Priors are introduced into goodness‐of‐fit tests, both for unknown parameters in the tested distribution and on the alternative density. Neyman–Pearson theory leads to the test with the highest expected power. To make the test practical, we seek priors that make it likely a priori that the power will be larger than the level of the test but not too close to one. As a result, priors are sample size dependent. We explore this procedure in particular for priors that are defined via a Gaussian process approximation for the logarithm of the alternative density. In the case of testing for the uniform distribution, we show that the optimal test is of the U‐statistic type and establish limiting distributions for the optimal test statistic, both under the null hypothesis and averaged over the alternative hypotheses. The optimal test statistic is shown to be of the Cramér–von Mises type for specific choices of the Gaussian process involved. The methodology when parameters in the tested distribution are unknown is discussed and illustrated in the case of testing for the von Mises distribution. The Canadian Journal of Statistics 47: 560–579; 2019 © 2019 Statistical Society of Canada

This chapter considers tests of fit based on the empirical distribution function (EDF). The EDF is a step function, calculated from the sample, which estimates the population distribution function. From the basic definitions of the supremum statistics and the quadratic statistics given above, suitable computing formulas must be found. A second group of statistics for censored samples is of the general Cramer-von Mises type. The modifications for all the statistics were calculated from points for finite n obtained by Monte Carlo methods. Green and Hegazy have shown that slight modifications of the basic EDF statistics can improve power in tests for normality against selected alternatives. Data may appear to be discrete either because the sample genuinely arises from a discrete distribution like the Binomial or Poisson, for example, in measurements of counts, or alternatively because originally continuous data may have been grouped.

This chapter examines another graphical method, related to probability plots, in which the order statistics X(i) are plotted on the vertical axis of the graph, against T(i), a suitable function of i, on the horizontal axis. The statistic is easier to calculate than W or W' but must be used with both tails; d'Agostino has shown by Monte Carlo studies that alternative distributions may produce large or small values of the statistic. The various techniques regression tests of normality and exponentiality have not been so extensively developed for other distributions, and the tests to follow, for the extreme-value, logistic, and Cauchy distributions, are all based on the simple correlation coefficients. In general, consistency of a test based on the ratio of two estimates of scale must depend critically on how these are chosen, and this question deserves closer examination.

This chapter gives tests for a uniform distribution with unknown limits, and summarizes tests for censored uniforms. It discusses the power of the various test statistics. However, before this, some general observations can be made on the appearance of the U-set and its effect on different test statistics. The uniform distribution plays a special role in goodness-of-fit testing. When a test is made for uniformity, the alternative is often that the sample comes from a distribution which gives spacings more irregular than those from a uniform sample. The statistics so far considered have been based on various methods of relating the order statistics or their spacings to the pattern expected of them. Omnibus tests are not designed for specific alternatives, but it is convenient to mention several of these, especially for the circle, before leaving this section. Neyman found an appropriate statistic, based on likelihood ratio methods, for testing this null hypothesis.

The distribution of a sum S of independent binomial random variables, each with different success probabilities, is discussed. An efficient algorithm is given to calculate the exact distribution by convolution. Two approximations are examined, one based on a method of Kolmogorov, and another based on fitting a distribution from the Pearson family. The Kolmogorov approximation is given as an algorithm, with a worked example. The Kolmogorov and Pearson approximations are compared for several given sets of binomials with different sample sizes and probabilities. Other methods of approximation are discussed and some compared numerically. The Kolmogorov approximation is found to be extremely accurate, and the Pearson curve approximation useful if extreme accuracy is not required.

AbstractIn this article, tests of fit based on Kolmogorov–Smirnov (KS) statistics and statistics similar to these are discussed. KS statistics belong to the wider class of empirical distribution function (EDF) statistics, so‐called because they are based on the EDF of a given sample.

A procedure to test fit to a distribution where a minimal sufficient statistic is available, is discussed for testing the Poisson distribution. The test is exact, and is compared with a simpler approximate test. A remarkable correlation between the p -values given by the exact and the approximate procedures is found, and shows the power of the computer over and above what is usually acknowledged.

