diff --git a/lib/node_modules/@stdlib/stats/incr/nanmvmr/README.md b/lib/node_modules/@stdlib/stats/incr/nanmvmr/README.md
new file mode 100644
index 000000000000..d9be6a4479ed
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/nanmvmr/README.md
@@ -0,0 +1,227 @@
+
+
+# incrnanmvmr
+
+> Compute a moving [variance-to-mean ratio][variance-to-mean-ratio] (VMR) incrementally.
+
+
+
+For a window of size `W`, the [unbiased sample variance][sample-variance] is defined as
+
+
+
+```math
+s^2 = \frac{1}{W-1} \sum_{i=0}^{W-1} ( x_i - \bar{x} )^2
+```
+
+
+
+
+
+and the [arithmetic mean][arithmetic-mean] is defined as
+
+
+
+```math
+\bar{x} = \frac{1}{W} \sum_{i=0}^{W-1} x_i
+```
+
+
+
+
+
+The [variance-to-mean ratio][variance-to-mean-ratio] (VMR) is thus defined as
+
+
+
+```math
+F = \frac{s^2}{\bar{x}}
+```
+
+
+
+
+
+
+
+
+
+
+
+## Usage
+
+```javascript
+var incrnanmvmr = require( '@stdlib/stats/incr/nanmvmr' );
+```
+
+#### incrnanmvmr( window\[, mean] )
+
+Returns an accumulator `function` which incrementally computes a moving [variance-to-mean ratio][variance-to-mean-ratio]. The `window` parameter defines the number of values over which to compute the moving [variance-to-mean ratio][variance-to-mean-ratio].
+
+```javascript
+var accumulator = incrnanmvmr( 3 );
+```
+
+If the mean is already known, provide a `mean` argument.
+
+```javascript
+var accumulator = incrnanmvmr( 3, 5.0 );
+```
+
+#### accumulator( \[x] )
+
+If provided an input value `x`, the accumulator function returns an updated accumulated value. If not provided an input value `x`, the accumulator function returns the current accumulated value.
+
+```javascript
+var accumulator = incrnanmvmr( 3 );
+
+var F = accumulator();
+// returns null
+
+// Fill the window...
+F = accumulator( 2.0 ); // [2.0]
+// returns 0.0
+
+F = accumulator( 1.0 ); // [2.0, 1.0]
+// returns ~0.33
+
+F = accumulator( 3.0 ); // [2.0, 1.0, 3.0]
+// returns 0.5
+
+// Window begins sliding...
+F = accumulator( 7.0 ); // [1.0, 3.0, 7.0]
+// returns ~2.55
+
+F = accumulator( 5.0 ); // [3.0, 7.0, 5.0]
+// returns ~0.80
+
+F = accumulator();
+// returns ~0.80
+```
+
+
+
+
+
+
+
+## Notes
+
+- 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 calculated from all provided values.
+
+- The following table summarizes how to interpret the [variance-to-mean ratio][variance-to-mean-ratio]:
+
+ | VMR | Description | Example Distribution |
+ | :---------------: | :-------------: | :--------------------------: |
+ | 0 | not dispersed | constant |
+ | 0 < VMR < 1 | under-dispersed | binomial |
+ | 1 | -- | Poisson |
+ | >1 | over-dispersed | geometric, negative-binomial |
+
+ Accordingly, one can use the [variance-to-mean ratio][variance-to-mean-ratio] to assess whether observed data can be modeled as a Poisson process. When observed data is "under-dispersed", observed data may be more regular than as would be the case for a Poisson process. When observed data is "over-dispersed", observed data may contain clusters (i.e., clumped, concentrated data).
+
+- The [variance-to-mean ratio][variance-to-mean-ratio] is typically computed on nonnegative values. The measure may lack meaning for data which can assume both positive and negative values.
+
+- The [variance-to-mean ratio][variance-to-mean-ratio] is also known as the **index of dispersion**, **dispersion index**, **coefficient of dispersion**, **relative variance**, and the [**Fano factor**][fano-factor].
+
+
+
+
+
+
+
+## Examples
+
+
+
+```javascript
+var randu = require( '@stdlib/random/base/randu' );
+var incrnanmvmr = require( '@stdlib/stats/incr/nanmvmr' );
+
+var accumulator;
+var v;
+var i;
+
+// Initialize an accumulator:
+accumulator = incrnanmvmr( 5 );
+
+// For each simulated datum, update the moving variance-to-mean ratio...
