6"&\      !"#$%&'()*+,-./0123456789: ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [\]^_`abcdefghijklmnopqrstuvwxyz{|}~$ ;Calculate rank of sample. Sample should be already sorted. Equivalence relation Vector to rank Split tagged vector Result of hypothesis testing #Data is compatible with hypothesis #Null hypothesis should be rejected 9Test type. Exact meaning depends on a specific test. But  generally it'4s tested whether some statistics is too big (small)  for ( or whether it too big or too small for  Significant if parameter is , not significant otherwiser !portable experimentalbos@serpentine.com=Just like unsafePerformIO, but we inline it. Big performance 9 gains as it exposes lots of things to further inlining. /Very  unsafe/;. In particular, you should do no memory allocation inside  an  block. On Hugs this is just unsafePerformIO. "portable experimentalbos@serpentine.com Compare two ( values for approximate equality, using  Dawson' s method. ;The required accuracy is specified in ULPs (units of least D precision). If the two numbers differ by the given number of ULPs  or less, this function returns True. $Number of ULPs of accuracy desired. portable experimentalbos@serpentine.com?The result of searching for a root of a mathematical function. A root was successfully found. 2The search failed to converge to within the given 7 error tolerance after the given number of iterations. /The function does not have opposite signs when 8 evaluated at the lower and upper bounds of the search. AReturns either the result of a search for a root, or the default  value if the search failed. Default value. Result of search for a root. ;Use the method of Ridders to compute a root of a function. BThe function must have opposite signs when evaluated at the lower C and upper bounds of the search (i.e. the root must be bracketed). Absolute error tolerance. 'Lower and upper bounds for the search. Function to find the roots of.    portable experimentalbos@serpentine.com Sort a vector. 'Sort a vector using a custom ordering. -Partially sort a vector, such that the least k elements will be  at the front.  The number k of least elements.  Return the indices of a vector. Zip a vector with its indices. 9Compute the minimum and maximum of a vector in one pass. 8Efficiently compute the next highest power of two for a A non-negative integer. If the given value is already a power of @ two, it is returned unchanged. If negative, zero is returned. portable experimentalbos@serpentine.comO(n)& Compute a histogram over a data set. *The result consists of a pair of vectors: # The lower bound of each interval. . * The number of samples within the interval. AInterval (bin) sizes are uniform, and the upper and lower bounds $ are chosen automatically using the  function. To specify $ these parameters directly, use the  function. #Number of bins (must be positive). Sample data (cannot be empty). O(n)& Compute a histogram over a data set. >Interval (bin) sizes are uniform, based on the supplied upper  and lower bounds. 6Number of bins. This value must be positive. A zero ( or negative value will cause an error. 6Lower bound on interval range. Sample data less than  this will cause an error. 7Upper bound on interval range. This value must not be : less than the lower bound. Sample data that falls above & the upper bound will cause an error.  Sample data. O(n); Compute decent defaults for the lower and upper bounds of C a histogram, based on the desired number of bins and the range of  the sample data. $The upper and lower bounds used are  (lo-d, hi+d), where &d = (maximum sample - minimum sample) / ((bins - 1) * 2)#Number of bins (must be positive). Sample data (cannot be empty). 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. GSample with weights. First element of sample is data, second is weight  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. O(e*r*s)3 Resample a data set repeatedly, with replacement, 2 computing each estimate over the resampled data. >This function is expensive; it has to do work proportional to  e*r*s, where e( is the number of estimation functions, r is ) the number of resamples to compute, and s is the number of  original samples. ;To improve performance, this function will make use of all D available CPUs. At least with GHC 7.0, parallel performance seems @ best if the parallel garbage collector is disabled (RTS option  -qg). Estimation functions.  Number of resamples to compute. Original sample. >Compute a statistical estimate repeatedly over a sample, each % time omitting a successive element.  Drop the kth element of a vector.    portable experimentalbos@serpentine.com!