diff --git a/lib/node_modules/@stdlib/stats/incr/mkurtosis/README.md b/lib/node_modules/@stdlib/stats/incr/mkurtosis/README.md
new file mode 100644
index 000000000000..21158ccf1f5f
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/mkurtosis/README.md
@@ -0,0 +1,228 @@
+
+
+# mkurtosis
+
+> Compute a moving [corrected sample excess kurtosis][sample-excess-kurtosis] incrementally.
+
+
+
+The [kurtosis][sample-excess-kurtosis] for a random variable `X` is defined as
+
+
+
+```math
+\mathop{\mathrm{Kurtosis}}[X] = \mathrm{E}\biggl[ \biggl( \frac{X - \mu}{\sigma} \biggr)^4 \biggr]
+```
+
+
+
+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
+
+
+
+```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}
+```
+
+
+
+The corrected (unbiased under normality) sample excess kurtosis is
+
+
+
+```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)}
+```
+
+
+
+where `n` is the number of values currently in the window (grows from 1 to `W` during the fill phase, then stays at `W`).
+
+
+
+
+
+
+
+## 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
+```
+
+
+
+
+
+
+
+## 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.
+
+
+
+
+
+
+
+## Examples
+
+
+
+```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() );
+```
+
+
+
+
+
+* * *
+
+
+
+## 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. .
+- 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
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+[sample-excess-kurtosis]: https://en.wikipedia.org/wiki/Kurtosis
+
+
+
+[@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
+
+
+
+
+
+
diff --git a/lib/node_modules/@stdlib/stats/incr/mkurtosis/benchmark/benchmark.js b/lib/node_modules/@stdlib/stats/incr/mkurtosis/benchmark/benchmark.js
new file mode 100644
index 000000000000..e27e181f42a7
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/mkurtosis/benchmark/benchmark.js
@@ -0,0 +1,92 @@
+/**
+* @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();
+});
diff --git a/lib/node_modules/@stdlib/stats/incr/mkurtosis/docs/repl.txt b/lib/node_modules/@stdlib/stats/incr/mkurtosis/docs/repl.txt
new file mode 100644
index 000000000000..67e88ed04fec
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/mkurtosis/docs/repl.txt
@@ -0,0 +1,50 @@
+
+{{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
+ --------
+
diff --git a/lib/node_modules/@stdlib/stats/incr/mkurtosis/docs/types/index.d.ts b/lib/node_modules/@stdlib/stats/incr/mkurtosis/docs/types/index.d.ts
new file mode 100644
index 000000000000..b56a6ba2aeaa
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/mkurtosis/docs/types/index.d.ts
@@ -0,0 +1,74 @@
+/*
+* @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.
+*/
+
+// TypeScript Version: 4.1
+
+///
+
+/**
+* If provided a value, returns an updated moving corrected sample excess
+* kurtosis; otherwise, returns the current moving corrected sample excess
+* kurtosis.
+*
+* ## Notes
+*
+* - 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 window, the accumulator
+* recovers.
+*
+* @param x - value
+* @returns moving corrected sample excess kurtosis or null
+*/
+type accumulator = ( x?: number ) => number | null;
+
+/**
+* Returns an accumulator function which incrementally computes a moving
+* corrected sample excess kurtosis.
+*
+* @param W - window size
+* @throws must provide a positive integer
+* @returns accumulator function
+*
+* @example
+* var accumulator = mkurtosis( 4 );
+*
+* var kurtosis = accumulator();
+* // returns null
+*
+* 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
+*/
+declare function mkurtosis( W: number ): accumulator;
+
+
+// EXPORTS //
+
+export = mkurtosis;
diff --git a/lib/node_modules/@stdlib/stats/incr/mkurtosis/docs/types/test.ts b/lib/node_modules/@stdlib/stats/incr/mkurtosis/docs/types/test.ts
new file mode 100644
index 000000000000..4b163c97ed58
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/mkurtosis/docs/types/test.ts
@@ -0,0 +1,65 @@
+/*
+* @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 mkurtosis = require( './index' );
+
+
+// TESTS //
+
+// The function returns an accumulator function...