+for ( i = 0; i < 100; i++ ) {
+ v = ( randu()> 0.2 ) ? NaN : randu() * 100.0;
+ accumulator( v );
+}
+console.log( accumulator() );
+```
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+[variance-to-mean-ratio]: https://en.wikipedia.org/wiki/Index_of_dispersion
+
+[arithmetic-mean]: https://en.wikipedia.org/wiki/Arithmetic_mean
+
+[sample-variance]: https://en.wikipedia.org/wiki/Variance
+
+[fano-factor]: https://en.wikipedia.org/wiki/Fano_factor
+
+
+
+[@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/vmr]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/stats/incr/vmr
+
+
+
+
+
+
diff --git a/lib/node_modules/@stdlib/stats/incr/nanmvmr/benchmark/benchmark.js b/lib/node_modules/@stdlib/stats/incr/nanmvmr/benchmark/benchmark.js
new file mode 100644
index 000000000000..dc52a35a0c1d
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/nanmvmr/benchmark/benchmark.js
@@ -0,0 +1,91 @@
+/**
+* @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 randu = require( '@stdlib/random/base/randu' );
+var pkg = require( './../package.json' ).name;
+var incrnanmvmr = require( './../lib' );
+
+
+// MAIN //
+
+bench( pkg, function benchmark( b ) {
+ var f;
+ var i;
+ b.tic();
+ for ( i = 0; i < b.iterations; i++ ) {
+ f = incrnanmvmr( (i%5)+1 );
+ 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 = incrnanmvmr( 5 );
+
+ b.tic();
+ for ( i = 0; i < b.iterations; i++ ) {
+ v = acc( randu() );
+ 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();
+});
+
+bench( pkg+'::accumulator,known_mean', function benchmark( b ) {
+ var acc;
+ var v;
+ var i;
+
+ acc = incrnanmvmr( 5, 0.5 );
+
+ b.tic();
+ for ( i = 0; i < b.iterations; i++ ) {
+ v = acc( randu() );
+ 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();
+});
diff --git a/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/img/equation_arithmetic_mean.svg b/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/img/equation_arithmetic_mean.svg
new file mode 100644
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--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/img/equation_arithmetic_mean.svg
@@ -0,0 +1,43 @@
+
\ No newline at end of file
diff --git a/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/img/equation_unbiased_sample_variance.svg b/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/img/equation_unbiased_sample_variance.svg
new file mode 100644
index 000000000000..f4722f986a70
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/img/equation_unbiased_sample_variance.svg
@@ -0,0 +1,61 @@
+
\ No newline at end of file
diff --git a/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/img/equation_variance_to_mean_ratio.svg b/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/img/equation_variance_to_mean_ratio.svg
new file mode 100644
index 000000000000..d7318aa42b21
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/img/equation_variance_to_mean_ratio.svg
@@ -0,0 +1,28 @@
+
\ No newline at end of file
diff --git a/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/repl.txt b/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/repl.txt
new file mode 100644
index 000000000000..388d1fb8cb26
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/repl.txt
@@ -0,0 +1,53 @@
+{{alias}}( W[, mean] )
+ Returns an accumulator function which incrementally computes a moving
+ variance-to-mean ratio (VMR) while ignoring NaN values.
+
+ The `W` parameter defines the number of values over which to compute the
+ moving variance-to-mean ratio.
+
+ If provided a value, the accumulator function returns an updated moving
+ variance-to-mean ratio. If not provided a value, the accumulator function
+ returns the current moving variance-to-mean ratio.
+
+ 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 calculated from all provided values.
+
+ NaN input values are ignored.
+
+ Parameters
+ ----------
+ W: integer
+ Window size.
+
+ mean: number (optional)
+ Known mean.
+
+ Returns
+ -------
+ acc: Function
+ Accumulator function.
+
+ Examples
+ --------
+ > var accumulator = {{alias}}( 3 );
+ > var F = accumulator()
+ null
+ > F = accumulator( 2.0 )
+ 0.0
+ > F = accumulator( NaN )
+ 0.0
+ > F = accumulator( 1.0 )
+ ~0.33
+ > F = accumulator( 3.0 )
+ 0.5
+ > F = accumulator( NaN )
+ 0.5
+ > F = accumulator( 7.0 )
+ ~2.55
+ > F = accumulator()
+ ~2.55
+
+ See Also
+ --------
+
diff --git a/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/types/index.d.ts b/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/types/index.d.ts
new file mode 100644
index 000000000000..eaee383b3bc1
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/types/index.d.ts
@@ -0,0 +1,79 @@
+/*
+* @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.
+*/
+
+// TypeScript Version: 4.1
+
+///
+
+/**
+* If provided a value, the accumulator function returns an updated accumulated value. If not provided a value, the accumulator function returns the current accumulated value.
+*
+* ## Notes
+*
+* - If provided `NaN` or a value which, when used in computations, results in `NaN`, the accumulated value is `NaN` for all future invocations.