The benefits of model predictive control (MPC) have been well established; however its application to reference tracking on digital servo drives (DSDs), which typically have very fast update rates, is limited by the computational power of present-day processors. This paper presents a novel MPC formulation, which provides a mechanism to trade-off online computation effort with tracking performance, while maintaining stability. This is achieved by introducing a trajectory horizon, which is distinct from the prediction and control horizons typically encountered in MPC formulations. It is shown that increasing the trajectory horizon inherently leads to improved tracking; however larger horizon lengths also have the unwanted effect of increasing online computation. The proposed MPC formulation is compatible with recently developed explicit MPC solutions, and hence the burden of online optimization is avoided. The new approach is successfully implemented on an industrial machine tool DSD, and in terms of tracking accuracy, is shown to outperform the incumbent approach of cascaded PID control.

Exact and approximate tests of fit are compared for testing that a given sample comes from the von Mises distribution. For the exact test, Gibbs sampling is used to generate samples from the conditional distribution of sample data, given the values of the sufficient statistics. The samples, called co-sufficient samples, are used to estimate the distribution of Watson’s statistic, and hence to find the exact p-value for the given sample. The test is compared to the approximate test using the parametric bootstrap. Two examples are analyzed, and the p-values of the two tests are compared. When more examples are examined, an unexpectedly high correlation is discovered between the two sets of p-values, suggesting a strong mathematical connection.

Cramér-von Mises (CvM) tests for the exponential distribution are discussed, particularly when the scale parameter must be estimated from the given sample. The methods used in the paper are based on components of the statistics; this leads to a matrix formulation discretizing previous analytic techniques. Some comparisons are made between the two methods. Plots are given of the eigenfunctions which arise in the analysis. Plots are also given of asymptotic powers of the tests statistics against local Weibull and Gamma alternatives; these show the Anderson-Darling statistic to be the most powerful. The power of the best component is found, and shown to be comparable to the Likelihood Ratio statistic and greater than that of the entire statistic. Some connections with Neyman smooth tests are pointed out.

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.

Higher-precision inferences about impending software failures can be achieved when the same software reliability model that fits failure data from the test interval also fits data from the field interval. If the test and field environments differ significantly in terms of how the software is used, then a single model for the pooled data may not be adequate. In this article we formulate the hypothesis of compatible test and field environments in terms of a statistical hypothesis and develop a Cramer-von Mises (CvM) test procedure within the context of a well-known nonhomogeneous Poisson process software reliability model. The CvM test has a compelling advantage over a previously proposed likelihood ratio test (LRT0), because it does not require specification of the class of alternatives, which are frequently unknown for real-life problems. Moreover, although there are existence issues with LRT0, the CvM test always exists. An asymptotic approximation for the p value of the CvM test is derived, and an algorithm for a small-sample bootstrap approximation is presented. A simulation study shows that the CvM test works well for the class of alternatives for which LRT0 also would work well and continues to work well for other alternatives for which LRT0 has no statistical motivation or otherwise has existence problems. Data from a real software project are used to illustrate the hypothesis testing procedures.

Abstract. Moraine ridges are commonly used to identify past glacier ice margins and so infer glacier mass balance changes in response to climatic variability. However, differences in the form of past ice margins and post‐depositional modification of moraine surfaces can complicate these geomorphic records. As a result, simple relationships, such as distance from current ice margin, or linear alignments, may not necessarily indicate moraines deposited contemporaneously. These disturbances can also modify the size distribution of lichen populations, providing a distinctive signature for surfaces with similar histories and a means of identifying contemporaneous moraine surfaces. In this paper, statistical analysis of lichen size distributions is used to identify moraine surfaces with similar histories from complex suites of Little Ice Age moraine fragments in the proglacial areas of Skálafellsjökull (including Sultartungnajökull) and Heinabergsjökull, southeast Iceland. The analysis is based on a novel use of the goodness‐of‐fit statistic, Watson's U2 which provides a measure of ‘closeness’ between two sample distributions. Moraine fragments with similar histories are identified using cluster analysis of the U2 closeness values. The spatial pattern of the clustered moraines suggests three distinct phases of moraine deposition at Skálafellsjökull and Heinabergsjökull, four phases at Sultartungnajökull and a digitate planform margin at Heinabergsjökull. These spatial patterns are corroborated with tephrochronology. The success of the U2 statistical analysis in identifying surfaces with similar histories using lichen size distributions suggests that the technique may be useful in augmenting lichenometric surface dating as well as differentiating between other surfaces that support lichen populations, such as rock avalanche deposits.