtosis]</span><span class="delimiter">: </span><span class="description">compute a corrected sample excess kurtosis incrementally (all data, no window).</span>
+-   <span class="package-name">[`@stdlib/stats/incr/mskewness`][@stdlib/stats/incr/mskewness]</span><span class="delimiter">: </span><span class="description">compute a moving corrected sample skewness incrementally.</span>
+-   <span class="package-name">[`@stdlib/stats/incr/mmean`][@stdlib/stats/incr/mmean]</span><span class="delimiter">: </span><span class="description">compute a moving arithmetic mean incrementally.</span>
+-   <span class="package-name">[`@stdlib/stats/incr/mvariance`][@stdlib/stats/incr/mvariance]</span><span class="delimiter">: </span><span class="description">compute a moving unbiased sample variance incrementally.</span>
+-   <span class="package-name">[`@stdlib/stats/incr/msummary`][@stdlib/stats/incr/msummary]</span><span class="delimiter">: </span><span class="description">compute a moving statistical summary 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
+
+<!-- <related-links> -->
+
+[@stdlib/stats/incr/kurtosis]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/stats/incr/kurtosis
+
+[@stdlib/stats/incr/mskewness]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/stats/incr/mskewness
+
+[@stdlib/stats/incr/mmean]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/stats/incr/mmean
+
+[@stdlib/stats/incr/mvariance]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/stats/incr/mvariance
+
+[@stdlib/stats/incr/msummary]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/stats/incr/msummary
+
+<!-- </related-links> -->
+
+</section>
+
+<!-- /.links -->

lib/node_modules/@stdlib/stats/incr/mkurtosis/benchmark/benchmark.js

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+/**
+* @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.
+*/
+
+'use strict';
+
+// MODULES //
+
+var bench = require( '@stdlib/bench' );
+var randu = require( '@stdlib/random/base/randu' );
+var format = require( '@stdlib/string/format' );
+var pkg = require( './../package.json' ).name;
+var mkurtosis = require( './../lib' );
+
+
+// MAIN //
+
+bench( pkg, function benchmark( b ) {
+	var f;
+	var i;
+	b.tic();
+	for ( i = 0; i < b.iterations; i++ ) {
+		f = mkurtosis( 20 );
+		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( format( '%s::accumulator,window=20', pkg ), function benchmark( b ) {
+	var acc;
+	var v;
+	var i;
+
+	acc = mkurtosis( 20 );
+
+	b.tic();
+	for ( i = 0; i < b.iterations; i++ ) {
+		v = acc( randu() );
+		if ( v !== v && v !== null ) {
+			b.fail( 'unexpected NaN' );
+		}
+	}
+	b.toc();
+	if ( v !== v && v !== null ) {
+		b.fail( 'unexpected NaN' );
+	}
+	b.pass( 'benchmark finished' );
+	b.end();
+});
+
+bench( format( '%s::accumulator,window=100', pkg ), function benchmark( b ) {
+	var acc;
+	var v;
+	var i;
+
+	acc = mkurtosis( 100 );
+
+	b.tic();
+	for ( i = 0; i < b.iterations; i++ ) {
+		v = acc( randu() );
+		if ( v !== v && v !== null ) {
+			b.fail( 'unexpected NaN' );
+		}
+	}
+	b.toc();
+	if ( v !== v && v !== null ) {
+		b.fail( 'unexpected NaN' );
+	}
+	b.pass( 'benchmark finished' );
+	b.end();
+});

lib/node_modules/@stdlib/stats/incr/mkurtosis/docs/repl.txt

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+
+{{alias}}( W )
+    Returns an accumulator function which incrementally computes a moving
+    corrected sample excess kurtosis.
+
+    The `W` parameter defines the number of values over which to compute the
+    moving corrected sample excess kurtosis.
+
+    If provided a value, the accumulator function returns an updated moving
+    corrected sample excess kurtosis. If not provided a value, the accumulator
+    function returns the current moving corrected sample excess kurtosis.
+
+    As `W` values are needed to fill the window buffer, the first `W-1`
+    returned values are calculated from smaller sample sizes. Until the window
+    is full, each returned value is the kurtosis of all values provided thus
+    far.
+
+    For fewer than 4 values, the accumulator returns `null`.
+
+    If provided `NaN` or a value which, when used in computations, results in
+    `NaN`, the accumulated value is `NaN` for, at most, `W` future invocations.
+    Once the `NaN` value leaves the sliding window, the accumulator recovers.
+
+    Parameters
+    ----------
+    W: integer
+        Window size. Must be a positive integer.
+
+    Returns
+    -------
+    acc: Function
+        Accumulator function.
+
+    Examples
+    --------
+    > var accumulator = {{alias}}( 4 );
+    > var v = accumulator( 2.0 )
+    null
+    > v = accumulator( 2.0 )
+    null
+    > v = accumulator( -4.0 )
+    null
+    > v = accumulator( -4.0 )
+    -6.0
+    > v = accumulator( 3.0 )
+    -5.652200677131425
+
+    See Also
+    --------
+