. e      !"#$%&'()*+,-./01 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E FGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdportable experimentalbos@serpentine.comefSort an array. -Partially sort an array, such that the least k elements will be  at the front.  The number k of least elements. 9Compute the minimum and maximum of an array in one pass. portable experimentalbos@serpentine.com>Weights for affecting the importance of elements of a sample. >A function that estimates a property of a sample, such as its  mean.  Sample data. portable experimentalbos@serpentine.com@A resample drawn randomly, with replacement, from a set of data B points. Distinct from a normal array to make it harder for your  humble author's brain to go wrong.  AResample a data set repeatedly, with replacement, computing each # estimate over the resampled data. gBCreate an array, using the given action to populate each element. >Compute a statistical estimate repeatedly over a sample, each % time omitting a successive element. h Drop the kth element of a vector.    portable experimentalbos@serpentine.com 7The interface shared by all probability distributions. 5Probability density function. The probability that a  stochastic variable x has the value X , i.e. P(x=X). :Cumulative distribution function. The probability that a  stochastic variable x is less than X , i.e. P(x<X). <Inverse of the cumulative distribution function. The value  X for which P(x<X). Approximate the value of X for which P(x>X)=p. ?This method uses a combination of Newton-Raphson iteration and D bisection with the given guess as a starting point. The upper and < lower bounds specify the interval in which the probability  distribution reaches the value p.  Probability p Initial guess Lower bound on interval Upper bound on interval      portable experimentalbos@serpentine.comijklportable experimentalbos@serpentine.comA very large number.  The largest m x such that 2**(x-1) is approximately  representable as a n.  sqrt 2  sqrt (2 * pi) 2 / sqrt pi 1 / sqrt 2 The smallest n larger than 1. portable experimentalbos@serpentine.com  Parameters a and b to the ! function.   Estimate the kth q*-quantile of a sample, using the weighted  average method. k, the desired quantile. q, the number of quantiles. x, the sample data. ! Estimate the kth q-quantile of a sample x , using the B continuous sample method with the given parameters. This is the C method used by most statistical software, such as R, Mathematica,  SPSS, and S.  Parameters a and b. k, the desired quantile. q, the number of quantiles. x, the sample data. "2California Department of Public Works definition, a=0, b=1. : Gives a linear interpolation of the empirical CDF. This / corresponds to method 4 in R and Mathematica. #Hazen's definition, a=0.5, b=0.5. This is claimed to be D popular among hydrologists. This corresponds to method 5 in R and  Mathematica. $9Definition used by the SPSS statistics application, with a=0,  b=0 (also known as Weibull'$s definition). This corresponds to  method 6 in R and Mathematica. %6Definition used by the S statistics application, with a=1,  b;=1. The interpolation points divide the sample range into n-1 @ intervals. This corresponds to method 7 in R and Mathematica. &Median unbiased definition, a=1/3, b=1/3. The resulting D quantile estimates are approximately median unbiased regardless of  the distribution of x). This corresponds to method 8 in R and  Mathematica. 'Normal unbiased definition, a=3/8, b=3/8. An approximately B unbiased estimate if the empirical distribution approximates the = normal distribution. This corresponds to method 9 in R and  Mathematica.  !"#$%&' !"#%$&'  !"#$%&'portable experimentalbos@serpentine.comopqr(#Arithmetic mean. This uses Welford's algorithm to provide @ numerical stability, using a single pass over the sample data. )?Harmonic mean. This algorithm performs a single pass over the  sample. *:Geometric mean of a sample containing no negative values. s+'Maximum likelihood estimate of a sample' s variance. ,Unbiased estimate of a sample' s variance. -;Standard deviation. This is simply the square root of the . maximum likelihood estimate of the variance. t.'Maximum likelihood estimate of a sample' s variance. /Unbiased estimate of a sample' s variance. 0;Standard deviation. This is simply the square root of the . maximum likelihood estimate of the variance. ()*+,-./0 ()*+,-./0 ()*+,-./0 portable experimentalbos@serpentine.com1uv2 (scale) parameter. 3123123123 portable experimentalbos@serpentine.com 4The normal distribution. wxyz{567|}~456746754567 portable experimentalbos@serpentine.com 89Evaluate a series of Chebyshev polynomials. Uses Clenshaw's  algorithm. Parameter of each function.  Coefficients of each polynomial  term, in increasing order. 9The binomial coefficient.  7 `choose` 3 == 35 :Compute the factorial function n !. Returns " if the E input is above 170 (above which the result cannot be represented by  a 64-bit n). ;@Compute the natural logarithm of the factorial function. Gives ! 16 decimal digits of precision. </Compute the incomplete gamma integral function (s,x).  