Package umontreal.iro.lecuyer.gof

This package contains tools for performing univariate goodness-of-fit (GOF) statistical tests.

See: Description

Class Summary

Class	Description
FBar	WARNING: All methods in this class are deprecated, except the method scan.
FDist	WARNING: Most methods in this class are deprecated.
GofFormat	This class contains methods used to format results of GOF test statistics, or to apply a series of tests simultaneously and format the results.
GofStat	This class provides methods to compute several types of EDF goodness-of-fit test statistics and to apply certain transformations to a set of observations.
GofStat.OutcomeCategoriesChi2	This class helps managing the partitions of possible outcomes into categories for applying chi-square tests.
KernelDensity	This class provides methods to compute a kernel density estimator from a set of n individual observations x0,…, xn-1, and returns its value at m selected points.

Package umontreal.iro.lecuyer.gof Description

This package contains tools for performing univariate goodness-of-fit (GOF) statistical tests. Static methods for computing (or approximating) the distribution function F(x) of certain GOF test statistics, as well as their complementary distribution function 1#1(x) = 1 - F(x), are implemented in classes umontreal.iro.lecuyer.gofFDist and umontreal.iro.lecuyer.gofFBar. Tools for computing the GOF test statistics and the corresponding p-values, and for formating the results, are provided in classes umontreal.iro.lecuyer.gofGofStat and umontreal.iro.lecuyer.gofGofFormat.

We are concerned here with GOF test statistics for testing the hypothesis H₀ that a sample of N observations X₁,..., X_N comes from a given univariate probability distribution F. We consider tests such as those of Kolmogorov-Smirnov, Anderson-Darling, Crámer-von Mises, etc. These test statistics generally measure, in different ways, the distance between a continuous distribution function F and the empirical distribution function (EDF) 2#2 of X₁,..., X_N. They are also called EDF test statistics. The observations X_i are usually transformed into U_i = F(X_i), which satisfy 0≤U_i≤1 and which follow the U(0, 1) distribution under H₀. (This is called the probability integral transformation.) Methods for applying this transformation, as well as other types of transformations, to the observations X_i or U_i are provided in umontreal.iro.lecuyer.gofGofStat.

Then the GOF tests are applied to the U_i sorted by increasing order. The corresponding p-values are easily computed by calling the appropriate static methods in umontreal.iro.lecuyer.gofFDist. If a GOF test statistic Y has a continuous distribution under H₀ and takes the value y, its (right) p-value is defined as p = P[Y≥y | H₀]. The test usually rejects H₀ if p is deemed too close to 0 (for a one-sided test) or too close to 0 or 1 (for a two-sided test).

In the case where Y has a discrete distribution under H₀, we distinguish the right p-value p_R = P[Y≥y | H₀] and the left p-value p_L = P[Y≤y | H₀]. We then define the p-value for a two-sided test as

p =	pR	if pR < pL,
p =	1 - pL	if pR≥pL and pL < 0.5,
p =	0.5	otherwise.

Why such a definition? Consider for example a Poisson random variable Y with mean 1 under H₀. If Y takes the value 0, the right p-value is p_R = P[Y≥0 | H₀] = 1. In the uniform case, this would obviously lead to rejecting H₀ on the basis that the p-value is too close to 1. However, P[Y = 0 | H₀] = 1/e 3#3 0.368, so it does not really make sense to reject H₀ in this case. In fact, the left p-value here is p_L = 0.368, and the p-value computed with the above definition is p = 1 - p_L [tex2html_wrap_inline293] 0.632. Note that if p_L is very small, in this definition, p becomes close to 1. If the left p-value was defined as p_L = 1 - p_R = P[Y < y | H₀], this would also lead to problems. In the example, one would have p_L = 0 in that case.

A very common type of test in the discrete case is the chi-square test, which applies when the possible outcomes are partitioned into a finite number of categories. Suppose there are k categories and that each observation belongs to category i with probability p_i, for 0≤i < k. If there are n independent observations, the expected number of observations in category i is e_i = np_i, and the chi-square test statistic is defined as

X² = 4#4(o_i - e_i)²/e_i

where o_i is the actual number of observations in category i. Assuming that all e_i's are large enough (a popular rule of thumb asks for e_i≥5 for each i), X² follows approximately the chi-square distribution with k - 1 degrees of freedom[#!tREA88a!#]. The class umontreal.iro.lecuyer.gofGofStat.OutcomeCategoriesChi2, a nested class defined inside the umontreal.iro.lecuyer.gofGofStat class, provides tools to automatically regroup categories in the cases where some e_i's are too small.

The class umontreal.iro.lecuyer.gofGofFormat contains methods used to format results of GOF test statistics, or to apply several such tests simultaneously to a given data set and format the results to produce a report that also contains the p-values of all these tests. A C version of this class is actually used extensively in the package TestU01, which applies statistical tests to random number generators[#!iLEC01t!#]. The class also provides tools to plot an empirical or theoretical distribution function, by creating a data file that contains a graphic plot in a format compatible with a given software.