# Kolmogorov-Smirnov Goodness-of-Fit Test

One-sample Kolmogorov-Smirnov goodness-of-fit test.

## Usage

``````var kstest = require( '@stdlib/math/stats/kstest' );
``````

### kstest( x, y[, ...params][, opts] )

For a numeric array or typed array `x`, a Kolmogorov-Smirnov goodness-of-fit is computed for the null hypothesis that the values of `x` come from the distribution specified by `y`. `y` can be either a string with the name of the distribution to test against, or a function. In the latter case, `y` is expected to be the cumulative distribution function (CDF) of the distribution to test against, with its first parameter being the value at which to evaluate the CDF and the remaining parameters constituting the parameters of the distribution. The parameters of the distribution are passed as additional arguments after `y` from `kstest` to the chosen CDF. The function returns an object holding the calculated test statistic `statistic` and the `pValue` of the test.

``````var factory = require( '@stdlib/random/base/uniform' ).factory;
var runif;
var out;
var x;
var i;

runif = factory( 0.0, 1.0, {
'seed': 8798
});

x = new Array( 100 );
for ( i = 0; i < x.length; i++ ) {
x[ i ] = runif();
}
out = kstest( x, 'uniform', 0.0, 1.0 );
// returns { 'pValue': ~0.703, 'statistic': ~0.069, ... }
``````

The returned object comes with a `.print()` method which when invoked will print a formatted output of the hypothesis test results.

``````console.log( out.print() );
/* e.g., =>
Kolmogorov-Smirnov goodness-of-fit test.

Null hypothesis: the CDF of `x` is equal equal to the reference CDF.

pValue: 0.7039
statistic: 0.0689

Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/
``````

The function accepts the following `options`:

• alpha: `number` in the interval `[0,1]` giving the significance level of the hypothesis test. Default: `0.05`.
• alternative: Either `two-sided`, `less` or `greater`. Indicates whether the alternative hypothesis is that the true distribution of `x` is not equal to the reference distribution specified by `y` (`two-sided`), whether it is `less` than the reference distribution or `greater` than the reference distribution. Default: `two-sided`.
• sorted: `boolean` indicating if the `x` array is in sorted order (ascending). Default: `false`.

By default, the test is carried out at a significance level of `0.05`. To choose a custom significance level, set the `alpha` option.

``````out = kstest( x, 'uniform', 0.0, 1.0, {
'alpha': 0.1
});
console.log( out.print() );
/* e.g., =>
Kolmogorov-Smirnov goodness-of-fit test.

Null hypothesis: the CDF of `x` is equal equal to the reference CDF.

pValue: 0.7039
statistic: 0.0689

Test Decision: Fail to reject null in favor of alternative at 10% significance level
*/
``````

By default, the function tests the null hypothesis that the true distribution of `x` and the reference distribution `y` are equal to each other against the alternative that they are not equal. To carry out a one-sided hypothesis test, set the `alternative` option to either `less` or `greater`.

``````var factory = require( '@stdlib/random/base/uniform' ).factory;
var runif;
var out;
var x;
var i;

runif = factory( 0.0, 1.0, {
'seed': 8798
});

x = new Array( 100 );
for ( i = 0; i < x.length; i++ ) {
x[ i ] = runif();
}

out = kstest( x, 'uniform', 0.0, 1.0, {
'alternative': 'less'
});
// returns { 'pValue': ~0.358, 'statistic': ~0.07, ... }

out = kstest( x, 'uniform', 0.0, 1.0, {
'alternative': 'greater'
});
// returns { 'pValue': ~0.907, 'statistic': ~0.02, ... }
``````

To perform the Kolmogorov-Smirnov test, the data has to be sorted in ascending order. If the data in `x` are already sorted, set the `sorted` option to `true` to speed up computation.

``````x = [ 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 ];

out = kstest( x, 'uniform', 0.0, 1.0, {
'sorted': true
});
// returns { 'pValue': ~1, 'statistic': 0.1, ... }
``````

## Examples

``````var kstest = require( '@stdlib/math/stats/kstest' );
var rnorm = require( '@stdlib/random/base/normal' ).factory({
'seed': 4839
});

var table;
var out;
var i;
var x;

// Values drawn from a Normal(3,1) distribution
x = new Array( 100 );
for ( i = 0; i < 100; i++ ) {
x[ i ] = rnorm( 3.0, 1.0 );
}

// Test against N(0,1)
out = kstest( x, 'normal', 0.0, 1.0 );
table = out.print();
/* e.g., returns
Kolmogorov-Smirnov goodness-of-fit test.

Null hypothesis: the CDF of `x` is equal to the reference CDF.

statistic: 0.847
pValue: 0

Test Decision: Reject null in favor of alternative at 5% significance level
*/

// Test against N(3,1)
out = kstest( x, 'normal', 3.0, 1.0 );
table = out.print();
/* e.g., returns
Kolmogorov-Smirnov goodness-of-fit test.

Null hypothesis: the CDF of `x` is equal to the reference CDF.

statistic: 0.0733
pValue: 0.6282

Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/
``````