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| <h2 title="Class BoundsOnBinomialProportions" class="title">Class BoundsOnBinomialProportions</h2> |
| </div> |
| <div class="contentContainer"> |
| <ul class="inheritance"> |
| <li><a href="https://docs.oracle.com/javase/8/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">java.lang.Object</a></li> |
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| <li>com.yahoo.sketches.BoundsOnBinomialProportions</li> |
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| <pre>public final class <span class="typeNameLabel">BoundsOnBinomialProportions</span> |
| extends <a href="https://docs.oracle.com/javase/8/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></pre> |
| <div class="block">Confidence intervals for binomial proportions. |
| |
| <p>This class computes an approximation to the Clopper-Pearson confidence interval |
| for a binomial proportion. Exact Clopper-Pearson intervals are strictly |
| conservative, but these approximations are not.</p> |
| |
| <p>The main inputs are numbers <i>n</i> and <i>k</i>, which are not the same as other things |
| that are called <i>n</i> and <i>k</i> in our sketching library. There is also a third |
| parameter, numStdDev, that specifies the desired confidence level.</p> |
| <ul> |
| <li><i>n</i> is the number of independent randomized trials. It is given and therefore known. |
| </li> |
| <li><i>p</i> is the probability of a trial being a success. It is unknown.</li> |
| <li><i>k</i> is the number of trials (out of <i>n</i>) that turn out to be successes. It is |
| a random variable governed by a binomial distribution. After any given |
| batch of <i>n</i> independent trials, the random variable <i>k</i> has a specific |
| value which is observed and is therefore known.</li> |
| <li><i>pHat</i> = <i>k</i> / <i>n</i> is an unbiased estimate of the unknown success |
| probability <i>p</i>.</li> |
| </ul> |
| |
| <p>Alternatively, consider a coin with unknown heads probability <i>p</i>. Where |
| <i>n</i> is the number of independent flips of that coin, and <i>k</i> is the number |
| of times that the coin comes up heads during a given batch of <i>n</i> flips. |
| This class computes a frequentist confidence interval [lowerBoundOnP, upperBoundOnP] for the |
| unknown <i>p</i>.</p> |
| |
| <p>Conceptually, the desired confidence level is specified by a tail probability delta.</p> |
| |
| <p>Ideally, over a large ensemble of independent batches of trials, |
| the fraction of batches in which the true <i>p</i> lies below lowerBoundOnP would be at most |
| delta, and the fraction of batches in which the true <i>p</i> lies above upperBoundOnP |
| would also be at most delta. |
| |
| <p>Setting aside the philosophical difficulties attaching to that statement, it isn't quite |
| true because we are approximating the Clopper-Pearson interval.</p> |
| |
| <p>Finally, we point out that in this class's interface, the confidence parameter delta is |
| not specified directly, but rather through a "number of standard deviations" numStdDev. |
| The library effectively converts that to a delta via delta = normalCDF (-1.0 * numStdDev).</p> |
| |
| <p>It is perhaps worth emphasizing that the library is NOT merely adding and subtracting |
| numStdDev standard deviations to the estimate. It is doing something better, that to some |
| extent accounts for the fact that the binomial distribution has a non-gaussian shape.</p> |
| |
| <p>In particular, it is using an approximation to the inverse of the incomplete beta function |
| that appears as formula 26.5.22 on page 945 of the "Handbook of Mathematical Functions" |
| by Abramowitz and Stegun.</p></div> |
| <dl> |
| <dt><span class="simpleTagLabel">Author:</span></dt> |
| <dd>Kevin Lang</dd> |
| </dl> |
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| <th class="colLast" scope="col">Method and Description</th> |
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| <td class="colFirst"><code>static double</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../com/yahoo/sketches/BoundsOnBinomialProportions.html#approximateLowerBoundOnP-long-long-double-">approximateLowerBoundOnP</a></span>(long n, |
| long k, |
| double numStdDevs)</code> |
| <div class="block">Computes lower bound of approximate Clopper-Pearson confidence interval for a binomial |
| proportion.</div> |
| </td> |
| </tr> |
| <tr id="i1" class="rowColor"> |
| <td class="colFirst"><code>static double</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../com/yahoo/sketches/BoundsOnBinomialProportions.html#approximateUpperBoundOnP-long-long-double-">approximateUpperBoundOnP</a></span>(long n, |
| long k, |
| double numStdDevs)</code> |
| <div class="block">Computes upper bound of approximate Clopper-Pearson confidence interval for a binomial |
| proportion.</div> |
| </td> |
| </tr> |
| <tr id="i2" class="altColor"> |
| <td class="colFirst"><code>static double</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../com/yahoo/sketches/BoundsOnBinomialProportions.html#erf-double-">erf</a></span>(double x)</code> |
| <div class="block">Computes an approximation to the erf() function.</div> |
| </td> |
| </tr> |
| <tr id="i3" class="rowColor"> |
| <td class="colFirst"><code>static double</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../com/yahoo/sketches/BoundsOnBinomialProportions.html#estimateUnknownP-long-long-">estimateUnknownP</a></span>(long n, |
| long k)</code> |
| <div class="block">Computes an estimate of an unknown binomial proportion.</div> |
| </td> |
| </tr> |
