M. A. Stephens

EDF Statistics for Goodness of Fit and Some Comparisons

Abstract This article offers a practical guide to goodness-of-fit tests using statistics based on the empirical distribution function (EDF). Five of the leading statistics are examined—those often labelled D, W 2, V, U 2, A 2—and three important situations: where the hypothesized distribution F(x) is completely specified and where F(x) represents the normal or exponential distribution with one or more parameters to be estimated from the data. EDF statistics are easily calculated, and the tests require only one line of significance points for each situation. They are also shown to be competitive in terms of power.

K-Sample Anderson–Darling Tests

Abstract Two k-sample versions of an Anderson–Darling rank statistic are proposed for testing the homogeneity of samples. Their asymptotic null distributions are derived for the continuous as well as the discrete case. In the continuous case the asymptotic distributions coincide with the (k – 1)-fold convolution of the asymptotic distribution for the Anderson–Darling one-sample statistic. The quality of this large sample approximation is investigated for small samples through Monte Carlo simulation. This is done for both versions of the statistic under various degrees of data rounding and sample size imbalances. Tables for carrying out these tests are provided, and their usage in combining independent one- or k-sample Anderson–Darling tests is pointed out. The test statistics are essentially based on a doubly weighted sum of integrated squared differences between the empirical distribution functions of the individual samples and that of the pooled sample. One weighting adjusts for the possibly different s...

Goodness-of-Fit Tests for the Generalized Pareto Distribution

Tests of fit are given for the generalized Pareto distribution (GPD) based on Cramér–von Mises statistics. Examples are given to illustrate the estimation techniques and the goodness-of-fit procedures. The tests are applied to the exceedances over given thresholds for 238 river flows in Canada; in general, the GPD provides an adequate fit. The tests are useful in deciding the threshold in such applications; this method is investigated and also the closeness of the GPD to some other distributions that might be used for long-tailed data.

The Kolmogorov-Smirnov Goodness-of-Fit Statistic with Discrete and Grouped Data

This paper considers the Kolmogorov-Smirnov two-sided goodness-of-fit statistic when applied to discrete or grouped data, The exact distribution of the statistic is tabulated for a given case and approximations discussed. The power of the test is compared with the power of the x 2 test, and the test is shown to have greater power for particular trend alternatives.

Distribution of a Sum of Weighted Chi-Square Variables

Abstract We consider distributions of quadratic forms of the type Qk = Σ k j = 1 cj (xj + aj )2, where the xj s are independent and identically distributed standard normal variables, and where cj and aj are nonnegative constants. Exact significance points of Qk , for selected values of cj and all aj = 0, have been published for k = 2, 3, 4, and 5. We give significance points for k = 6, 8, and 10. We propose and assess two new approximations to Qk : (1) fitting a Pearson curve with the same first four moments as Qk ; and (2) fitting Qk = Awr , where w has the χ2 p distribution, and where A, r, and p are determined by the first three moments of Qk.

Cramer-von Mises statistics for discrete distributions

Cramer-von Mises statistics are developed for use in testing for discrete distributions, and tables are given for tests for the discrete uniform distribution. The Cram6r-von Mises family of goodness-of-fit statistics is a well-known group of statistics used to test fit to a continuous distribution. In this article we extend the family to provide tests for discrete distributions. The statistics examined are the analogues of those called Cramer-von Mises, Watson, and Anderson-Darling, namely W2, U2 and A2 respectively, and their components. We provide formulae for the test statistics, and asymptotic percentage points for the test for a uniform distribution with k cells. The tests are based on the empirical distribution function (EDF) of the sample. They are closely related to Pearsons X2 test, and to Neyman-Barton smooth tests; in particular, all the tests can be broken down into components, as has been observed by many authors. It is suggested that A2 be used to test the overall null hypothesis in general, and U2 for the particular case where observations are counts around a circle. Their components can be used to test for particular types of departure from the null. In Section 2, we define the test statistics and give the general distribution theory. In Section 3 the solution of the uniform case is given, together with two examples; in Section 4 modified versions of the statistics are discussed. In Section 5 power studies are given which show that A2 is a good omnibus test statistic. Finally, in Section 6 we discuss the use of components as individual test statistics and demonstrate the use of a graphical procedure called the Z-plot to determine, when a statistic is found to be significant, the type of departure from the null.

Kolmogorov Statistics for Tests of Fit for the Extreme-Value and Weibull Distributions

Abstract Percentage points are given for the Kolmogorov-Smirnov statistics D +, D -, and D and for the Kuiper statistic V for testing fit to the extreme-value distribution with unknown parameters. The statistics may also be used for tests for the Weibull distribution.

Approximations to Density Functions Using Pearson Curves

Abstract This article is an expository paper to demonstrate the usefulness of Pearson curves in density estimation especially for those unaware of this early development in statistics. It is shown how to fit the curves and how very good approximate percentage points can be obtained for intractable distributions when the first four moments (or three moments and one endpoint) are known exactly (not estimated from sample data). The effectiveness of this method in density estimation is illustrated in three somewhat disparate contexts and reference is given to others. In general, the Pearson curves give an excellent approximation to the long tail of a distribution, the tail most often needed in practical work.

Tests of Fit for the Laplace Distribution, With Applications

Tests are given for the Laplace or double exponential distribution. The test statistics are based on the empirical distribution function and include the families of Cramér-von Mises and Kolmogorov-Smirnov. Asymptotic theory is given, and asymptotic points are calculated, for the Cramér-von Mises family, and Monte Carlo points for finite samples are given for all the statistics. Power studies suggest that the Watson statistic is the most powerful for the common problem of testing Laplace against other symmetric distributions. An application of the Laplace distribution is in LAD (or L1) regression. This is also discussed in the article, with two examples.

A New and Efficient Estimation Method for the Generalized Pareto Distribution

The generalized Pareto distribution (GPD) is widely used to model extreme values, for example, exceedences over thresholds, in modeling floods. Existing methods for estimating parameters have theoretical or computational defects. An efficient new estimator is proposed, which is computationally easy, free from the problems observed in traditional approaches, and performs well compared with existing estimators. A numerical example involving heights of waves is used to illustrate the various methods and tests of fit are performed to compare them.