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* @license Apache-2.0
*
* Copyright (c) 2018 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 isnan = require( '@stdlib/math/base/assert/is-nan' );
var gamma = require( '@stdlib/math/base/special/gamma' );
var pow = require( '@stdlib/math/base/special/pow' );
var PINF = require( '@stdlib/constants/float64/pinf' );
// MAIN //
/**
* Returns the excess kurtosis for a Fréchet distribution with shape `alpha`, scale `s`, and location `m`.
*
* @param {PositiveNumber} alpha - shape parameter
* @param {PositiveNumber} s - scale parameter
* @param {number} m - location parameter
* @returns {number} excess kurtosis
*
* @example
* var y = kurtosis( 5.0, 2.0, 0.0 );
* // returns ~45.092
*
* @example
* var y = kurtosis( 5.0, 2.0, -5.0 );
* // returns ~45.092
*
* @example
* var y = kurtosis( 3.8, 1.0, 0.0 );
* // returns Infinity
*
* @example
* var y = kurtosis( NaN, 1.0, 0.0 );
* // returns NaN
*
* @example
* var y = kurtosis( 1.0, NaN, 0.0 );
* // returns NaN
*
* @example
* var y = kurtosis( 1.0, 1.0, NaN );
* // returns NaN
*/
function kurtosis( alpha, s, m ) {
var out;
var g1;
var g2;
var g3;
var g4;
if (
isnan( alpha ) ||
isnan( s ) ||
isnan( m ) ||
alpha <= 0.0 ||
s <= 0.0
) {
return NaN;
}
if ( alpha <= 4.0 ) {
// Case: 0 < alpha <= 4
return PINF;
}
g1 = gamma( 1.0 - ( 1.0/alpha ) );
g2 = gamma( 1.0 - ( 2.0/alpha ) );
g3 = gamma( 1.0 - ( 3.0/alpha ) );
g4 = gamma( 1.0 - ( 4.0/alpha ) );
out = ( g4 - ( 4.0*g3*g1 ) + ( 3.0*g2*g2 ) ) / pow( g2 - ( g1*g1 ), 2.0 );
out -= 6.0;
return out;
}
// EXPORTS //
module.exports = kurtosis;
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