+{
+ mkurtosis( 4 ); // $ExpectType accumulator
+}
+
+// The compiler throws an error if the function is provided a non-number argument...
+{
+ mkurtosis( '5' ); // $ExpectError
+ mkurtosis( true ); // $ExpectError
+ mkurtosis( false ); // $ExpectError
+ mkurtosis( null ); // $ExpectError
+ mkurtosis( undefined ); // $ExpectError
+ mkurtosis( [] ); // $ExpectError
+ mkurtosis( {} ); // $ExpectError
+ mkurtosis( ( x: number ): number => x ); // $ExpectError
+}
+
+// The compiler throws an error if the function is not provided any arguments...
+{
+ mkurtosis(); // $ExpectError
+}
+
+// The function returns an accumulator which returns a number or null...
+{
+ const acc = mkurtosis( 4 );
+
+ acc(); // $ExpectType number | null
+ acc( 3.14 ); // $ExpectType number | null
+}
+
+// The compiler throws an error if the accumulator is provided invalid arguments...
+{
+ const acc = mkurtosis( 4 );
+
+ acc( '5' ); // $ExpectError
+ acc( true ); // $ExpectError
+ acc( false ); // $ExpectError
+ acc( null ); // $ExpectError
+ acc( [] ); // $ExpectError
+ acc( {} ); // $ExpectError
+ acc( ( x: number ): number => x ); // $ExpectError
+}
diff --git a/lib/node_modules/@stdlib/stats/incr/mkurtosis/examples/index.js b/lib/node_modules/@stdlib/stats/incr/mkurtosis/examples/index.js
new file mode 100644
index 000000000000..6e14f1effa97
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/mkurtosis/examples/index.js
@@ -0,0 +1,43 @@
+/**
+* @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';
+
+var randu = require( '@stdlib/random/base/randu' );
+var mkurtosis = require( './../lib' );
+
+var accumulator;
+var kurtosis;
+var v;
+var i;
+
+// Initialize a moving kurtosis accumulator with a window of 20:
+accumulator = mkurtosis( 20 );
+
+// For each simulated datum, update the moving corrected sample excess kurtosis:
+console.log( '\nValue\tKurtosis\n' );
+for ( i = 0; i < 100; i++ ) {
+ v = randu() * 100.0;
+ kurtosis = accumulator( v );
+ if ( kurtosis === null ) {
+ console.log( '%d\t%s', v.toFixed( 4 ), kurtosis );
+ } else {
+ console.log( '%d\t%d', v.toFixed( 4 ), kurtosis.toFixed( 4 ) );
+ }
+}
+console.log( '\nFinal moving kurtosis: %d\n', accumulator() );
diff --git a/lib/node_modules/@stdlib/stats/incr/mkurtosis/lib/index.js b/lib/node_modules/@stdlib/stats/incr/mkurtosis/lib/index.js
new file mode 100644
index 000000000000..c4b689385e93
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/mkurtosis/lib/index.js
@@ -0,0 +1,54 @@
+/**
+* @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';
+
+/**
+* Compute a moving corrected sample excess kurtosis incrementally.
+*
+* @module @stdlib/stats/incr/mkurtosis
+*
+* @example
+* var mkurtosis = require( '@stdlib/stats/incr/mkurtosis' );
+*
+* 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
+*/
+
+// MODULES //
+
+var main = require( './main.js' );
+
+
+// EXPORTS //
+
+module.exports = main;
diff --git a/lib/node_modules/@stdlib/stats/incr/mkurtosis/lib/main.js b/lib/node_modules/@stdlib/stats/incr/mkurtosis/lib/main.js
new file mode 100644
index 000000000000..de7670a4a10e
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/mkurtosis/lib/main.js
@@ -0,0 +1,312 @@
+/**
+* @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 isnan = require( '@stdlib/math/base/assert/is-nan' );
+var isPositiveInteger = require( '@stdlib/assert/is-positive-integer' ).isPrimitive;
+var Float64Array = require( '@stdlib/array/float64' );
+var format = require( '@stdlib/string/format' );
+
+
+// MAIN //
+
+/**
+* Returns an accumulator function which incrementally computes a moving corrected sample excess kurtosis.
