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  4. <title>The Anderson-Darling Test</title>
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    24. 

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    25. 

  25. <div class="section">
    26. 

  26. <div class="titlepage"><div><div><h2 class="title" style="clear: both">
    27. 

  27. <a name="math_toolkit.anderson_darling"></a><a class="link" href="anderson_darling.html" title="The Anderson-Darling Test">The Anderson-Darling Test</a>
    28. 

  28. </h2></div></div></div>
    29. 

  29. <h4>
    30. 

  30. <a name="math_toolkit.anderson_darling.h0"></a>
    31. 

  31.       <span class="phrase"><a name="math_toolkit.anderson_darling.synopsis"></a></span><a class="link" href="anderson_darling.html#math_toolkit.anderson_darling.synopsis">Synopsis</a>
    32. 

  32.     </h4>
    33. 

  33. <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">anderson_darling</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
    34. 

  34. 

  35. <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="special">{</span> <span class="identifier">statistics</span> <span class="special">{</span>
    36. 

  36. 

  37. <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">&gt;</span>
    38. 

  38. <span class="keyword">auto</span> <span class="identifier">anderson_darling_normality_statistic</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">,</span>
    39. 

  39.                                           <span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">mu</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">(),</span>
    40. 

  40.                                           <span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">sd</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">());</span>
    41. 

  41. 

  42. <span class="special">}}}</span>
    43. 

  43. </pre>
    44. 

  44. <h4>
    45. 

  45. <a name="math_toolkit.anderson_darling.h1"></a>
    46. 

  46.       <span class="phrase"><a name="math_toolkit.anderson_darling.background"></a></span><a class="link" href="anderson_darling.html#math_toolkit.anderson_darling.background">Background</a>
    47. 

  47.     </h4>
    48. 

  48. <p>
    49. 

  49.       The Anderson-Darling test for normality asks if a given sequence of numbers
    50. 

  50.       are drawn from a normal distribution by computing an integral over the empirical
    51. 

  51.       cumulative distribution function. The test statistic <span class="emphasis"><em>A</em></span><sup>2</sup> is
    52. 

  52.       given by
    53. 

  53.     </p>
    54. 

  54. <p>
    55. 

  55.       <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/anderson_darling_definition.svg" width="346" height="56"></object></span>
    56. 

  56.     </p>
    57. 

  57. <p>
    58. 

  58.       where <span class="emphasis"><em>F</em></span><sub><span class="emphasis"><em>n</em></span></sub> is the empirical cumulative
    59. 

  59.       distribution and <span class="emphasis"><em>F</em></span> is the CDF of the normal distribution.
    60. 

  60.     </p>
    61. 

  61. <p>
    62. 

  62.       The value returned by the routine is <span class="emphasis"><em>A</em></span><sup>2</sup>.
    63. 

  63.     </p>
    64. 

  64. <p>
    65. 

  65.       If <span class="emphasis"><em>A</em></span><sup>2</sup>/n converges to zero as <span class="emphasis"><em>n</em></span> goes
    66. 

  66.       to infinity, then the hypothesis that the data is normally distributed is supported
    67. 

  67.       by the test.
    68. 

  68.     </p>
    69. 

  69. <p>
    70. 

  70.       If <span class="emphasis"><em>A</em></span><sup>2</sup>/n converges to a finite positive value as <span class="emphasis"><em>n</em></span>
    71. 

  71.       goes to infinity, then the hypothesis is not supported by the test.
    72. 

  72.     </p>
    73. 

  73. <p>
    74. 

  74.       An example usage is demonstrated below:
    75. 

  75.     </p>
    76. 

  76. <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">vector</span><span class="special">&gt;</span>
    77. 

  77. <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">random</span><span class="special">&gt;</span>
    78. 

  78. <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
    79. 

  79. <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">anderson_darling</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
    80. 

  80. <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">anderson_darling_normality_statistic</span><span class="special">;</span>
    81. 

  81. <span class="identifier">std</span><span class="special">::</span><span class="identifier">random_device</span> <span class="identifier">rd</span><span class="special">;</span>
    82. 

  82. <span class="identifier">std</span><span class="special">::</span><span class="identifier">normal_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">dis</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span>
    83. 

  83. <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">v</span><span class="special">(</span><span class="number">8192</span><span class="special">);</span>
    84. 

  84. <span class="keyword">for</span> <span class="special">(</span><span class="keyword">auto</span> <span class="special">&amp;</span> <span class="identifier">x</span> <span class="special">:</span> <span class="identifier">v</span><span class="special">)</span> <span class="special">{</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">dis</span><span class="special">(</span><span class="identifier">rd</span><span class="special">);</span> <span class="special">}</span>
    85. 