+* - NaN values are ignored.
+*
+* @param x - value
+* @returns accumulated value or null
+*/
+type accumulator = ( x?: number ) => number | null;
+
+/**
+* Returns an accumulator function which incrementally computes a moving variance-to-mean ratio (VMR) while ignoring NaN values.
+*
+* @param W - window size
+* @param mean - mean value
+* @throws first argument must be a positive integer
+* @returns accumulator function
+*
+* @example
+* var accumulator = incrnanmvmr( 3 );
+*
+* var F = accumulator();
+* // returns null
+*
+* F = accumulator( 2.0 );
+* // returns 0.0
+*
+* F = accumulator( NaN );
+* // returns 0.0
+*
+* F = accumulator( 1.0 );
+* // returns ~0.33
+*
+* F = accumulator( 3.0 );
+* // returns 0.5
+*
+* F = accumulator( NaN );
+* // returns 0.5
+*
+* F = accumulator( 7.0 );
+* // returns ~2.55
+*
+* F = accumulator();
+* // returns ~2.55
+*
+* @example
+* var accumulator = incrnanmvmr( 3, 2.0 );
+*/
+declare function incrnanmvmr( W: number, mean?: number ): accumulator;
+
+
+// EXPORTS //
+
+export = incrnanmvmr;
diff --git a/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/types/test.ts b/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/types/test.ts
new file mode 100644
index 000000000000..29577d71b496
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/nanmvmr/docs/types/test.ts
@@ -0,0 +1,77 @@
+/*
+* @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.
+*/
+
+import incrnanmvmr = require( './index' );
+
+
+// TESTS //
+
+// The function returns an accumulator function...
+{
+ incrnanmvmr( 3 ); // $ExpectType accumulator
+ incrnanmvmr( 3, 0.0 ); // $ExpectType accumulator
+}
+
+// The compiler throws an error if the function is provided a first argument that is not a number...
+{
+ incrnanmvmr( '5' ); // $ExpectError
+ incrnanmvmr( true ); // $ExpectError
+ incrnanmvmr( false ); // $ExpectError
+ incrnanmvmr( null ); // $ExpectError
+ incrnanmvmr( [] ); // $ExpectError
+ incrnanmvmr( {} ); // $ExpectError
+ incrnanmvmr( ( x: number ): number => x ); // $ExpectError
+}
+
+// The compiler throws an error if the function is provided a second argument that is not a number...
+{
+ incrnanmvmr( 2, '5' ); // $ExpectError
+ incrnanmvmr( 2, true ); // $ExpectError
+ incrnanmvmr( 2, false ); // $ExpectError
+ incrnanmvmr( 2, null ); // $ExpectError
+ incrnanmvmr( 2, [] ); // $ExpectError
+ incrnanmvmr( 2, {} ); // $ExpectError
+ incrnanmvmr( 2, ( x: number ): number => x ); // $ExpectError
+}
+
+// The compiler throws an error if the function is provided an invalid number of arguments...
+{
+ incrnanmvmr(); // $ExpectError
+ incrnanmvmr( 2, 3, 4 ); // $ExpectError
+}
+
+// The function returns an accumulator function which returns an accumulated result...
+{
+ const acc = incrnanmvmr( 3 );
+
+ acc(); // $ExpectType number | null
+ acc( 3.14 ); // $ExpectType number | null
+}
+
+// The compiler throws an error if the returned accumulator function is provided invalid arguments...
+{
+ const acc = incrnanmvmr( 3 );
+
+ 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/nanmvmr/examples/index.js b/lib/node_modules/@stdlib/stats/incr/nanmvmr/examples/index.js
new file mode 100644
index 000000000000..dce324a7dde2
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/nanmvmr/examples/index.js
@@ -0,0 +1,38 @@
+/**
+* @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';
+
+var randu = require( '@stdlib/random/base/randu' );
+var incrnanmvmr = require( './../lib' );
+
+var accumulator;
+var F;
+var v;
+var i;
+
+// Initialize an accumulator:
+accumulator = incrnanmvmr( 5 );
+
+// For each simulated datum, update the moving variance-to-mean ratio...
+console.log( '\nValue\tVMR\n' );
+for ( i = 0; i < 100; i++ ) {
+ v = ( randu() < 0.2 ) ? NaN : randu() * 100.0;
+ F = accumulator( v );
+ console.log( '%s\t%s', ( Number.isNaN(v) ) ? 'NaN' : v.toFixed( 4 ), ( Number.isNaN(F) && F === null ) ? 'NaN' : F.toFixed( 4 ) );
+}
diff --git a/lib/node_modules/@stdlib/stats/incr/nanmvmr/lib/index.js b/lib/node_modules/@stdlib/stats/incr/nanmvmr/lib/index.js
new file mode 100644
index 000000000000..9f2efa7f6f99
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/nanmvmr/lib/index.js
@@ -0,0 +1,63 @@
+/**
+* @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';
+
+/**
+* Compute a moving variance-to-mean ratio (VMR) incrementally, ignoring NaN values.