+*
+* ## Method
+*
+* The algorithm maintains running central moment accumulators `M2`, `M3`,
+* and `M4` using Welford's online algorithm. When the window is full, each
+* new datum triggers an O(1) downdate (removing the oldest value) followed
+* by an O(1) update (adding the newest value).
+*
+* ### Welford update (adding value `x`, going from `n-1` to `n` values)
+*
+* ```text
+* delta = x - mean_prev
+* deltaN = delta / n
+* term1 = delta * deltaN * (n - 1)
+* M4 += term1 * deltaN^2 * (n^2 - 3n + 3) + 6*deltaN^2*M2 - 4*deltaN*M3
+* M3 += term1 * deltaN * (n - 2) - 3*deltaN*M2
+* M2 += term1
+* mean += deltaN
+* ```
+*
+* ### Welford downdate (removing value `xo`, going from `n` to `n-1` values)
+*
+* ```text
+* mean_new = (n*mean - xo) / (n - 1)
+* delta = xo - mean_new
+* deltaN = delta / n
+* term1 = delta * deltaN * (n - 1)
+* M2_new = M2 - term1
+* M3_new = M3 - term1*deltaN*(n-2) + 3*deltaN*M2_new
+* M4_new = M4 - term1*deltaN^2*(n^2-3n+3) - 6*deltaN^2*M2_new + 4*deltaN*M3_new
+* ```
+*
+* The corrected sample excess kurtosis is
+*
+* ```text
+* G_2 = (n-1)/((n-2)*(n-3)) * ((n+1)*g2 + 6)
+* ```
+*
+* where `g2 = n*M4/M2^2 - 3`.
+*
+* ## Notes
+*
+* - The kurtosis is only defined for a window containing at least 4 values.
+* For fewer values the accumulator returns `null`.
+*
+* - Input values are **not** type-checked. If a `NaN` enters the window,
+* the accumulator returns `NaN` until the `NaN` leaves the window, at
+* which point the central moments are recomputed from the circular buffer.
+*
+* ## 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].
+*
+* [@joanes:1998]: http://dx.doi.org/10.1111/1467-9884.00122
+*
+* @param {PositiveInteger} W - window size
+* @throws {TypeError} must provide a positive integer
+* @returns {Function} accumulator function
+*
+* @example
+* var accumulator = mkurtosis( 4 );
+*
+* var kurtosis = accumulator();
+* // returns null
+*
+* 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
+*/
+function mkurtosis( W ) {
+ var deltaN2;
+ var deltaN;
+ var delta;
+ var term1;
+ var mean;
+ var nans;
+ var buf;
+ var M2;
+ var M3;
+ var M4;
+ var g2;
+ var N;
+ var n;
+ var i;
+
+ if ( !isPositiveInteger( W ) ) {
+ throw new TypeError( format( 'invalid argument. Must provide a positive integer. Value: `%s`.', W ) );
+ }
+
+ buf = new Float64Array( W );
+ mean = 0.0;
+ M2 = 0.0;
+ M3 = 0.0;
+ M4 = 0.0;
+ N = 0;
+ i = -1;
+ nans = 0;
+
+ return accumulator;
+
+ /**
+ * If provided a value, returns an updated moving corrected sample excess kurtosis; otherwise, returns the current moving corrected sample excess kurtosis.