-The Wilcoxon matched-pairs signed-rank test. OThe value returned is the pair (T+, T-). T+ is the sum of positive ranks (the M ranks of the differences where the first parameter is higher) whereas T- is ` the sum of negative ranks (the ranks of the differences where the second parameter is higher). I These values mean little by themselves, and should be combined with the wilcoxonSignificant 5 function in this module to get a meaningful result. UThe samples are zipped together: if one is longer than the other, both are truncated * to the the length of the shorter sample. -Note that: wilcoxonMatchedPairSignedRank == ((6x, y) -> (y, x)) . flip wilcoxonMatchedPairSignedRank ;The coefficients for x^0, x^1, x^2, etc, in the expression  p9rod_{r=1}^s (1 + x^r). See the Mitic paper for details. We can define:  f(1) = 1 + x  f(r) = (1 + x^r)*f(r-1)  = f(r-1) + x^r * f(r-1) ; The effect of multiplying the equation by x^r is to shift * all the coefficients by r down the list. 2This list will be processed lazily from the head. "LTests whether a given result from a Wilcoxon signed-rank matched-pairs test $ is significant at the given level. IThis function can perform a one-tailed or two-tailed test. If the first  parameter to this function is &, the test is performed two-tailed to K check if the two samples differ significantly. If the first parameter is  G, the check is performed one-tailed to decide whether the first sample & (i.e. the first sample you passed to !) is F greater than the second sample (i.e. the second sample you passed to  !-). If you wish to perform a one-tailed test N in the opposite direction, you can either pass the parameters in a different  order to !-, or simply swap the values in the resulting , pair before passing them to this function. 1Perform one-tailed test (see description above). <The sample size from which the (T+,T-) values were derived. )The p-value at which to test (e.g. 0.05) The (T+, T-) values from !. Return  if the sample was too  small to make a decision. #HObtains the critical value of T to compare against, given a sample size H and a p-value (significance level). Your T value must be less than or H equal to the return of this function in order for the test to work out M significant. If there is a Nothing return, the sample size is too small to  make a decision. wilcoxonSignificant tests the return value of !  for you, so you should use wilcoxonSignificant for determining test results. N However, this function is useful, for example, for generating lookup tables + for Wilcoxon signed rank critical values. NThe return values of this function are generated using the method detailed in  the paper "6Critical Values for the Wilcoxon Signed Rank Statistic", Peter M Mitic, The Mathematica Journal, volume 6, issue 3, 1996, which can be found  here:  Phttp://www.mathematica-journal.com/issue/v6i3/article/mitic/contents/63mitic.pdf. Y According to that paper, the results may differ from other published lookup tables, but O (Mitic claims) the values obtained by this function will be the correct ones. The sample size ?The p-value (e.g. 0.05) for which you want the critical value. )The critical value (of T), or Nothing if / the sample is too small to make a decision. $HWorks out the significance level (p-value) of a T value, given a sample F size and a T value from the Wilcoxon signed-rank matched-pairs test. See the notes on wilcoxonCriticalValue for how this is calculated. The sample size 4The value of T for which you want the significance. The significance (p-value). %=The Wilcoxon matched-pairs signed-rank test. The samples are < zipped together: if one is longer than the other, both are 4 truncated to the the length of the shorter sample. CFor one-tailed test it tests whether first sample is significantly @ greater than the second. For two-tailed it checks whether they  significantly differ Check ! and  " for additional information. Perform one-tailed test. )The p-value at which to test (e.g. 0.05)  First sample Second sample Return  if the sample was too  small to make a decision. !"#$% %!"$#!"#$%portable experimentalbos@serpentine.com&@Type class for distributions with variance. If distibution have ? finite variance for all valid parameter values it should be  instance of this type class. Minimal complete definition is ' or ( '();Type class for distributions with variance. If variance is , undefined for some parameter values both * and  + should return Nothing. Minimal complete definition is * or + *+,=Type class for distributions with mean. If distribution have ? finite mean for all valid values of parameters it should be  instance of this type class. -.(Type class for distributions with mean. / should return   if it'&s undefined for current value of data /0%Continuous probability distributuion 16Probability density function. Probability that random  variable X$ lies in the infinitesimal interval  [x,x+x ) equal to  density(x)"x 2;Inverse of the cumulative distribution function. The value  x for which P(X"dx) = p. If probability is outside  of [0,1] range function should call  3#Discrete probability distribution. 4Probability of n-th outcome. 5=Type class common to all distributions. Only c.d.f. could be 8 defined for both discrete and continous distributions. 6:Cumulative distribution function. The probability that a  random variable X is less or equal than x,  i.e. P(X"dx). 