Package umontreal.iro.lecuyer.gof

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

See:
Description

Class Summary

FBar	This class is similar to FDist, except that it provides static methods to compute or approximate the complementary distribution function of X, which we define as bar(F)(x) = P[X >= x], instead of F(x) = P[X <= x].
FDist	This class provides methods to compute (or approximate) the distribution functions of special types of goodness-of-fit test statistics.
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. Methods for computing (or approximating) the distribution function F(x) of certain GOF test statistics, as well as their complementary distribution function bar(F)(x) = 1 - F(x), are implemented in classes of package probdist. Tools for computing the GOF test statistics and the corresponding p-values, and for formating the results, are provided in classes GofStat and GofFormat.

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) hat(F)_N 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 GofStat.

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 methods in the classes of package probdist. 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 = 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 = 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 = ∑_i=0^k-1(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. The class GofStat.OutcomeCategoriesChi2, a nested class defined inside the GofStat class, provides tools to automatically regroup categories in the cases where some e_i's are too small.

The class GofFormat 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. 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.