+ *
+ * @private
+ * @param {number} [x] - new value
+ * @returns {(number|null)} corrected sample excess kurtosis or null
+ */
+ function accumulator( x ) {
+ var xo;
+
+ if ( arguments.length === 0 ) {
+ if ( nans > 0 ) {
+ return NaN;
+ }
+ n = ( N < W ) ? N : W;
+ if ( n < 4 ) {
+ return null;
+ }
+ g2 = ( n * M4 / ( M2 * M2 ) ) - 3.0;
+ return ( (n-1) / ( (n-2)*(n-3) ) ) * ( ( (n+1)*g2 ) + 6.0 );
+ }
+
+ // Advance the circular buffer index:
+ i = ( i + 1 ) % W;
+
+ if ( N < W ) {
+ // Fill phase: window is not yet full.
+ buf[ i ] = x;
+ N += 1;
+ if ( isnan( x ) ) {
+ nans += 1;
+ }
+ if ( nans > 0 ) {
+ return NaN;
+ }
+
+ // Welford update (add x, going from N-1 to N):
+ n = N;
+ delta = x - mean;
+ deltaN = delta / n;
+ deltaN2 = deltaN * deltaN;
+ term1 = delta * deltaN * ( n - 1 );
+
+ M4 += term1 * deltaN2 * ( ( n*n ) - ( 3*n ) + 3 );
+ M4 += 6.0 * deltaN2 * M2;
+ M4 -= 4.0 * deltaN * M3;
+
+ M3 += term1 * deltaN * ( n - 2 );
+ M3 -= 3.0 * deltaN * M2;
+
+ M2 += term1;
+ mean += deltaN;
+
+ if ( n < 4 ) {
+ return null;
+ }
+ g2 = ( n * M4 / ( M2 * M2 ) ) - 3.0;
+ return ( (n-1) / ( (n-2)*(n-3) ) ) * ( ( (n+1)*g2 ) + 6.0 );
+ }
+
+ // Slide phase: window is full (N >= W).
+ xo = buf[ i ];
+ buf[ i ] = x;
+
+ // Track NaN count changes:
+ if ( isnan( xo ) ) {
+ nans -= 1;
+ }
+ if ( isnan( x ) ) {
+ nans += 1;
+ }
+
+ // If a NaN just left the window and none remain, recompute from scratch
+ // Because NaN corrupted the running moment state:
+ if ( nans === 0 && isnan( xo ) ) {
+ recompute();
+ } else if ( nans > 0 ) {
+ // NaN still present or just entered; skip moment update:
+ return NaN;
+ } else {
+ // Normal case: downdate xo, then update x.
+ n = W;
+
+ // Downdate: remove xo from (n, mean, M2, M3, M4).
+ mean = ( ( n * mean ) - xo ) / ( n - 1 );
+
+ // Recover delta/deltaN as they were when xo was originally added:
+ delta = xo - mean;
+ deltaN = delta / n;
+ deltaN2 = deltaN * deltaN;
+ term1 = delta * deltaN * ( n - 1 );
+
+ M2 -= term1;
+ M3 -= term1 * deltaN * ( n - 2 );
+ M3 += 3.0 * deltaN * M2;
+ M4 -= term1 * deltaN2 * ( ( n*n ) - ( 3*n ) + 3 );
+ M4 -= 6.0 * deltaN2 * M2;
+ M4 += 4.0 * deltaN * M3;
+
+ // Update: add x to (n-1) values to recover a full window of n:
+ delta = x - mean;
+ deltaN = delta / n;
+ deltaN2 = deltaN * deltaN;
+ term1 = delta * deltaN * ( n - 1 );
+
+ M4 += term1 * deltaN2 * ( ( n*n ) - ( 3*n ) + 3 );
+ M4 += 6.0 * deltaN2 * M2;
+ M4 -= 4.0 * deltaN * M3;
+
+ M3 += term1 * deltaN * ( n - 2 );
+ M3 -= 3.0 * deltaN * M2;
+
+ M2 += term1;
+ mean += deltaN;
+ }
+
+ n = W;
+ if ( n < 4 ) {
+ return null;
+ }
+ g2 = ( n * M4 / ( M2 * M2 ) ) - 3.0;
+ return ( (n-1) / ( (n-2)*(n-3) ) ) * ( ( (n+1)*g2 ) + 6.0 );
+ }
+
+ /**
+ * Recomputes central moments M2, M3, M4 and the mean from the circular buffer.