7One'(s complement of cumulative distibution:  * complCumulative d x = 1 - cumulative d x It'%s useful when one is interested in P(X"ex) and < expression on the right side begin to lose precision. This ; function have default implementation but implementors are 3 encouraged to provide more precise implementation 8Approximate 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.  Distribution  Probability p Initial guess Lower bound on interval Upper bound on interval 9)Sum probabilities in inclusive interval. &'()*+,-./012345678956734012./,-)*+&'(89&'('()*+*+,--.//012123445676789 portable experimentalbos@serpentine.com:Cauchy-Lorentz distribution. ;:Central value of Cauchy-Lorentz distribution which is its & mode and median. Distribution doesn't have mean so function  is named after median. <2Scale parameter of Cauchy-Lorentz distribution. It's : different from variance and specify half width at half  maximum (HWHM). =Cauchy distribution Central point Scale parameter (FWHM) >:;<=>:;<;<=>:;<;<=> portable experimentalbos@serpentine.com?@ACreate geometric distribution.  Success rate ?@A?@A@?@@A portable experimentalbos@serpentine.comBUniform distribution CCreate uniform distribution. BCBCBC portable experimentalbos@serpentine.com DA very large number. EF The largest  x such that 2**(x-1) is approximately  representable as a . G sqrt 2H  sqrt (2 * pi)I 2 / sqrt piJ 1 / sqrt 2K The smallest   such that 1 +  "` 1. L log(sqrt((2*pi))MPositive infinity. NNegative infinity. ONot a number. DEFGHIJKLMNO KDEJILFGHMNO DEFGHIJKLMNO portable experimentalbos@serpentine.com P Parameters a and b to the S function. QRO(n log n). 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. SO(n log n). Estimate the kth q-quantile of a sample x, E using the continuous sample method with the given parameters. This = is the 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. TO(n log n). Estimate the range between q-quantiles 1 and  q-1 of a sample x., using the continuous sample method with the  given parameters. @For instance, the interquartile range (IQR) can be estimated as  follows: 5 midspread medianUnbiased 4 (U.fromList [1,1,2,2,3])  ==> 1.333333  Parameters a and b. q, the number of quantiles. x, the sample data. U2California 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. VHazen'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. W9Definition used by the SPSS statistics application, with a=0,  b=0 (also known as Weibull'$s definition). This corresponds to  method 6 in R and Mathematica. X6Definition used by the S statistics application, with a=1,  b;=1. The interpolation points divide the sample range into n-1 @ intervals. This corresponds to method 7 in R and Mathematica. YMedian unbiased definition, a=1/3, b=1/3. The resulting D quantile estimates are approximately median unbiased regardless of  the distribution of x). This corresponds to method 8 in R and  Mathematica. ZNormal unbiased definition, a=3/8, b=3/8. An approximately B unbiased estimate if the empirical distribution approximates the = normal distribution. This corresponds to method 9 in R and  Mathematica. PQRSTUVWXYZ RPQSTUVXWYZ PQQRSTUVWXYZportable experimentalbos@serpentine.com[O(n)8 Range. The difference between the largest and smallest  elements of a sample. \O(n)$ Arithmetic mean. This uses Welford's algorithm to provide @ numerical stability, using a single pass over the sample data. ]O(n)< Arithmetic mean for weighted sample. It uses a single-pass ( algorithm analogous to the one used by \. ^O(n)< Harmonic mean. This algorithm performs a single pass over  the sample. _O(n); Geometric mean of a sample containing no negative values. ` Compute the k3th central moment of a sample. The central moment - is also known as the moment about the mean. EThis function performs two passes over the sample, so is not subject  to stream fusion. @For samples containing many values very close to the mean, this E function is subject to inaccuracy due to catastrophic cancellation. a Compute the kth and j th central moments of a sample. EThis function performs two passes over the sample, so is not subject  to stream fusion. @For samples containing many values very close to the mean, this E function is subject to inaccuracy due to catastrophic cancellation. b;Compute the skewness of a sample. This is a measure of the  asymmetry of its distribution. *A sample with negative skew is said to be  left-skewed . Most of D its mass is on the right of the distribution, with the tail on the  left.  % skewness $ U.to [1,100,101,102,103]  ==> -1.497681449918257 *A sample with positive skew is said to be  right-skewed.   