incrskewness

Compute a corrected sample skewness incrementally.

The skewness for a random variable X is defined as

upper S k e w n e s s left-bracket upper X right-bracket equals normal upper E left-bracket left-parenthesis StartFraction upper X minus mu Over sigma EndFraction right-parenthesis cubed right-bracket

For a sample of n values, the sample skewness is

b 1 equals StartFraction m 3 Over s cubed EndFraction equals StartStartFraction StartFraction 1 Over n EndFraction sigma-summation Underscript i equals 0 Overscript n minus 1 Endscripts left-parenthesis x Subscript i Baseline minus x overbar right-parenthesis cubed OverOver left-parenthesis StartFraction 1 Over n minus 1 EndFraction sigma-summation Underscript i equals 0 Overscript n minus 1 Endscripts left-parenthesis x Subscript i Baseline minus x overbar right-parenthesis squared right-parenthesis Superscript 3 slash 2 Baseline EndEndFraction

where m_3 is the sample third central moment and s is the sample standard deviation.

An alternative definition for the sample skewness which includes an adjustment factor (and is the implemented definition) is

upper G 1 equals StartFraction n squared Over left-parenthesis n minus 1 right-parenthesis left-parenthesis n minus 2 right-parenthesis EndFraction StartFraction m 3 Over s cubed EndFraction equals StartFraction StartRoot n left-parenthesis n minus 1 right-parenthesis EndRoot Over n minus 2 EndFraction StartStartFraction StartFraction 1 Over n EndFraction sigma-summation Underscript i equals 0 Overscript n minus 1 Endscripts left-parenthesis x Subscript i Baseline minus x overbar right-parenthesis cubed OverOver left-parenthesis StartFraction 1 Over n EndFraction sigma-summation Underscript i equals 0 Overscript n minus 1 Endscripts left-parenthesis x Subscript i Baseline minus x overbar right-parenthesis squared right-parenthesis Superscript 3 slash 2 Baseline EndEndFraction

Usage

var incrskewness = require( '@stdlib/stats/incr/skewness' );

incrskewness()

Returns an accumulator function which incrementally computes a corrected sample skewness.

var accumulator = incrskewness();

accumulator( [x] )

If provided an input value x, the accumulator function returns an updated corrected sample skewness. If not provided an input value x, the accumulator function returns the current corrected sample skewness.

var accumulator = incrskewness();

var skewness = accumulator();
// returns null

skewness = accumulator( 2.0 );
// returns null

skewness = accumulator( -5.0 );
// returns null

skewness = accumulator( -10.0 );
// returns ~0.492

skewness = accumulator();
// returns ~0.492

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 incrskewness = require( '@stdlib/stats/incr/skewness' );

var accumulator;
var v;
var i;

// Initialize an accumulator:
accumulator = incrskewness();

// For each simulated datum, update the corrected sample skewness...
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.