+*
+* @module @stdlib/stats/incr/nanmvmr
+*
+* @example
+* var incrnanmvmr = require( '@stdlib/stats/incr/nanmvmr' );
+*
+* var accumulator = incrnanmvmr( 3 );
+*
+* var F = accumulator();
+* // returns null
+*
+* F = accumulator( 2.0 );
+* // returns 0.0
+*
+* F = accumulator( NaN );
+* // returns 0.0
+*
+* F = accumulator( 1.0 );
+* // returns ~0.33
+*
+* F = accumulator( 3.0 );
+* // returns 0.5
+*
+* F = accumulator( NaN );
+* // returns 0.5
+*
+* F = accumulator( 7.0 );
+* // returns ~2.55
+*
+* F = accumulator();
+* // returns ~2.55
+*/
+
+// MODULES //
+
+var main = require( './main.js' );
+
+
+// EXPORTS //
+
+module.exports = main;
diff --git a/lib/node_modules/@stdlib/stats/incr/nanmvmr/lib/main.js b/lib/node_modules/@stdlib/stats/incr/nanmvmr/lib/main.js
new file mode 100644
index 000000000000..70d229c05a5f
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/nanmvmr/lib/main.js
@@ -0,0 +1,170 @@
+/**
+* @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 incrmvmr = require( '@stdlib/stats/incr/mvmr' );
+var isnan = require( '@stdlib/math/base/assert/is-nan' );
+
+
+// MAIN //
+
+/**
+* Returns an accumulator function which incrementally computes a moving variance-to-mean ratio (VMR) while ignoring NaN values.
+*
+* ## Method
+*
+* - Let \\(W\\) be a window of \\(N\\) elements over which we want to compute a variance-to-mean ratio (VMR).
+*
+* - The difference between the unbiased sample variance in a window \\(W_i\\) and the unbiased sample variance in a window \\(W_{i+1})\\) is given by
+*
+* ```tex
+* \Delta s^2 = s_{i+1}^2 - s_{i}^2
+* ```
+*
+* - If we multiply both sides by \\(N-1\\),
+*
+* ```tex
+* (N-1)(\Delta s^2) = (N-1)s_{i+1}^2 - (N-1)s_{i}^2
+* ```
+*
+* - If we substitute the definition of the unbiased sample variance having the form
+*
+* ```tex
+* \begin{align*}
+* s^2 &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} (x_i - \bar{x})^2 \biggr) \\
+* &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} (x_i^2 - 2\bar{x}x_i + \bar{x}^2) \biggr) \\
+* &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} x_i^2 - 2\bar{x} \sum_{i=1}^{N} x_i + \sum_{i=1}^{N} \bar{x}^2) \biggr) \\
+* &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} x_i^2 - \frac{2N\bar{x}\sum_{i=1}^{N} x_i}{N} + N\bar{x}^2 \biggr) \\
+* &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} x_i^2 - 2N\bar{x}^2 + N\bar{x}^2 \biggr) \\
+* &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} x_i^2 - N\bar{x}^2 \biggr)
+* \end{align*}
+* ```
+*
+* we return
+*
+* ```tex
+* (N-1)(\Delta s^2) = \biggl(\sum_{k=1}^N x_k^2 - N\bar{x}_{i+1}^2 \biggr) - \biggl(\sum_{k=0}^{N-1} x_k^2 - N\bar{x}_{i}^2 \biggr)
+* ```
+*
+* - This can be further simplified by recognizing that subtracting the sums reduces to \\(x_N^2 - x_0^2\\); in which case,
+*
+* ```tex
+* \begin{align*}
+* (N-1)(\Delta s^2) &= x_N^2 - x_0^2 - N\bar{x}_{i+1}^2 + N\bar{x}_{i}^2 \\
+* &= x_N^2 - x_0^2 - N(\bar{x}_{i+1}^2 - \bar{x}_{i}^2) \\
+* &= x_N^2 - x_0^2 - N(\bar{x}_{i+1} - \bar{x}_{i})(\bar{x}_{i+1} + \bar{x}_{i})
+* \end{align*}
+* ```
+*
+* - Recognizing that the difference of means can be expressed
+*
+* ```tex
+* \bar{x}_{i+1} - \bar{x}_i = \frac{1}{N} \biggl( \sum_{k=1}^N x_k - \sum_{k=0}^{N-1} x_k \biggr) = \frac{x_N - x_0}{N}
+* ```
+*
+* and substituting into the equation above
+*
+* ```tex
+* (N-1)(\Delta s^2) = x_N^2 - x_0^2 - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i})