Uses Algorithm AS 239 by Shea. s x =,Compute the logarithm of the gamma function (x ). Uses  Algorithm AS 245 by Macleod. Gives an accuracy of 10 &12 significant decimal digits, except  for small regions around x = 1 and x = 2, where the function * goes to zero. For greater accuracy, use >. Returns ") if the input is outside of the range (0 < x  "d 1e305). >-Compute the logarithm of the gamma function, (x ). Uses a  Lanczos approximation. This function is slower than =, but gives 14 or more 7 significant decimal digits of accuracy, except around x = 1 and  x' = 2, where the function goes to zero. Returns ") if the input is outside of the range (0 < x  "d 1e305). 89:;<=>89:;<=>89:;<=> portable experimentalbos@serpentine.com ?The binomial distribution. @Number of trials. A Probability. BNumber of trials.  Probability. ?@AB?@AB@A?@A@AB portable experimentalbos@serpentine.comCThe gamma distribution. DShape parameter, k. EScale parameter, . CDECDEDECDEDEportable experimentalbos@serpentine.com FGHIJm l k FGHIJFGHIJGHIFGHIGHIJportable experimentalbos@serpentine.comKLKLKLKLportable experimentalbos@serpentine.comM8The convolution kernel. Its parameters are as follows:  Scaling factor, 1/nh  Bandwidth, h ' A point at which to sample the input, p  One sample value, v N*The width of the convolution kernel used. OPoints from the range of a . PQR0Bandwidth estimator for an Epanechnikov kernel. S+Bandwidth estimator for a Gaussian kernel. TCCompute the optimal bandwidth from the observed data for the given  kernel. U>Choose a uniform range of points at which to estimate a sample's  probability density function. 7If you are using a Gaussian kernel, multiply the sample' s bandwidth * by 3 before passing it to this function. AIf this function is passed an empty vector, it returns values of ! positive and negative infinity. Number of points to select, n Sample bandwidth, h  Input data VAEpanechnikov kernel for probability density function estimation. W=Gaussian kernel for probability density function estimation. X<Kernel density estimator, providing a non-parametric way of * estimating the PDF of a random variable. Kernel function  Bandwidth, h  Sample data Points at which to estimate YBA helper for creating a simple kernel density estimation function < with automatically chosen bandwidth and estimation points. Bandwidth function Kernel function EBandwidth scaling factor (3 for a Gaussian kernel, 1 for all others) &Number of points at which to estimate  Sample data Z;Simple Epanechnikov kernel density estimator. Returns the D uniformly spaced points from the sample range at which the density < function was estimated, and the estimates at those points. &Number of points at which to estimate [ASimple Gaussian kernel density estimator. Returns the uniformly C spaced points from the sample range at which the density function 3 was estimated, and the estimates at those points. &Number of points at which to estimate MNOPQRSTUVWXYZ[Z[OPQUNTRSMVWXYMNOPQPQRSTUVWXYZ[portable experimentalbos@serpentine.com \.A point and interval estimate computed via an . ]^Point estimate. _>Lower bound of the estimate interval (i.e. the lower bound of  the confidence interval). `>Upper bound of the estimate interval (i.e. the upper bound of  the confidence interval). a.Confidence level of the confidence intervals. bBBias-corrected accelerated (BCA) bootstrap. This adjusts for both 2 bias and skewness in the resampled distribution. Confidence level  Sample data  Estimators Resampled data \]^_`ab\]^_`ab\]^_`a]^_`abportable experimentalbos@serpentine.comc?Compute the autocovariance of a sample, i.e. the covariance of 1 the sample against a shifted version of itself. d@Compute the autocorrelation function of a sample, and the upper < and lower bounds of confidence intervals for each element. Note;: The calculation of the 95% confidence interval assumes a  stationary Gaussian process. cdcdcd !"#$%&'()*+,-./0012345678 9:;<=>? @ A B C D E B F G H I J K L M N O P Q R STUVWEXAYZ[[\]^_`abcdefgghijklmnoopqr"#$stustvwwxxyz { | }  ~  " # $        " # $  r " # $ "#$"#$xstatistics-0.2Statistics.FunctionStatistics.TypesStatistics.ResamplingStatistics.Distribution!Statistics.Distribution.GeometricStatistics.ConstantsStatistics.QuantileStatistics.Sample#Statistics.Distribution.ExponentialStatistics.Distribution.NormalStatistics.Math Statistics.Distribution.BinomialStatistics.Distribution.Gamma&Statistics.Distribution.HypergeometricStatistics.Distribution.PoissonStatistics.KernelDensityStatistics.Resampling.BootstrapStatistics.Autocorrelationsort partialSortminMaxWeights EstimatorSampleResample fromResampleresample jackknifeVariancevarianceMeanmean Distribution probability cumulativeinversefindRootGeometricDistribution pdSuccess fromSuccessm_huge m_max_expm_sqrt_2 m_sqrt_2_pi m_2_sqrt_pi m_1_sqrt_2 m_epsilon ContParam weightedAvg continuousBycadpwhazenspsssmedianUnbiasednormalUnbiased harmonicMean geometricMeanvarianceUnbiasedstdDev fastVariancefastVarianceUnbiased fastStdDevExponentialDistribution fromLambda fromSampleNormalDistributionstandard fromParams chebyshevchoose factorial logFactorialincompleteGammalogGamma logGammaLBinomialDistributionbdTrials bdProbabilitybinomialGammaDistributiongdShapegdScaleHypergeometricDistributionhdMhdLhdKPoissonDistributionKernel BandwidthPoints fromPointsepanechnikovBW gaussianBW bandwidth choosePointsepanechnikovKernelgaussianKernel estimatePDF simplePDFepanechnikovPDF gaussianPDFEstimateestPoint estLowerBound estUpperBoundestConfidenceLevel bootstrapBCAautocovarianceautocorrelationMMcreateUdropAtGDghc-prim GHC.TypesIntDoubleT1T robustVarfastVarEDedLambdaND ndPdfDenom ndCdfDenomLFCBDHDPDpdLambda errorShort:<estimate