# incrkurtosis

Compute a corrected sample excess kurtosis incrementally.

The kurtosis for a random variable `X`

is defined as

Using a univariate normal distribution as the standard of comparison, the excess kurtosis is the kurtosis minus `3`

.

For a sample of `n`

values, the sample excess kurtosis is

where `m_4`

is the sample fourth central moment and `m_2`

is the sample second central moment.

The previous equation is, however, a biased estimator of the population excess kurtosis. An alternative estimator which is unbiased under normality is

## Usage

```
var incrkurtosis = require( '@stdlib/stats/incr/kurtosis' );
```

#### incrkurtosis()

Returns an accumulator `function`

which incrementally computes a corrected sample excess kurtosis.

```
var accumulator = incrkurtosis();
```

#### accumulator( [x] )

If provided an input value `x`

, the accumulator function returns an updated corrected sample excess kurtosis. If not provided an input value `x`

, the accumulator function returns the current corrected sample excess kurtosis.

```
var accumulator = incrkurtosis();
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
```

## Notes

- 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**all**future invocations. If non-numeric inputs are possible, you are advised to type check and handle accordingly**before**passing the value to the accumulator function.

## Examples

```
var randu = require( '@stdlib/random/base/randu' );
var incrkurtosis = require( '@stdlib/stats/incr/kurtosis' );
var accumulator;
var v;
var i;
// Initialize an accumulator:
accumulator = incrkurtosis();
// For each simulated datum, update the 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.