+* ```
+*
+* - Rearranging terms gives us the update equation for the unbiased sample variance
+*
+* ```tex
+* \begin{align*}
+* (N-1)(\Delta s^2) &= (x_N - x_0)(x_N + x_0) - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i}) \\
+* &= (x_N - x_0)(x_N + x_0 - \bar{x}_{i+1} - \bar{x}_{i}) \\
+* &= (x_N - x_0)(x_N - \bar{x}_{i+1} + x_0 - \bar{x}_{i})
+* \end{align*}
+* ```
+*
+* @param {PositiveInteger} W - window size
+* @param {number} [mean] - mean value
+* @throws {TypeError} first argument must be a positive integer
+* @throws {TypeError} second argument must be a number
+* @returns {Function} accumulator function
+*
+* @example
+* var accumulator = incrnanmvmr( 3 );
+*
+* var F = accumulator();
+* // returns null
+*
+* F = accumulator( 2.0 );
+* // returns 0.0
+*
+* F = accumulator( NaN );
+* // returns 0.0
+*
+* F = accumulator( 1.0 );
+* // returns ~0.33
+*
+* F = accumulator( 3.0 );
+* // returns 0.5
+*
+* F = accumulator( NaN );
+* // returns 0.5
+*
+* F = accumulator( 7.0 );
+* // returns ~2.55
+*
+* F = accumulator();
+* // returns ~2.55
+*
+* @example
+* var accumulator = incrnanmvmr( 3, 2.0 );
+*/
+function incrnanmvmr( W, mean ) {
+ var acc;
+
+ // Initialize the wrapped accumulator:
+ if ( arguments.length > 1 ) {
+ acc = incrmvmr( W, mean );
+ } else {
+ acc = incrmvmr( W );
+ }
+
+ return accumulator;
+
+ /**
+ * If provided a value, the accumulator function returns an updated accumulated value. If not provided a value, the accumulator function returns the current accumulated value.
+ * NaN input values are ignored.
+ *
+ * @private
+ * @param {number} [x] - input value
+ * @returns {(number|null)} accumulated value or null
+ */
+ function accumulator( x ) {
+ if ( arguments.length === 0 ) {
+ return acc();
+ }
+ if ( isnan( x ) === false ) {
+ return acc( x );
+ }
+
+ return acc();
+ }
+}
+
+
+// EXPORTS //
+
+module.exports = incrnanmvmr;
diff --git a/lib/node_modules/@stdlib/stats/incr/nanmvmr/package.json b/lib/node_modules/@stdlib/stats/incr/nanmvmr/package.json
new file mode 100644
index 000000000000..2b1225f78deb
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/nanmvmr/package.json
@@ -0,0 +1,78 @@
+{
+ "name": "@stdlib/stats/incr/nanmvmr",
+ "version": "0.0.0",
+ "description": "Compute a moving variance-to-mean ratio (VMR) incrementally, ignoring NaN values.",
+ "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": {},
+ "devDependencies": {},
+ "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",
+ "variance",
+ "sample",
+ "sample variance",
+ "unbiased",
+ "dispersion",
+ "index of dispersion",
+ "fano factor",
+ "fano",
+ "dispersion index",
+ "relative variance",
+ "vmr",
+ "mean",
+ "ratio",
+ "incremental",
+ "accumulator",
+ "sliding window",
+ "sliding",
+ "window",
+ "moving"
+ ]
+}
diff --git a/lib/node_modules/@stdlib/stats/incr/nanmvmr/test/test.js b/lib/node_modules/@stdlib/stats/incr/nanmvmr/test/test.js
new file mode 100644
index 000000000000..7de2e459f9bc
--- /dev/null
+++ b/lib/node_modules/@stdlib/stats/incr/nanmvmr/test/test.js
@@ -0,0 +1,534 @@
+/**
+* @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 tape = require( 'tape' );
+var randu = require( '@stdlib/random/base/randu' );
+var abs = require( '@stdlib/math/base/special/abs' );
+var EPS = require( '@stdlib/constants/float64/eps' );
+var incrnanmvmr = require( './../lib' );
+
+
+// TESTS //
+