skewness $ U.to [1,2,3,4,100]  ==> 1.4975367033335198 A sample'!s skewness is not defined if its d is zero. EThis function performs two passes over the sample, so is not subject  to stream fusion. @For samples containing many values very close to the mean, this E function is subject to inaccuracy due to catastrophic cancellation. c?Compute the excess kurtosis of a sample. This is a measure of  the " peakedness"1 of its distribution. A high kurtosis indicates  that more of the sample''s variance is due to infrequent severe : deviations, rather than more frequent modest deviations. A sample'(s excess kurtosis is not defined if its d is  zero. EThis function performs two passes over the sample, so is not subject  to stream fusion. @For samples containing many values very close to the mean, this E function is subject to inaccuracy due to catastrophic cancellation. d'Maximum likelihood estimate of a sample's variance. Also known 6 as the population variance, where the denominator is n. eUnbiased estimate of a sample's variance. Also known as the + sample variance, where the denominator is n-1. fACalculate mean and maximum likelihood estimate of variance. This @ function should be used if both mean and variance are required ) since it will calculate mean only once. g7Calculate mean and unbiased estimate of variance. This @ function should be used if both mean and variance are required ) since it will calculate mean only once. h;Standard deviation. This is simply the square root of the $ unbiased estimate of the variance. i.Weighted variance. This is biased estimation. j'Maximum likelihood estimate of a sample' s variance. kUnbiased estimate of a sample' s variance. l;Standard deviation. This is simply the square root of the . maximum likelihood estimate of the variance. [\]^_`abcdefghijkl[\]^_`abcdefghijkl[\]^_`abcdefghijklportable experimentalbos@serpentine.com mno$Create an exponential distribution.  (scale) parameter. pBCreate exponential distribution from sample. No tests are made to ( check whether it truly is exponential. mnopmnopnmnnopportable experimentalbos@serpentine.com qThe normal distribution. rJStandard normal distribution with mean equal to 0 and variance equal to 1 s,Create normal distribution from parameters. CIMPORTANT: prior to 0.10 release second parameter was variance not  standard deviation. Mean of distribution #Standard deviation of distribution t4Create distribution using parameters estimated from A sample. Variance is estimated using maximum likelihood method  (biased estimation). qrstqstrqrstportable experimentalbos@serpentine.com  u8Evaluate a Chebyshev polynomial of the first kind. Uses  Clenshaw' s algorithm. Parameter of each function. ;Coefficients of each polynomial term, in increasing order. v?Evaluate a Chebyshev polynomial of the first kind. Uses Broucke's F ECHEB algorithm, and his convention for coefficient handling, and so  gives different results than u for the same inputs. Parameter of each function. ;Coefficients of each polynomial term, in increasing order.  )Quickly compute the natural logarithm of n w k, with  no checking. w!Compute the binomial coefficient n `w` k. For  values of k2 > 30, this uses an approximation for performance E reasons. The approximation is accurate to 12 decimal places in the  worst case  Example:  7 `choose` 3 == 35 xCompute the factorial function n !. Returns " if the E input is above 170 (above which the result cannot be represented by  a 64-bit ). y@Compute the natural logarithm of the factorial function. Gives ! 16 decimal digits of precision. z7Compute the normalized lower incomplete gamma function  (s,x). Normalization means that  (s,"$)=1. Uses Algorithm AS 239 by Shea. s x {%Inverse incomplete gamma function. It's approximately inverse of  z for the same s. So following equality  approximately holds:  / invIncompleteGamma s . incompleteGamma s = id For invIncompleteGamma s p s must be positive and p must be  in [0,1] range. |,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).  ,Compute the log gamma correction factor for x "e 10. This : correction factor is suitable for an alternate (but less % numerically accurate) definition of |: Elgg x = 0.5 * log(2*pi) + (x-0.5) * log x - x + logGammaCorrection x ~4Compute the natural logarithm of the beta function. =Regularized incomplete beta function. Uses algorithm AS63 by  Majumder abd Bhattachrjee. p > 0 q > 0 x, must lie in [0,1] range .Regularized incomplete beta function. Same as   but also takes value of lo logarithm of beta function p > 0 q > 0 x, must lie in [0,1] range  >Compute inverse of regularized incomplete beta function. Uses G initial approximation from AS109 and Halley method to solve equation. p q a %Compute the natural logarithm of 1 + x. This is accurate even  for values of x near zero, where use of log(1+x) would lose  precision. ACalculate the error term of the Stirling approximation. This is ' only defined for non-negative values. : stirlingError @n@ = @log(n!) - log(sqrt(2*pi*n)*(n/e)^n) Evaluate the deviance term x log(x/ np) + np - x.  x npO(log n)5 Compute the logarithm in base 2 of the given value. uvwxyz{|}~w~uvxy|}z{uvwxyz{|}~portable experimentalbos@serpentine.com The binomial distribution. Number of trials.  Probability. :Construct binomial distribution. Number of trials must be + positive and probability must be in [0,1] range Number of trials.  