  85. <span class="identifier">std</span><span class="special">::</span><span class="identifier">sort</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span>
    86. 

  86. <span class="keyword">double</span> <span class="identifier">presumed_mean</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span>
    87. 

  87. <span class="keyword">double</span> <span class="identifier">presumed_standard_deviation</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span>
    88. 

  88. <span class="keyword">double</span> <span class="identifier">Asq</span> <span class="special">=</span> <span class="identifier">anderson_darling_normality_statistic</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">presumed_mean</span><span class="special">,</span> <span class="identifier">presumed_standard_deviation</span><span class="special">);</span>
    89. 

  89. <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"A^2/n = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">Asq</span><span class="special">/</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
    90. 

  90. <span class="number">5.39e-05</span> <span class="comment">// should be small . . .</span>
    91. 

  91. <span class="comment">// Now use an incorrect hypothesis:</span>
    92. 

  92. <span class="identifier">presumed_mean</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span>
    93. 

  93. <span class="identifier">Asq</span> <span class="special">=</span> <span class="identifier">anderson_darling_normality_statistic</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">presumed_mean</span><span class="special">,</span> <span class="identifier">presumed_standard_deviation</span><span class="special">);</span>
    94. 

  94. <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"A^2/n = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">Asq</span><span class="special">/</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
    95. 

  95. <span class="number">7.41</span> <span class="comment">// should be somewhat large . . .</span>
    96. 

  96. </pre>
    97. 

  97. <p>
    98. 

  98.       The Anderson-Darling normality requires sorted data. If the data are not sorted
    99. 

  99.       an exception is thrown.
    100. 

  100.     </p>
    101. 

  101. <p>
    102. 

  102.       If you simply wish to know whether or not data is normally distributed, and
    103. 

  103.       not whether it is normally distributed with a presumed mean and variance, then
    104. 

  104.       you can call the function without the final two arguments, and the mean and
    105. 

  105.       variance will be estimated from the data themselves:
    106. 

  106.     </p>
    107. 

  107. <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">Asq</span> <span class="special">=</span> <span class="identifier">anderson_darling_normality_statistic</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
    108. 

  108. </pre>
    109. 

  109. <p>
    110. 

  110.       The following graph demonstrates the convergence of the test statistic. Each
    111. 

  111.       data point represents a vector of length <span class="emphasis"><em>n</em></span> which is filled
    112. 

  112.       with normally distributed data. The test statistic is computed over this vector,
    113. 

  113.       divided by <span class="emphasis"><em>n</em></span>, and passed to the natural logarithm. This
    114. 

  114.       exhibits the (admittedly slow) convergence of the integral to zero when the
    115. 

  115.       hypothesis is true.
    116. 

  116.     </p>
    117. 

  117. <p>
    118. 

  118.       <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/anderson_darling_simulation.svg"></object></span>
    119. 

  119.     </p>
    120. 

  120. <h4>
    121. 

  121. <a name="math_toolkit.anderson_darling.h2"></a>
    122. 

  122.       <span class="phrase"><a name="math_toolkit.anderson_darling.performance"></a></span><a class="link" href="anderson_darling.html#math_toolkit.anderson_darling.performance">Performance</a>
    123. 

  123.     </h4>
    124. 

  124. <pre class="programlisting"><span class="special">---------------------------------------------------------------</span>
    125. 

  125. <span class="identifier">Benchmark</span>                                              <span class="identifier">Time</span>
    126. 

  126. <span class="special">---------------------------------------------------------------</span>
    127. 

  127. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8</span>                <span class="number">224</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">136.509</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    128. 

  128. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16</span>               <span class="number">435</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">140.254</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    129. 

  129. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">32</span>               <span class="number">898</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">135.995</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    130. 

  130. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">64</span>              <span class="number">1773</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">137.675</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    131. 

  131. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">128</span>             <span class="number">3455</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">141.338</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    132. 

  132. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">256</span>             <span class="number">7001</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.488</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    133. 

  133. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">512</span>            <span class="number">13996</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.551</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    134. 

  134. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">1024</span>           <span class="number">28129</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">138.868</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    135. 

  135. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">2048</span>           <span class="number">55723</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">140.206</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    136. 

  136. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">4096</span>          <span class="number">112008</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.501</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    137. 

  137. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8192</span>          <span class="number">224643</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.11</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    138. 

  138. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16384</span>         <span class="number">450320</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">138.791</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    139. 

  139. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">32768</span>         <span class="number">896409</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.45</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    140. 

