time-to-botec

Benchmark sampling in different programming languages
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README.md (4489B)

      1 <!--
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      3 @license Apache-2.0
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      5 Copyright (c) 2018 The Stdlib Authors.
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      7 Licensed under the Apache License, Version 2.0 (the "License");
      8 you may not use this file except in compliance with the License.
      9 You may obtain a copy of the License at
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     11    http://www.apache.org/licenses/LICENSE-2.0
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     13 Unless required by applicable law or agreed to in writing, software
     14 distributed under the License is distributed on an "AS IS" BASIS,
     15 WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
     16 See the License for the specific language governing permissions and
     17 limitations under the License.
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     19 -->
     20 
     21 # Kurtosis
     22 
     23 > [Beta prime][betaprime-distribution] distribution [excess kurtosis][kurtosis].
     24 
     25 <!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->
     26 
     27 <section class="intro">
     28 
     29 The [excess kurtosis][kurtosis] for a [beta prime][betaprime-distribution] random variable with first shape parameter `α` and second shape parameter `β` is
     30 
     31 <!-- <equation class="equation" label="eq:betaprime_kurtosis" align="center" raw="\operatorname{Kurt}\left( X \right) = 6{\frac{\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}}" alt="Kurtosis for a beta prime distribution."> -->
     32 
     33 <div class="equation" align="center" data-raw-text="\operatorname{Kurt}\left( X \right) = 6{\frac{\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}}" data-equation="eq:betaprime_kurtosis">
     34     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/betaprime/kurtosis/docs/img/equation_betaprime_kurtosis.svg" alt="Kurtosis for a beta prime distribution.">
     35     <br>
     36 </div>
     37 
     38 <!-- </equation> -->
     39 
     40 when `α > 0` and `β > 4`. Otherwise, the excess kurtosis is not defined.
     41 
     42 </section>
     43 
     44 <!-- /.intro -->
     45 
     46 <!-- Package usage documentation. -->
     47 
     48 <section class="usage">
     49 
     50 ## Usage
     51 
     52 ```javascript
     53 var kurtosis = require( '@stdlib/stats/base/dists/betaprime/kurtosis' );
     54 ```
     55 
     56 #### kurtosis( alpha, beta )
     57 
     58 Returns the [excess kurtosis][kurtosis] of a [beta prime][betaprime-distribution] distribution with parameters `alpha` (first shape parameter) and `beta` (second shape parameter).
     59 
     60 ```javascript
     61 var v = kurtosis( 2.0, 5.0 );
     62 // returns 54.0
     63 
     64 v = kurtosis( 4.0, 12.0 );
     65 // returns ~5.764
     66 
     67 v = kurtosis( 12.0, 6.0 );
     68 // returns ~19.49
     69 ```
     70 
     71 If provided `NaN` as any argument, the function returns `NaN`.
     72 
     73 ```javascript
     74 var v = kurtosis( NaN, 5.0 );
     75 // returns NaN
     76 
     77 v = kurtosis( 2.0, NaN );
     78 // returns NaN
     79 ```
     80 
     81 If provided `alpha <= 0`, the function returns `NaN`.
     82 
     83 ```javascript
     84 var v = kurtosis( 0.0, 5.0 );
     85 // returns NaN
     86 
     87 v = kurtosis( -1.0, 5.0 );
     88 // returns NaN
     89 ```
     90 
     91 If provided `beta <= 4`, the function returns `NaN`.
     92 
     93 ```javascript
     94 var v = kurtosis( 1.0, 3.5 );
     95 // returns NaN
     96 
     97 v = kurtosis( 1.0, 2.0 );
     98 // returns NaN
     99 
    100 v = kurtosis( 1.0, -1.0 );
    101 // returns NaN
    102 ```
    103 
    104 </section>
    105 
    106 <!-- /.usage -->
    107 
    108 <!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
    109 
    110 <section class="notes">
    111 
    112 </section>
    113 
    114 <!-- /.notes -->
    115 
    116 <!-- Package usage examples. -->
    117 
    118 <section class="examples">
    119 
    120 ## Examples
    121 
    122 <!-- eslint no-undef: "error" -->
    123 
    124 ```javascript
    125 var randu = require( '@stdlib/random/base/randu' );
    126 var EPS = require( '@stdlib/constants/float64/eps' );
    127 var kurtosis = require( '@stdlib/stats/base/dists/betaprime/kurtosis' );
    128 
    129 var alpha;
    130 var beta;
    131 var v;
    132 var i;
    133 
    134 for ( i = 0; i < 10; i++ ) {
    135     alpha = ( randu()*10.0 ) + EPS;
    136     beta = ( randu()*10.0 ) + 4.0 + EPS;
    137     v = kurtosis( alpha, beta );
    138     console.log( 'α: %d, β: %d, Kurt(X;α,β): %d', alpha.toFixed( 4 ), beta.toFixed( 4 ), v.toFixed( 4 ) );
    139 }
    140 ```
    141 
    142 </section>
    143 
    144 <!-- /.examples -->
    145 
    146 <!-- Section to include cited references. If references are included, add a horizontal rule *before* the section. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
    147 
    148 <section class="references">
    149 
    150 </section>
    151 
    152 <!-- /.references -->
    153 
    154 <!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
    155 
    156 <section class="links">
    157 
    158 [betaprime-distribution]: https://en.wikipedia.org/wiki/Beta_prime_distribution
    159 
    160 [kurtosis]: https://en.wikipedia.org/wiki/Kurtosis
    161 
    162 </section>
    163 
    164 <!-- /.links -->