+tape( 'main export is a function', function test( t ) {
+ t.ok( true, __filename );
+ t.strictEqual( typeof incrnanmvmr, 'function', 'main export is a function' );
+ t.end();
+});
+
+tape( 'the function throws an error if not provided a positive integer for the window size', function test( t ) {
+ var values;
+ var i;
+
+ values = [
+ '5',
+ -5.0,
+ 0.0,
+ 3.14,
+ true,
+ null,
+ void 0,
+ NaN,
+ [],
+ {},
+ function noop() {}
+ ];
+
+ for ( i = 0; i < values.length; i++ ) {
+ t.throws( badValue( values[i] ), TypeError, 'throws an error when provided '+values[i] );
+ }
+ t.end();
+
+ function badValue( value ) {
+ return function badValue() {
+ incrnanmvmr( value );
+ };
+ }
+});
+
+tape( 'the function throws an error if not provided a positive integer for the window size (known mean)', function test( t ) {
+ var values;
+ var i;
+
+ values = [
+ '5',
+ -5.0,
+ 0.0,
+ 3.14,
+ true,
+ null,
+ void 0,
+ NaN,
+ [],
+ {},
+ function noop() {}
+ ];
+
+ for ( i = 0; i < values.length; i++ ) {
+ t.throws( badValue( values[i] ), TypeError, 'throws an error when provided '+values[i] );
+ }
+ t.end();
+
+ function badValue( value ) {
+ return function badValue() {
+ incrnanmvmr( value, 3.0 );
+ };
+ }
+});
+
+tape( 'the function throws an error if not provided a number as the mean value', function test( t ) {
+ var values;
+ var i;
+
+ values = [
+ '5',
+ true,
+ false,
+ null,
+ void 0,
+ [],
+ {},
+ function noop() {}
+ ];
+
+ for ( i = 0; i < values.length; i++ ) {
+ t.throws( badValue( values[i] ), TypeError, 'throws an error when provided '+values[i] );
+ }
+ t.end();
+
+ function badValue( value ) {
+ return function badValue() {
+ incrnanmvmr( 3, value );
+ };
+ }
+});
+
+tape( 'the function returns an accumulator function', function test( t ) {
+ t.equal( typeof incrnanmvmr( 3 ), 'function', 'returns a function' );
+ t.end();
+});
+
+tape( 'the function returns an accumulator function (known mean)', function test( t ) {
+ t.equal( typeof incrnanmvmr( 3, 3.0 ), 'function', 'returns a function' );
+ t.end();
+});
+
+tape( 'the accumulator function computes a moving variance-to-mean ratio incrementally', function test( t ) {
+ var expected;
+ var actual;
+ var data;
+ var acc;
+ var N;
+ var i;
+
+ data = [ 2.0, 3.0, 4.0, -1.0, 3.0, 1.0 ];
+ N = data.length;
+
+ expected = [
+ 0.0/2.0,
+ 0.5/2.5,
+ 1.0/3.0,
+ 7.0/2.0,
+ 7.0/2.0,
+ 4.0/1.0
+ ];
+
+ acc = incrnanmvmr( 3 );
+
+ actual = [];
+ for ( i = 0; i < N; i++ ) {
+ actual.push( acc( data[ i ] ) );
+ }
+ t.deepEqual( actual, expected, 'returns expected results' );
+ t.end();
+});
+
+tape( 'the accumulator function computes a moving variance-to-mean ratio incrementally (known mean)', function test( t ) {
+ var expected;
+ var actual;
+ var data;
+ var acc;
+ var N;
+ var i;
+
+ data = [ 2.0, 3.0, 4.0, -1.0, 3.0, 1.0 ];
+ N = data.length;
+
+ acc = incrnanmvmr( 3, 2.0 );
+
+ actual = [];
+ for ( i = 0; i < N; i++ ) {
+ actual.push( acc( data[ i ] ) );
+ }
+ expected = [
+ 0.0/2.0,
+ 0.5/2.0,
+ 1.6666666666666667/2.0,
+ 4.666666666666667/2.0,
+ 4.666666666666667/2.0,
+ 3.6666666666666665/2.0
+ ];
+ t.deepEqual( actual, expected, 'returns expected results' );
+ t.end();
+});
+
+tape( 'if not provided an input value, the accumulator function returns the current accumulated value', function test( t ) {
+ var expected;
+ var actual;
+ var delta;
+ var data;
+ var tol;
+ var acc;
+ var i;
+
+ data = [ 2.0, 3.0, 10.0 ];