Probability. portable experimentalbos@serpentine.comChi-squared distribution !Get number of degrees of freedom AConstruct chi-squared distribution. Number of degrees of freedom  must be positive. portable experimentalbos@serpentine.com Student-T distribution #portable experimentalbos@serpentine.com2An unchecked, non-integer-valued version of Loader's saddle point  algorithm. portable experimentalbos@serpentine.comThe gamma distribution. Shape parameter, k. Scale parameter, . @Create gamma distribution. Both shape and scale parameters must  be positive. Shape parameter. k Scale parameter, .  !"portable experimentalbos@serpentine.com#Create Poisson distribution. portable experimentalbos@serpentine.com $%&m l k '(portable experimentalbos@serpentine.comStudent-T distribution )FCreate Student-T distribution. Number of parameters must be positive. *+,portable experimentalbos@serpentine.comDiscrete cosine transform. #Inverse discrete cosine transform.  Inverse fast Fourier transform. 3Radix-2 decimation-in-time fast Fourier transform. -./portable experimentalbos@serpentine.comBGaussian kernel density estimator for one-dimensional data, using  the method of Botev et al. -The result is a pair of vectors, containing: B The coordinates of each mesh point. The mesh interval is chosen C to be 20% larger than the range of the sample. (To specify the  mesh interval, use .) ' Density estimates at each mesh point. ?The number of mesh points to use in the uniform discretization  of the interval  (min,max)#. If this value is not a power of 6 two, then it is rounded up to the next power of two. BGaussian kernel density estimator for one-dimensional data, using  the method of Botev et al. -The result is a pair of vectors, containing: % The coordinates of each mesh point. ' Density estimates at each mesh point. ?The number of mesh points to use in the uniform discretization  of the interval  (min,max)#. If this value is not a power of 6 two, then it is rounded up to the next power of two.  Lower bound (min) of the mesh range.  Upper bound (max) of the mesh range. portable experimentalbos@serpentine.com 0O(n) Collect the n simple powers of a sample.  Functions computed over a sample'#s simple powers require at least a  certain number (or order) of powers to be collected.  To compute the kth  , at least k simple powers  must be collected.  For the ', at least 2 simple powers are needed.  For $, we need at least 3 simple powers.  For ), at least 4 simple powers are required. +This function is subject to stream fusion. n, the number of powers, where n >= 2. 5The order (number) of simple powers collected from a sample.  Compute the k,th central moment of a sample. The central 4 moment is also known as the moment about the mean. 'Maximum likelihood estimate of a sample's variance. Also known 6 as the population variance, where the denominator is n . This is * the second central moment of the sample. BThis is less numerically robust than the variance function in the  Statistics.Sample/ module, but the number is essentially free to / compute if you have already collected a sample's simple powers.  Requires  with  at least 2. ;Standard deviation. This is simply the square root of the . maximum likelihood estimate of the variance. Unbiased estimate of a sample's variance. Also known as the + sample variance, where the denominator is n-1.  Requires  with  at least 2. ;Compute the skewness of a sample. This is a measure of the  asymmetry of its distribution. *A sample with negative skew is said to be  left-skewed . Most of D its mass is on the right of the distribution, with the tail on the  left.  0 skewness . powers 3 $ U.to [1,100,101,102,103]  ==> -1.497681449918257 *A sample with positive skew is said to be  right-skewed.  * skewness . powers 3 $ U.to [1,2,3,4,100]  ==> 1.4975367033335198 A sample'!s skewness is not defined if its  is zero.  Requires  with  at least 3. ?Compute the excess kurtosis of a sample. This is a measure of  the " peakedness"1 of its distribution. A high kurtosis indicates  that the sample',s variance is due more to infrequent severe 0 deviations than to frequent modest deviations. A sample'(s excess kurtosis is not defined if its  is  zero.  Requires  with  at least 4. 'The number of elements in the original Sample. This is the  sample's zeroth simple power. $The sum of elements in the original Sample. This is the  sample's first simple power. 