| <tr id="i4" class="altColor"> |
| <td class="colFirst"><code>static double</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../com/yahoo/sketches/BoundsOnBinomialProportions.html#normalCDF-double-">normalCDF</a></span>(double x)</code> |
| <div class="block">Computes an approximation to normalCDF(x).</div> |
| </td> |
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| <h3>Methods inherited from class java.lang.<a href="https://docs.oracle.com/javase/8/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></h3> |
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| <h4>approximateLowerBoundOnP</h4> |
| <pre>public static double approximateLowerBoundOnP(long n, |
| long k, |
| double numStdDevs)</pre> |
| <div class="block">Computes lower bound of approximate Clopper-Pearson confidence interval for a binomial |
| proportion. |
| |
| <p>Implementation Notes:<br> |
| The approximateLowerBoundOnP is defined with respect to the right tail of the binomial |
| distribution.</p> |
| <ul> |
| <li>We want to solve for the <i>p</i> for which sum<sub><i>j,k,n</i></sub>bino(<i>j;n,p</i>) |
| = delta.</li> |
| <li>We now restate that in terms of the left tail.</li> |
| <li>We want to solve for the p for which sum<sub><i>j,0,(k-1)</i></sub>bino(<i>j;n,p</i>) |
| = 1 - delta.</li> |
| <li>Define <i>x</i> = 1-<i>p</i>.</li> |
| <li>We want to solve for the <i>x</i> for which I<sub><i>x(n-k+1,k)</i></sub> = 1 - delta.</li> |
| <li>We specify 1-delta via numStdDevs through the right tail of the standard normal |
| distribution.</li> |
| <li>Smaller values of numStdDevs correspond to bigger values of 1-delta and hence to smaller |
| values of delta. In fact, usefully small values of delta correspond to negative values of |
| numStdDevs.</li> |
| <li>return <i>p</i> = 1-<i>x</i>.</li> |
| </ul></div> |
| <dl> |
| <dt><span class="paramLabel">Parameters:</span></dt> |
| <dd><code>n</code> - is the number of trials. Must be non-negative.</dd> |
| <dd><code>k</code> - is the number of successes. Must be non-negative, and cannot exceed n.</dd> |
| <dd><code>numStdDevs</code> - the number of standard deviations defining the confidence interval</dd> |
| <dt><span class="returnLabel">Returns:</span></dt> |
| <dd>the lower bound of the approximate Clopper-Pearson confidence interval for the |
| unknown success probability.</dd> |
| </dl> |
| </li> |
| </ul> |
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| <h4>approximateUpperBoundOnP</h4> |
| <pre>public static double approximateUpperBoundOnP(long n, |
| long k, |
| double numStdDevs)</pre> |
| <div class="block">Computes upper bound of approximate Clopper-Pearson confidence interval for a binomial |
| proportion. |
| |
| <p>Implementation Notes:<br> |
| The approximateUpperBoundOnP is defined with respect to the left tail of the binomial |
| distribution.</p> |
| <ul> |
| <li>We want to solve for the <i>p</i> for which sum<sub><i>j,0,k</i></sub>bino(<i>j;n,p</i>) |
| = delta.</li> |
| <li>Define <i>x</i> = 1-<i>p</i>.</li> |
| <li>We want to solve for the <i>x</i> for which I<sub><i>x(n-k,k+1)</i></sub> = delta.</li> |
| <li>We specify delta via numStdDevs through the right tail of the standard normal |
| distribution.</li> |
| <li>Bigger values of numStdDevs correspond to smaller values of delta.</li> |
| <li>return <i>p</i> = 1-<i>x</i>.</li> |
| </ul></div> |
| <dl> |
| <dt><span class="paramLabel">Parameters:</span></dt> |
| <dd><code>n</code> - is the number of trials. Must be non-negative.</dd> |
| <dd><code>k</code> - is the number of successes. Must be non-negative, and cannot exceed <i>n</i>.</dd> |
| <dd><code>numStdDevs</code> - the number of standard deviations defining the confidence interval</dd> |
| <dt><span class="returnLabel">Returns:</span></dt> |
| <dd>the upper bound of the approximate Clopper-Pearson confidence interval for the |
| unknown success probability.</dd> |
| </dl> |
| </li> |
| </ul> |
| <a name="estimateUnknownP-long-long-"> |
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| <h4>estimateUnknownP</h4> |
| <pre>public static double estimateUnknownP(long n, |
| long k)</pre> |
| <div class="block">Computes an estimate of an unknown binomial proportion.</div> |
| <dl> |
| <dt><span class="paramLabel">Parameters:</span></dt> |
| <dd><code>n</code> - is the number of trials. Must be non-negative.</dd> |
| <dd><code>k</code> - is the number of successes. Must be non-negative, and cannot exceed n.</dd> |
| <dt><span class="returnLabel">Returns:</span></dt> |
| <dd>the estimate of the unknown binomial proportion.</dd> |
| </dl> |
| </li> |
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| <h4>erf</h4> |
| <pre>public static double erf(double x)</pre> |
| <div class="block">Computes an approximation to the erf() function.</div> |
| <dl> |
| <dt><span class="paramLabel">Parameters:</span></dt> |
| <dd><code>x</code> - is the input to the erf function</dd> |
| <dt><span class="returnLabel">Returns:</span></dt> |
| <dd>returns erf(x), accurate to roughly 7 decimal digits.</dd> |
| </dl> |
| </li> |
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| <h4>normalCDF</h4> |
| <pre>public static double normalCDF(double x)</pre> |
| <div class="block">Computes an approximation to normalCDF(x).</div> |
| <dl> |
| <dt><span class="paramLabel">Parameters:</span></dt> |
| <dd><code>x</code> - is the input to the normalCDF function</dd> |
| <dt><span class="returnLabel">Returns:</span></dt> |
| <dd>returns the approximation to normalCDF(x).</dd> |
| </dl> |
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