@stdlib/stats/base/dists/invgamma/README.md

Standard library statistical functions.

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# Inverse Gamma

> Inverse gamma distribution.

<section class="usage">

## Usage

```javascript
var invgamma = require( '@stdlib/stats/base/dists/invgamma' );
```

#### invgamma

Inverse gamma distribution.

```javascript
var dist = invgamma;
// returns {...}
```

The namespace contains the following distribution functions:

<!-- <toc pattern="*+(cdf|pdf|mgf|quantile)*"> -->

<div class="namespace-toc">

-   <span class="signature">[`cdf( x, alpha, beta )`][@stdlib/stats/base/dists/invgamma/cdf]</span><span class="delimiter">: </span><span class="description">inverse Gamma distribution cumulative distribution function.</span>
-   <span class="signature">[`logpdf( x, alpha, beta )`][@stdlib/stats/base/dists/invgamma/logpdf]</span><span class="delimiter">: </span><span class="description">evaluate the natural logarithm of the probability density function (PDF) for an inverse gamma distribution.</span>
-   <span class="signature">[`pdf( x, alpha, beta )`][@stdlib/stats/base/dists/invgamma/pdf]</span><span class="delimiter">: </span><span class="description">inverse gamma distribution probability density function (PDF).</span>
-   <span class="signature">[`quantile( p, alpha, beta )`][@stdlib/stats/base/dists/invgamma/quantile]</span><span class="delimiter">: </span><span class="description">inverse gamma distribution quantile function.</span>

</div>

<!-- </toc> -->

The namespace contains the following functions for calculating distribution properties:

<!-- <toc pattern="*+(entropy|kurtosis|mean|median|mode|skewness|stdev|variance)*"> -->

<div class="namespace-toc">

-   <span class="signature">[`entropy( alpha, beta )`][@stdlib/stats/base/dists/invgamma/entropy]</span><span class="delimiter">: </span><span class="description">inverse gamma distribution differential entropy.</span>
-   <span class="signature">[`kurtosis( alpha, beta )`][@stdlib/stats/base/dists/invgamma/kurtosis]</span><span class="delimiter">: </span><span class="description">inverse gamma distribution excess kurtosis.</span>
-   <span class="signature">[`mean( alpha, beta )`][@stdlib/stats/base/dists/invgamma/mean]</span><span class="delimiter">: </span><span class="description">inverse gamma distribution expected value.</span>
-   <span class="signature">[`mode( alpha, beta )`][@stdlib/stats/base/dists/invgamma/mode]</span><span class="delimiter">: </span><span class="description">inverse gamma distribution mode.</span>
-   <span class="signature">[`skewness( alpha, beta )`][@stdlib/stats/base/dists/invgamma/skewness]</span><span class="delimiter">: </span><span class="description">inverse gamma distribution skewness.</span>
-   <span class="signature">[`stdev( alpha, beta )`][@stdlib/stats/base/dists/invgamma/stdev]</span><span class="delimiter">: </span><span class="description">inverse gamma distribution standard deviation.</span>
-   <span class="signature">[`variance( alpha, beta )`][@stdlib/stats/base/dists/invgamma/variance]</span><span class="delimiter">: </span><span class="description">inverse gamma distribution variance.</span>

</div>

<!-- </toc> -->

The namespace contains a constructor function for creating a [inverse gamma][invgamma-distribution] distribution object.

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<div class="namespace-toc">

-   <span class="signature">[`InvGamma( [alpha, beta] )`][@stdlib/stats/base/dists/invgamma/ctor]</span><span class="delimiter">: </span><span class="description">inverse gamma distribution constructor.</span>