+ *
+ * @private
+ */
+ function recompute() {
+ var v;
+ var k;
+
+ mean = 0.0;
+ M2 = 0.0;
+ M3 = 0.0;
+ M4 = 0.0;
+
+ // Iterate over buffer from oldest (index (i+1)%W) to newest (index i):
+ for ( k = 1; k <= W; k++ ) {
+ v = buf[ ( i + k ) % W ];
+
+ // Welford update for position k (1-indexed):
+ n = k;
+ delta = v - mean;
+ deltaN = delta / n;
+ deltaN2 = deltaN * deltaN;
+ term1 = delta * deltaN * ( n - 1 );
+
+ M4 += term1 * deltaN2 * ( ( n*n ) - ( 3*n ) + 3 );
+ M4 += 6.0 * deltaN2 * M2;
+ M4 -= 4.0 * deltaN * M3;
+
+ M3 += term1 * deltaN * ( n - 2 );
+ M3 -= 3.0 * deltaN * M2;
+
+ M2 += term1;
+ mean += deltaN;
+ }
+ }
+}
+
+
+// EXPORTS //
+
+module.exports = mkurtosis;
diff --git a/lib/node_modules/@stdlib/stats/incr/mkurtosis/package.json b/lib/node_modules/@stdlib/stats/incr/mkurtosis/package.json
new file mode 100644
index 000000000000..175032dd67ef
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/mkurtosis/package.json
@@ -0,0 +1,79 @@
+{
+ "name": "@stdlib/stats/incr/mkurtosis",
+ "version": "0.0.0",
+ "description": "Compute a moving corrected sample excess kurtosis incrementally.",
+ "license": "Apache-2.0",
+ "author": {
+ "name": "The Stdlib Authors",
+ "url": "https://github.com/stdlib-js/stdlib/graphs/contributors"
+ },
+ "contributors": [
+ {
+ "name": "The Stdlib Authors",
+ "url": "https://github.com/stdlib-js/stdlib/graphs/contributors"
+ }
+ ],
+ "main": "./lib",
+ "directories": {
+ "benchmark": "./benchmark",
+ "doc": "./docs",
+ "example": "./examples",
+ "lib": "./lib",
+ "test": "./test"
+ },
+ "types": "./docs/types",
+ "scripts": {},
+ "homepage": "https://github.com/stdlib-js/stdlib",
+ "repository": {
+ "type": "git",
+ "url": "git://github.com/stdlib-js/stdlib.git"
+ },
+ "bugs": {
+ "url": "https://github.com/stdlib-js/stdlib/issues"
+ },
+ "dependencies": {
+ "@stdlib/array/float64": "^0.0.x",
+ "@stdlib/assert/is-positive-integer": "^0.0.x",
+ "@stdlib/math/base/assert/is-nan": "^0.0.x",
+ "@stdlib/string/format": "^0.0.x"
+ },
+ "devDependencies": {
+ "@stdlib/constants/float64/eps": "^0.0.x",
+ "@stdlib/math/base/special/abs": "^0.0.x",
+ "@stdlib/math/base/special/max": "^0.0.x",
+ "@stdlib/random/base/randu": "^0.0.x"
+ },
+ "engines": {
+ "node": ">=0.10.0",
+ "npm": ">2.7.0"
+ },
+ "os": [
+ "aix",
+ "darwin",
+ "freebsd",
+ "linux",
+ "macos",
+ "openbsd",
+ "sunos",
+ "win32",
+ "windows"
+ ],
+ "keywords": [
+ "stdlib",
+ "stdmath",
+ "statistics",
+ "stats",
+ "mathematics",
+ "math",
+ "kurtosis",
+ "sample kurtosis",
+ "moving",
+ "window",
+ "sliding",
+ "shape",
+ "kurt",
+ "corrected",
+ "incremental",
+ "accumulator"
+ ]
+}
diff --git a/lib/node_modules/@stdlib/stats/incr/mkurtosis/test/test.js b/lib/node_modules/@stdlib/stats/incr/mkurtosis/test/test.js
new file mode 100644
index 000000000000..022c3ba981e2
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/mkurtosis/test/test.js
@@ -0,0 +1,350 @@
+/**
+* @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 tape = require( 'tape' );
+var abs = require( '@stdlib/math/base/special/abs' );
+var max = require( '@stdlib/math/base/special/max' );
+var EPS = require( '@stdlib/constants/float64/eps' );
+var isnan = require( '@stdlib/math/base/assert/is-nan' );
+var mkurtosis = require( './../lib' );
+
+
+// FUNCTIONS //
+
+/**
+* Computes the corrected sample excess kurtosis for an array.