  140. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">65536</span>        <span class="number">1797800</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.058</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    141. 

  141. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">131072</span>       <span class="number">3604995</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">138.698</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    142. 

  142. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">262144</span>       <span class="number">7235625</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">138.207</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    143. 

  143. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">524288</span>      <span class="number">14502815</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">137.904</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    144. 

  144. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">1048576</span>     <span class="number">29058087</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">137.659</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    145. 

  145. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">2097152</span>     <span class="number">58470439</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">136.824</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    146. 

  146. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">4194304</span>    <span class="number">117476365</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">136.201</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    147. 

  147. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8388608</span>    <span class="number">239887895</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">133.397</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    148. 

  148. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16777216</span>   <span class="number">488787211</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">130.94</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    149. 

  149. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;</span><span class="identifier">_BigO</span>           <span class="number">28.96</span> <span class="identifier">N</span>         <span class="number">28.96</span> <span class="identifier">N</span>
    150. 

  150. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span>               <span class="number">470</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">129.733</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    151. 

  151. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16</span>              <span class="number">911</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">133.989</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    152. 

  152. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32</span>             <span class="number">1773</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">137.723</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    153. 

  153. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span>             <span class="number">3368</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">144.966</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    154. 

  154. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">128</span>            <span class="number">6627</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">147.357</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    155. 

  155. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span>           <span class="number">12458</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">156.777</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    156. 

  156. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span>           <span class="number">23060</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">169.395</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    157. 

  157. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span>          <span class="number">44529</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">175.45</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    158. 

  158. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span>          <span class="number">88735</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">176.087</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    159. 

  159. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span>         <span class="number">175583</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">177.978</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    160. 

  160. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span>         <span class="number">348042</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">179.577</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    161. 

  161. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span>        <span class="number">701439</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">178.206</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    162. 

  162. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span>       <span class="number">1394597</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">179.262</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    163. 

  163. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span>       <span class="number">2777943</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">179.994</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    164. 

  164. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span>      <span class="number">5571455</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">179.487</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    165. 

  165. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span>     <span class="number">11161456</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">179.193</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    166. 

  166. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span>     <span class="number">22048950</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.417</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    167. 

  167. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span>    <span class="number">44094409</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.429</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    168. 

  168. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2097152</span>    <span class="number">88300185</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.199</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    169. 

  169. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4194304</span>   <span class="number">176140378</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.678</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    170. 

  170. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8388608</span>   <span class="number">352102955</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.769</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    171. 

  171. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16777216</span>  <span class="number">706160246</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.267</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    172. 

  172. <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span>          <span class="number">42.06</span> <span class="identifier">N</span>
    173. 

  173. </pre>
    174. 

  174. <h4>
    175. 

  175. <a name="math_toolkit.anderson_darling.h3"></a>
    176. 

  176.       <span class="phrase"><a name="math_toolkit.anderson_darling.caveats"></a></span><a class="link" href="anderson_darling.html#math_toolkit.anderson_darling.caveats">Caveats</a>
    177. 

  177.     </h4>
    178. 

  178. <p>
    179. 

  179.       Some authors, including <a href="https://www.itl.nist.gov/div898/handbook/eda/section3/eda35e.htm" target="_top">NIST</a>,
    180. 

  180.       give the following definition of the Anderson-Darling test statistic:
    181. 

  181.     </p>
    182. 

  182. <p>
    183. 

  183.       <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/alternative_anderson_darling_definition.svg" width="461" height="54"></object></span>
    184. 

  184.     </p>
    185. 

  185. <p>
    186. 

  186.       This is an approximation to the quadrature sum we use as our definition. Boost.Math
    187. 

  187.       <span class="emphasis"><em>does not compute this quantity</em></span>. (However, with a sufficiently
    188. 

  188.       large amount of data the two definitions seem to agree to two digits, so the
    189. 

  189.       importance of making a clear distinction between the two is unclear.) Our computation
    190. 

  190.       of the Anderson-Darling test statistic agrees with Mathematica.
    191. 

  191.     </p>
    192. 

  192. </div>
    193. 

  193. <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
    194. 

  194. <td align="left"></td>
    195. 

  195. <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
    196. 

  196.       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
    197. 

  197.       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
    198. 

  198.       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
    199. 

  199.       Daryle Walker and Xiaogang Zhang<p>
    200. 

  200.         Distributed under the Boost Software License, Version 1.0. (See accompanying
    201. 

  201.         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
    202. 

  202.       </p>
    203. 

  203. </div></td>
    204. 

  204. </tr></table>
    205. 

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