</div>

<!-- </toc> -->

```javascript
var InvGamma = require( '@stdlib/stats/base/dists/invgamma' ).InvGamma;

var dist = new InvGamma( 2.0, 4.0 );

var y = dist.cdf( 0.5 );
// returns ~0.003
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var invgammaRandomFactory = require( '@stdlib/random/base/invgamma' ).factory;
var filledarrayBy = require( '@stdlib/array/filled-by' );
var variance = require( '@stdlib/stats/base/variance' );
var linspace = require( '@stdlib/array/base/linspace' );
var gamma = require( '@stdlib/stats/base/dists/gamma' );
var mean = require( '@stdlib/stats/base/mean' );
var abs = require( '@stdlib/math/base/special/abs' );
var invgamma = require( '@stdlib/stats/base/dists/invgamma' );

// Define the shape and scale parameters:
var alpha = 5.0; // shape parameter (α)
var beta = 1.0;  // scale parameter (β)

// Generate an array of x values:
var x = linspace( 0.01, 3.0, 100 );

// Compute the PDF for each x:
var invgammaPDF = invgamma.pdf.factory( alpha, beta );
var pdf = filledarrayBy( x.length, 'float64', invgammaPDF );

// Compute the CDF for each x:
var invgammaCDF = invgamma.cdf.factory( alpha, beta );
var cdf = filledarrayBy( x.length, 'float64', invgammaCDF );

// Output the PDF and CDF values:
console.log( 'x values: %s', x );
console.log( 'PDF values: %s', pdf );
console.log( 'CDF values: %s', cdf );

// Compute statistical properties:
var theoreticalMean = invgamma.mean( alpha, beta );
var theoreticalVariance = invgamma.variance( alpha, beta );
var theoreticalSkewness = invgamma.skewness( alpha, beta );
var theoreticalKurtosis = invgamma.kurtosis( alpha, beta );

console.log( 'Theoretical Mean: %s', theoreticalMean );
console.log( 'Theoretical Variance: %s', theoreticalVariance );
console.log( 'Skewness: %s', theoreticalSkewness );
console.log( 'Kurtosis: %s', theoreticalKurtosis );

// Generate random samples from the inverse gamma distribution:
var rinvGamma = invgammaRandomFactory( alpha, beta );
var n = 1000;
var samples = filledarrayBy( n, 'float64', rinvGamma );

// Compute sample mean and variance:
var sampleMean = mean( n, samples, 1 );
var sampleVariance = variance( n, 1, samples, 1 );

console.log( 'Sample Mean: %s', sampleMean );
console.log( 'Sample Variance: %s', sampleVariance );

// Compare sample statistics to theoretical values:
console.log( 'Difference in Mean: %s', abs( theoreticalMean - sampleMean ) );
console.log( 'Difference in Variance: %s', abs( theoreticalVariance - sampleVariance ) );

// Demonstrate the relationship between inverse gamma and gamma distributions:
var y = 0.5;
var invGammaCDF = invgamma.cdf( y, alpha, beta );
var gammaCDF = 1.0 - gamma.cdf( 1.0 / y, alpha, 1.0 / beta );

console.log( 'Inverse Gamma CDF at y = %s: %s', y, invGammaCDF );
console.log( '1 - Gamma CDF at 1/y = %s: %s', 1 / y, gammaCDF );
console.log( 'Difference: %s', abs( invGammaCDF - gammaCDF ) );
```

</section>

<!-- /.examples -->

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<section class="related">

</section>

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<section class="links">

[invgamma-distribution]: https://en.wikipedia.org/wiki/Inverse-gamma_distribution

<!-- <toc-links> -->

[@stdlib/stats/base/dists/invgamma/ctor]: https://github.com/stdlib-js/stats/tree/main/base/dists/invgamma/ctor

[@stdlib/stats/base/dists/invgamma/entropy]: https://github.com/stdlib-js/stats/tree/main/base/dists/invgamma/entropy

[@stdlib/stats/base/dists/invgamma/kurtosis]: https://github.com/stdlib-js/stats/tree/main/base/dists/invgamma/kurtosis

[@stdlib/stats/base/dists/invgamma/mean]: https://github.com/stdlib-js/stats/tree/main/base/dists/invgamma/mean

[@stdlib/stats/base/dists/invgamma/mode]: https://github.com/stdlib-js/stats/tree/main/base/dists/invgamma/mode

[@stdlib/stats/base/dists/invgamma/skewness]: https://github.com/stdlib-js/stats/tree/main/base/dists/invgamma/skewness

[@stdlib/stats/base/dists/invgamma/stdev]: https://github.com/stdlib-js/stats/tree/main/base/dists/invgamma/stdev

[@stdlib/stats/base/dists/invgamma/variance]: https://github.com/stdlib-js/stats/tree/main/base/dists/invgamma/variance

[@stdlib/stats/base/dists/invgamma/cdf]: https://github.com/stdlib-js/stats/tree/main/base/dists/invgamma/cdf

[@stdlib/stats/base/dists/invgamma/logpdf]: https://github.com/stdlib-js/stats/tree/main/base/dists/invgamma/logpdf

[@stdlib/stats/base/dists/invgamma/pdf]: https://github.com/stdlib-js/stats/tree/main/base/dists/invgamma/pdf

[@stdlib/stats/base/dists/invgamma/quantile]: https://github.com/stdlib-js/stats/tree/main/base/dists/invgamma/quantile

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</section>

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