+*
+* @private
+* @param {Array} arr - array of numbers
+* @returns {number|null} kurtosis or null
+*/
+function naiveKurtosis( arr ) {
+ var deltaN2;
+ var deltaN;
+ var delta;
+ var term1;
+ var mean;
+ var n1;
+ var M2;
+ var M3;
+ var M4;
+ var g2;
+ var n;
+ var i;
+
+ n = arr.length;
+ if ( n < 4 ) {
+ return null;
+ }
+ mean = 0.0;
+ M2 = 0.0;
+ M3 = 0.0;
+ M4 = 0.0;
+ for ( i = 0; i < n; i++ ) {
+ n1 = i + 1;
+ delta = arr[ i ] - mean;
+ deltaN = delta / n1;
+ deltaN2 = deltaN * deltaN;
+ term1 = delta * deltaN * i;
+ M4 += term1 * deltaN2 * ( ( n1*n1 ) - ( 3*n1 ) + 3 );
+ M4 += 6.0 * deltaN2 * M2;
+ M4 -= 4.0 * deltaN * M3;
+ M3 += term1 * deltaN * ( n1 - 2 );
+ M3 -= 3.0 * deltaN * M2;
+ M2 += term1;
+ mean += deltaN;
+ }
+ g2 = ( n * M4 / ( M2 * M2 ) ) - 3.0;
+ return ( (n-1) / ( (n-2)*(n-3) ) ) * ( ( (n+1)*g2 ) + 6.0 );
+}
+
+
+// TESTS //
+
+tape( 'main export is a function', function test( t ) {
+ t.ok( true, __filename );
+ t.strictEqual( typeof mkurtosis, 'function', 'main export is a function' );
+ t.end();
+});
+
+tape( 'the function throws if not provided a positive integer for the window size', function test( t ) {
+ var values;
+ var i;
+
+ values = [
+ '5',
+ -5,
+ 0,
+ 3.14,
+ true,
+ false,
+ null,
+ void 0,
+ NaN,
+ [],
+ {},
+ function noop() {}
+ ];
+
+ for ( i = 0; i < values.length; i++ ) {
+ t.throws( badValue( values[ i ] ), TypeError, 'throws a type error when provided ' + values[ i ] );
+ }
+ t.end();
+
+ function badValue( value ) {
+ return function badValue() {
+ mkurtosis( value );
+ };
+ }
+});
+
+tape( 'the function returns an accumulator function', function test( t ) {
+ t.strictEqual( typeof mkurtosis( 4 ), 'function', 'returns a function' );
+ t.end();
+});
+
+tape( 'if not provided an input value, the accumulator function returns the current moving corrected sample excess kurtosis', function test( t ) {
+ var data;
+ var acc;
+ var i;
+
+ data = [ -10.0, -10.0, 10.0, 10.0 ];
+ acc = mkurtosis( 4 );
+ for ( i = 0; i < data.length; i++ ) {
+ acc( data[ i ] );
+ }
+ t.strictEqual( acc(), -6.0, 'returns current kurtosis without advancing state' );
+ t.end();
+});
+
+tape( 'the accumulator returns `null` until at least 4 datums have been provided', function test( t ) {
+ var acc;
+ var out;
+
+ acc = mkurtosis( 6 );
+
+ out = acc();
+ t.strictEqual( out, null, 'returns null (0 values)' );
+
+ out = acc( 2.0 );
+ t.strictEqual( out, null, 'returns null (1 value)' );
+
+ out = acc( 8.0 );
+ t.strictEqual( out, null, 'returns null (2 values)' );
+
+ out = acc( -4.0 );
+ t.strictEqual( out, null, 'returns null (3 values)' );
+
+ out = acc( 3.0 );
+ t.notEqual( out, null, 'does not return null (4 values)' );
+
+ t.end();
+});
+
+tape( 'the accumulator returns `null` for all invocations when window size is less than 4', function test( t ) {
+ var acc;
+ var out;
+ var i;
+
+ // W = 3: kurtosis is always undefined.