+ acc = incrnanmvmr( 3 );
+ for ( i = 0; i < data.length-1; i++ ) {
+ acc( data[ i ] );
+ }
+ t.equal( acc(), 0.5/2.5, 'returns expected value' );
+
+ acc( data[ data.length-1 ] );
+
+ expected = 19.0/5.0;
+ actual = acc();
+ delta = abs( actual - expected );
+ tol = EPS * expected;
+
+ t.equal( delta < tol, true, 'expected: '+expected+'. actual: '+actual+'. tol: '+tol+'. delta: '+delta+'.' );
+ t.end();
+});
+
+tape( 'if not provided an input value, the accumulator function returns the current accumulated value (known mean)', function test( t ) {
+ var expected;
+ var actual;
+ var delta;
+ var data;
+ var tol;
+ var acc;
+ var i;
+
+ data = [ 2.0, 3.0, 10.0 ];
+ acc = incrnanmvmr( 3, 5.0 );
+ for ( i = 0; i < data.length-1; i++ ) {
+ acc( data[ i ] );
+ }
+ t.equal( acc(), 6.5/5.0, 'returns expected value' );
+
+ acc( data[ data.length-1 ] );
+
+ expected = 12.666666666666666/5.0;
+ actual = acc();
+ delta = abs( actual - expected );
+ tol = EPS * expected;
+
+ t.equal( delta < tol, true, 'expected: '+expected+'. actual: '+actual+'. tol: '+tol+'. delta: '+delta+'.' );
+ t.end();
+});
+
+tape( 'if data has yet to be provided, the accumulator function returns `null`', function test( t ) {
+ var acc = incrnanmvmr( 3 );
+ t.equal( acc(), null, 'returns null' );
+ t.end();
+});
+
+tape( 'if data has yet to be provided, the accumulator function returns `null` (known mean)', function test( t ) {
+ var acc = incrnanmvmr( 3, 3.0 );
+ t.equal( acc(), null, 'returns null' );
+ t.end();
+});
+
+tape( 'if only one datum has been provided and the mean is unknown, the accumulator function returns `0`', function test( t ) {
+ var acc = incrnanmvmr( 3 );
+ acc( 2.0 );
+ t.equal( acc(), 0.0, 'returns 0' );
+ t.end();
+});
+
+tape( 'if only one datum has been provided and the mean is known, the accumulator function may not return `0`', function test( t ) {
+ var acc = incrnanmvmr( 3, 30 );
+ acc( 2.0 );
+ t.notEqual( acc(), 0.0, 'does not return 0' );
+ t.end();
+});
+
+tape( 'if the window size is `1` and the mean is unknown, the accumulator function always returns `0`', function test( t ) {
+ var acc;
+ var F;
+ var i;
+
+ acc = incrnanmvmr( 1 );
+ for ( i = 0; i < 100; i++ ) {
+ F = acc( randu() * 100.0 );
+ t.equal( F, 0.0, 'returns 0' );
+ }
+ t.end();
+});
+
+tape( 'if the window size is `1` and the mean is known, the accumulator function may not always return `0`', function test( t ) {
+ var acc;
+ var F;
+ var i;
+
+ acc = incrnanmvmr( 1, 500.0 ); // mean is outside the range of simulated values so the variance should never be zero
+ for ( i = 0; i < 100; i++ ) {
+ F = acc( randu() * 100.0 );
+ t.notEqual( F, 0.0, 'does not return 0' );
+ }
+ t.end();
+});
+
+tape( 'if provided `NaN`, the accumulated value is not affected (unknown mean)', function test( t ) {
+ var expected;
+ var data;
+ var acc;
+ var v;
+ var i;
+
+ acc = incrnanmvmr( 3 );
+
+ data = [
+ NaN, // NaN (ignored)
+ 3.14, // 3.14
+ 3.14, // 3.14, 3.14
+ NaN, // 3.14, 3.14 (NaN ignored)
+ 3.14, // 3.14, 3.14, 3.14
+ 3.14, // 3.14, 3.14, 3.14
+ 3.14, // 3.14, 3.14, 3.14
+ NaN, // 3.14, 3.14, 3.14 (NaN ignored)
+ 3.14, // 3.14, 3.14, 3.14
+ 3.14, // 3.14, 3.14, 3.14
+ 3.14, // 3.14, 3.14, 3.14
+ NaN, // 3.14, 3.14, 3.14 (NaN ignored)
+ 3.14, // 3.14, 3.14, 3.14
+ 3.14, // 3.14, 3.14, 3.14
+ NaN, // 3.14, 3.14, 3.14 (NaN ignored)
+ NaN, // 3.14, 3.14, 3.14 (NaN ignored)
