úÎA`=ñ;      !"#$%&'()*+,-./0123456789:DeriveDataTypeable provisionalEdward Kmett <ekmett@gmail.com> Safe-Infered ;<=>?@;<=>?@Haskell 2011 + TypeFamilies provisionalEdward Kmett <ekmett@gmail.com> Safe-Infered,Sample quantile estimators AL-estimators are linear combinations of order statistics used by robust statistics. YCalculate the result of applying an L-estimator after sorting list into order statistics ]Calculate the result of applying an L-estimator to a *pre-sorted* vector of order statistics ^get a vector of the coefficients of an L estimator when applied to an input of a given length ,calculate the breakdown % of an L-estimator 8The average of all of the order statistics. Not robust.  breakdown mean = 0% 4The sum of all of the order statistics. Not robust.  breakdown total = 0% 7Calculate a trimmed L-estimator. If the sample size isn'/t evenly divided, linear interpolation is used  as described in  7http://en.wikipedia.org/wiki/Trimmed_mean#Interpolation cCalculates an interpolated winsorized L-estimator in a manner analogous to the trimmed estimator. ; Unlike trimming, winsorizing replaces the extreme values. cCalculates an interpolated winsorized L-estimator in a manner analogous to the trimmed estimator. ; Unlike trimming, winsorizing replaces the extreme values. \Jackknifes the statistic by removing each sample in turn and recalculating the L-estimator,  requires at least 2 samples! &The most robust L-estimator possible.  breakdown median = 50 `Compute a quantile using the given estimation strategy to interpolate when an exact quantile isn' t available 5Compute a quantile with traditional direct averaging terciles 1 and 2 # breakdown t1 = breakdown t2 = 33% terciles 1 and 2 # breakdown t1 = breakdown t2 = 33% @quantiles, with breakdown points 25%, 50%, and 25% respectively @quantiles, with breakdown points 25%, 50%, and 25% respectively @quantiles, with breakdown points 25%, 50%, and 25% respectively quintiles 1 through 4 !quintiles 1 through 4 "quintiles 1 through 4 #quintiles 1 through 4 $ , breakdown (percentile n) = min n (100 - n) ( " midhinge = trimmed 0.25 midrange  breakdown midhinge = 25% )Tukey' s trimean  breakdown trimean = 25 * The maximum value in the sample + The minimum value in the sample ,  midrange = lmax - lmin  breakdown midrange = 0% -interquartile range  breakdown iqr = 25%  iqr = trimmed 0.25 midrange .interquartile mean  iqm = trimmed 0.25 mean /8Direct estimator for the second L-moment given a sample 0`The Harrell-Davis quantile estimate. Uses multiple order statistics to approximate the quantile  to reduce variance. 1/Inverse of the empirical distribution function 2%.. with averaging at discontinuities 3KThe observation numbered closest to Np. NB: does not yield a proper median 4aLinear interpolation of the empirical distribution function. NB: does not yield a proper median. 5„.. with knots midway through the steps as used in hydrology. This is the simplest continuous estimator that yields a correct median 6eLinear interpolation of the expectations of the order statistics for the uniform distribution on [0,1] 7_Linear interpolation of the modes for the order statistics for the uniform distribution on [0,1] 8ELinear interpolation of the approximate medans for order statistics. 9|The resulting quantile estimates are approximately unbiased for the expected order statistics if x is normally distributed. :iWhen rounding h, this yields the order statistic with the least expected square deviation relative to p. 8  !"#$%&'()*+,-./0123456789:AB6  !"#$%&'()*+,-./0123456789:6 )(-./+*,&' #"!$%0123456789:5  !"#$%&'()*+,-./0123456789:ABC        !"#$%&'()*+,-./0123456789:;<=>?@ABCDorder-statistics-0.1Statistics.Distribution.BetaStatistics.OrderBetaDistributionBDbdAlphabdBeta betaDistr EstimatorEstimateLrunL@@@!@# breakdownmeantotaltrimmed winsorised winsorized jackknifedmedian quantileByquantiletercilet1t2quartileq1q2q3quintilequ1qu2qu3qu4 percentilepermille nthSmallest nthLargestmidhingetrimeanlmaxlminmidrangeiqriqmlscale hdquantiler1r2r3r4r5r6r7r8r9r10$fMaybeVarianceBetaDistribution$fContDistrBetaDistribution$fVarianceBetaDistribution$fMaybeMeanBetaDistribution$fMeanBetaDistribution$fDistributionBetaDistribution$fVectorSpaceL$fAdditiveGroupL