+ acc = mkurtosis( 3 );
+ for ( i = 0; i < 10; i++ ) {
+ out = acc( i * 1.0 );
+ t.strictEqual( out, null, 'returns null for W=3 (iteration '+i+')' );
+ }
+ t.end();
+});
+
+tape( 'the accumulator function incrementally computes a moving corrected sample excess kurtosis (window: 4, matches incrkurtosis for first window)', function test( t ) {
+ var expected;
+ var actual;
+ var delta;
+ var data;
+ var tol;
+ var acc;
+ var i;
+
+ // Using window=4: the first 4 results should match incrkurtosis exactly.
+ data = [ 2.0, 2.0, -4.0, -4.0 ];
+ expected = [
+ null,
+ null,
+ null,
+ -6.0
+ ];
+
+ acc = mkurtosis( 4 );
+ for ( i = 0; i < data.length; i++ ) {
+ actual = acc( data[ i ] );
+ if ( expected[ i ] === null ) {
+ t.strictEqual( actual, null, 'returns null' );
+ } else {
+ delta = abs( actual - expected[ i ] );
+ tol = 2.0 * EPS * abs( expected[ i ] );
+ t.ok( delta <= tol, 'within tolerance. actual: '+actual+'. E: '+expected[i]+'. Δ: '+delta+'. tol: '+tol );
+ }
+ }
+ t.end();
+});
+
+tape( 'the accumulator correctly slides the window (W=4)', function test( t ) {
+ var expected;
+ var actual;
+ var delta;
+ var data;
+ var tol;
+ var acc;
+ var i;
+
+ data = [ 2.0, 2.0, -4.0, -4.0, 1.5, -10.0, 5.0, 3.5, -1.0 ];
+ acc = mkurtosis( 4 );
+
+ for ( i = 0; i < data.length; i++ ) {
+ actual = acc( data[ i ] );
+
+ if ( i < 3 ) {
+ t.strictEqual( actual, null, 'returns null for i='+i );
+ } else {
+ // Expected: kurtosis of the last 4 values.