+ NaN, // 3.14, 3.14, 3.14 (NaN ignored)
+ NaN, // 3.14, 3.14, 3.14 (NaN ignored)
+ 3.14 // 3.14, 3.14, 3.14
+ ];
+ expected = [
+ null,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0
+ ];
+ for ( i = 0; i < data.length; i++ ) {
+ v = acc( data[ i ] );
+ t.equal( v, expected[ i ], 'returns expected value for window '+i );
+ t.equal( acc(), expected[ i ], 'returns expected value for window '+i );
+ }
+ t.end();
+});
+
+tape( 'if provided `NaN`, the accumulated value is not affected (known mean)', function test( t ) {
+ var expected;
+ var data;
+ var acc;
+ var v;
+ var i;
+
+ acc = incrnanmvmr( 3, 3.14 );
+
+ data = [
+ NaN, // NaN (ignored)
+ 3.14, // 3.14
+ 3.14, // 3.14, 3.14
+ NaN, // 3.14, 3.14 (NaN ignored)
+ 3.14, // 3.14, 3.14, 3.14
+ 3.14, // 3.14, 3.14, 3.14
+ 3.14, // 3.14, 3.14, 3.14
+ NaN, // 3.14, 3.14, 3.14 (NaN ignored)
+ 3.14, // 3.14, 3.14, 3.14
+ 3.14, // 3.14, 3.14, 3.14
+ 3.14, // 3.14, 3.14, 3.14
+ NaN, // 3.14, 3.14, 3.14 (NaN ignored)
+ 3.14, // 3.14, 3.14, 3.14
+ 3.14, // 3.14, 3.14, 3.14
+ NaN, // 3.14, 3.14, 3.14 (NaN ignored)
+ NaN, // 3.14, 3.14, 3.14 (NaN ignored)
+ NaN, // 3.14, 3.14, 3.14 (NaN ignored)
+ NaN, // 3.14, 3.14, 3.14 (NaN ignored)
+ 3.14 // 3.14, 3.14, 3.14
+ ];
+ expected = [
+ null,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0
+ ];
+ for ( i = 0; i < data.length; i++ ) {
+ v = acc( data[ i ] );
+ t.equal( v, expected[ i ], 'returns expected value for window '+i );
+ t.equal( acc(), expected[ i ], 'returns expected value for window '+i );
+ }
+ t.end();
+});
+
+tape( 'if provided `NaN`, the accumulated value is not affected (unknown mean, W=1)', function test( t ) {
+ var expected;
+ var data;
+ var acc;
+ var v;
+ var i;
+
+ acc = incrnanmvmr( 1 );
+
+ data = [
+ NaN, // NaN (ignored)
+ 3.14, // 3.14
+ 3.14, // 3.14
+ NaN, // 3.14 (NaN ignored)
+ 3.14, // 3.14
+ 3.14, // 3.14
+ 3.14, // 3.14
+ NaN, // 3.14 (NaN ignored)
+ 3.14, // 3.14
+ 3.14, // 3.14
+ 3.14, // 3.14
+ NaN, // 3.14 (NaN ignored)
+ 3.14, // 3.14
+ 3.14, // 3.14
+ NaN, // 3.14 (NaN ignored)
+ NaN, // 3.14 (NaN ignored)
+ NaN, // 3.14 (NaN ignored)
+ NaN, // 3.14 (NaN ignored)
+ 3.14 // 3.14
+ ];
+ expected = [
+ null,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0
+ ];
+ for ( i = 0; i < data.length; i++ ) {
+ v = acc( data[ i ] );
+ t.equal( v, expected[ i ], 'returns expected value for window '+i );
+ t.equal( acc(), expected[ i ], 'returns expected value for window '+i );
+ }
+ t.end();
+});
+
+tape( 'if provided `NaN`, the accumulated value is not affected (known mean, W=1)', function test( t ) {
+ var expected;
+ var data;
+ var acc;
+ var v;
+ var i;
+
+ acc = incrnanmvmr( 1, 3.14 );
+
+ data = [
+ NaN, // NaN (ignored)
+ 3.14, // 3.14
+ 3.14, // 3.14
+ NaN, // 3.14 (NaN ignored)
+ 3.14, // 3.14
+ 3.14, // 3.14
+ 3.14, // 3.14
+ NaN, // 3.14 (NaN ignored)
+ 3.14, // 3.14
+ 3.14, // 3.14
+ 3.14, // 3.14
+ NaN, // 3.14 (NaN ignored)
+ 3.14, // 3.14
+ 3.14, // 3.14
+ NaN, // 3.14 (NaN ignored)
+ NaN, // 3.14 (NaN ignored)
+ NaN, // 3.14 (NaN ignored)
+ NaN, // 3.14 (NaN ignored)
+ 3.14 // 3.14
+ ];
+ expected = [
+ null,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0,
+ 0.0
+ ];
+ for ( i = 0; i < data.length; i++ ) {
+ v = acc( data[ i ] );
+ t.equal( v, expected[ i ], 'returns expected value for window '+i );
+ t.equal( acc(), expected[ i ], 'returns expected value for window '+i );
+ }
+ t.end();
+});