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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 pow = require( '@stdlib/math/base/special/pow' );
// MAIN //
/**
* Returns the excess kurtosis of a Pareto (Type I) distribution.
*
* @param {PositiveNumber} alpha - shape parameter
* @param {PositiveNumber} beta - scale parameter
* @returns {PositiveNumber} excess kurtosis
*
* @example
* var v = kurtosis( 5.0, 1.0 );
* // returns ~70.8
*
* @example
* var v = kurtosis( 7.0, 12.0 );
* // returns ~24.857
*
* @example
* var v = kurtosis( 8.0, 2.0 );
* // returns ~19.725
*
* @example
* var v = kurtosis( 1.0, -0.1 );
* // returns NaN
*
* @example
* var v = kurtosis( -0.1, 1.0 );
* // returns NaN
*
* @example
* var v = kurtosis( 2.0, NaN );
* // returns NaN
*
* @example
* var v = kurtosis( NaN, 2.0 );
* // returns NaN
*/
function kurtosis( alpha, beta ) {
var out;
if (
isnan( alpha ) ||
alpha <= 4.0 ||
isnan( beta ) ||
beta <= 0.0
) {
return NaN;
}
out = 6.0 * ( pow( alpha, 3.0 ) + pow( alpha, 2.0 ) - ( 6.0*alpha ) - 2.0 );
out /= alpha * ( alpha-3.0 ) * ( alpha-4.0 );
return out;
}
// EXPORTS //
module.exports = kurtosis;
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