0The arithmetic mean of elements in the original Sample. >This is less numerically robust than the mean function in the  Statistics.Sample/ module, but the number is essentially free to / compute if you have already collected a sample's simple powers. portable experimentalbos@serpentine.comThe Wilcoxon Rank Sums Test. NThis test calculates the sum of ranks for the given two samples. The samples Q are ordered, and assigned ranks (ties are given their average rank), then these # ranks are summed for each sample. The return value is (W , W  ) where W ) is the sum of ranks of the first sample  and W O is the sum of ranks of the second sample. This test is trivially transformed > into the Mann-Whitney U test. You will probably want to use  R and the related functions for testing significance, but this function is exposed  for completeness. The Mann-Whitney U Test. AThis is sometimes known as the Mann-Whitney-Wilcoxon U test, and L confusingly many sources state that the Mann-Whitney U test is the same as  the Wilcoxon'&s rank sum test (which is provided as ). 5 The Mann-Whitney U is a simple transform of Wilcoxon's rank sum test. EAgain confusingly, different sources state reversed definitions for U   and U B, so it is worth being explicit about what this function returns. ! Given two samples, the first, xs  , of size n  and the second, xs ,  of size n , this function returns (U , U )  where U  = W  - (n (n +1))/2  and U  = W  - (n (n +1))/2,  where (W , W ) is the return value of wilcoxonRankSums xs1 xs2. !Some sources instead state that U  and U & should be the other way round, often  expressing this using U ' = n n  - U  (since U  + U  = n n ). All of which you probably don'(t care about if you just feed this into . ECalculates the critical value of Mann-Whitney U for the given sample  sizes and significance level. LThis function returns the exact calculated value of U for all sample sizes; N it does not use the normal approximation at all. Above sample size 20 it is R generally recommended to use the normal approximation instead, but this function = will calculate the higher critical values if you need them. LThe algorithm to generate these values is a faster, memoised version of the > simple unoptimised generating function given in section 2 of "The Mann Whitney ) Wilcoxon Distribution Using Linked Lists" The sample size ?The p-value (e.g. 0.05) for which you want the critical value. The critical value (of U). 1;Calculates whether the Mann Whitney U test is significant. NIf both sample sizes are less than or equal to 20, the exact U critical value  (as calculated by  ) is used. If either sample is ; larger than 20, the normal approximation is used instead. MIf you use a one-tailed test, the test indicates whether the first sample is Q significantly larger than the second. If you want the opposite, simply reverse - the order in both the sample size and the (U , U  ) pairs. 1Perform one-tailed test (see description above).  The samples' size from which the (U ,U ) values were derived. )The p-value at which to test (e.g. 0.05) The (U , U ) values from . Return  if the sample was too  small to make a decision. 9Perform Mann-Whitney U Test for two samples and required A significance. For additional information check documentation of   and . This is just a helper  function. DOne-tailed test checks whether first sample is significantly larger C than second. Two-tailed whether they are significantly different. 1Perform one-tailed test (see description above). )The p-value at which to test (e.g. 0.05)  First sample Second sample Return  if the sample was too small to  make a decision.  $portable experimentalbos@serpentine.com!"#$%portable experimentalbos@serpentine.com 23.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). .Confidence level of the confidence intervals. 7Multiply the point, lower bound, and upper bound in an   by the given value. Value to multiply by. 4BBias-corrected accelerated (BCA) bootstrap. This adjusts for both 2 bias and skewness in the resampled distribution. Confidence level  Sample data  Estimators Resampled data portable experimentalbos@serpentine.com8The 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 *The width of the convolution kernel used. Points from the range of a Sample. 0Bandwidth estimator for an Epanechnikov kernel. +Bandwidth estimator for a Gaussian kernel. =Compute the optimal bandwidth from the observed data for the  given kernel. DThis function uses an estimate based on the standard deviation of a ? sample (due to Deheuvels), which performs reasonably well for D unimodal distributions but leads to oversmoothing for more complex  ones. >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 AEpanechnikov kernel for probability density function estimation. =Gaussian kernel for probability density function estimation. <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 BA 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 ;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  Data sample 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  Data sample 5portable experimentalbos@serpentine.com?Compute the autocovariance of a sample, i.e. the covariance of 1 the sample against a shifted version of itself. @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. 6%&'()*+",--./0123456789:;<=>?@@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\ ] ^ _ ` a b c d e f g h i j k l m n o p q r s s t u v w x y z { |;P}~JKJKP   !   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