+ expected = naiveKurtosis( data.slice( max( 0, i-3 ), i+1 ) );
+ delta = abs( actual - expected );
+ tol = ( 2.0 * EPS * abs( expected ) ) + 1.0e-10;
+ t.ok( delta <= tol, 'i='+i+' actual: '+actual+' expected: '+expected+' Δ: '+delta+' tol: '+tol );
+ }
+ }
+ t.end();
+});
+
+tape( 'the accumulator correctly slides the window (W=6)', function test( t ) {
+ var expected;
+ var actual;
+ var delta;
+ var data;
+ var tol;
+ var acc;
+ var i;
+
+ data = [ 3.0, -1.0, 7.0, 2.5, -4.0, 6.0, 1.0, -2.0, 8.0, 5.0 ];
+ acc = mkurtosis( 6 );
+
+ for ( i = 0; i < data.length; i++ ) {
+ actual = acc( data[ i ] );
+ if ( i < 3 ) {
+ t.strictEqual( actual, null, 'returns null for i='+i );
+ } else if ( i < 6 ) {
+ // Window not full yet; compute over first i+1 values:
+ expected = naiveKurtosis( data.slice( 0, i+1 ) );
+ if ( expected === null ) {
+ t.strictEqual( actual, null, 'returns null for i='+i );
+ } else {
+ delta = abs( actual - expected );
+ tol = ( 2.0 * EPS * abs( expected ) ) + 1.0e-10;
+ t.ok( delta <= tol, 'i='+i+' actual: '+actual+' expected: '+expected+' Δ: '+delta+' tol: '+tol );
+ }
+ } else {
+ // Window full — last 6 values:
+ expected = naiveKurtosis( data.slice( i-5, i+1 ) );
+ delta = abs( actual - expected );
+ tol = ( 2.0 * EPS * abs( expected ) ) + 1.0e-10;
+ t.ok( delta <= tol, 'i='+i+' actual: '+actual+' expected: '+expected+' Δ: '+delta+' tol: '+tol );
+ }
+ }
+ t.end();
+});
+
+tape( 'the accumulator returns `NaN` when a NaN is in the window', function test( t ) {
+ var data;
+ var acc;
+ var v;
+ var i;
+
+ // NaN at position 2 (0-indexed), window=4.
+ data = [ 1.0, 2.0, NaN, -2.0, -3.0, 4.0, 5.0 ];
+ acc = mkurtosis( 4 );
+ for ( i = 0; i < data.length; i++ ) {
+ v = acc( data[ i ] );
+ if ( i >= 3 && i <= 5 ) {
+ // NaN in the window: positions 2-5.
+ t.ok( isnan( v ), 'returns NaN while NaN is in the window (i='+i+')' );
+ }
+ }
+ t.end();
+});
+
+tape( 'the accumulator recovers after a NaN exits the window', function test( t ) {
+ var data;
+ var acc;
+ var v;
+ var i;
+
+ // W=4: NaN at index 0 leaves after 4 slides.
+ data = [ NaN, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0 ];
+ acc = mkurtosis( 4 );
+
+ // Consume NaN and 3 good values:
+ for ( i = 0; i < 4; i++ ) {
+ v = acc( data[ i ] );
+
+ // Returns NaN or null while NaN is present or data is insufficient:
+ t.ok( isnan( v ) || v === null, 'returns NaN or null (i='+i+')' );
+ }
+
+ // Fifth value pushes NaN out of the window:
+ v = acc( data[ 4 ] );
+ t.ok( !isnan( v ), 'recovered from NaN (i=4), got: ' + v );
+ t.notEqual( v, null, 'does not return null after recovery' );
+
+ t.end();
+});
+
+tape( 'the accumulator handles a window entirely filled with identical values (kurtosis undefined: M2=0)', function test( t ) {
+ var acc;
+ var v;
+ var i;
+
+ acc = mkurtosis( 4 );
+ for ( i = 0; i < 8; i++ ) {
+ v = acc( 5.0 );
+ if ( i < 3 ) {
+ t.strictEqual( v, null, 'returns null for i='+i );
+ } else {
+ // M2=0 leads to division by zero; null, NaN, or Infinity are all acceptable.
+ t.ok( v === null || isnan( v ) || !isFinite( v ), 'returns null, NaN, or Infinity for degenerate window (i='+i+')' );
+ }
+ }
+ t.end();
+});
+
+tape( 'the accumulator works with window size W=4 and exactly 4 values', function test( t ) {
+ var acc;
+ var v;
+
+ acc = mkurtosis( 4 );
+
+ acc( -10.0 );
+ acc( -10.0 );
+ acc( 10.0 );
+ v = acc( 10.0 );
+
+ t.strictEqual( v, -6.0, 'returns -6.0 for symmetric data' );
+ t.end();
+});