The main intent of this paper is to introduce a new statistical procedure for testing a complete sample for normality. The test statistic is obtained by dividing the square of an appropriate linear combination of the sample order statistics by the usual symmetric estimate of variance. This ratio is both scale and origin invariant and hence the statistic is appropriate for a test of the composite hypothesis of normality. Testing for distributional assumptions in general and for normality in particular has been a major area of continuing statistical research-both theoretically and practically. A possible cause of such sustained interest is that many statistical procedures have been derived based on particular distributional assumptions-especially that of normality. Although in many cases the techniques are more robust than the assumptions underlying them, still a knowledge that the underlying assumption is incorrect may temper the use and application of the methods. Moreover, the study of a body of data with the stimulus of a distributional test may encourage consideration of, for example, normalizing transformations and the use of alternate methods such as distribution-free techniques, as well as detection of gross peculiarities such as outliers or errors. The test procedure developed in this paper is defined and some of its analytical properties described in ? 2. Operational information and tables useful in employing the test are detailed in ? 3 (which may be read independently of the rest of the paper). Some examples are given in ? 4. Section 5 consists of an extract from an empirical sampling study of the comparison of the effectiveness of various alternative tests. Discussion and concluding remarks are given in ?6. 2. THE W TEST FOR NORMALITY (COMPLETE SAMPLES) 2 1. Motivation and early work This study was initiated, in part, in an attempt to summarize formally certain indications of probability plots. In particular, could one condense departures from statistical linearity of probability plots into one or a few 'degrees of freedom' in the manner of the application of analysis of variance in regression analysis? In a probability plot, one can consider the regression of the ordered observations on the expected values of the order statistics from a standardized version of the hypothesized distribution-the plot tending to be linear if the hypothesis is true. Hence a possible method of testing the distributional assumptionis by means of an analysis of variance type procedure. Using generalized least squares (the ordered variates are correlated) linear and higher-order

/pdf/an-analysis-of-variance-test-for-normality-complete-samples-2h979ba301.pdf

An Analysis of Variance Test for Normality (Complete Samples)

https://www.jstor.org/stable/pdfplus/1912517.pdf

Likelihood ratio statistics for autoregressive time series with a unit root

This paper considers tests for parameter instability and structural change with unknown change point. The results apply to a wide class of parametric models that are suitable for estimation by generalized method of moments procedures. The asymptotic distributions of the test statistics considered here are nonstandard because the change point parameter only appears under the alternative hypothesis and not under the null. The tests considered here are shown to have nontrivial asymptotic local power against all alternatives for which the parameters are nonconstant. The tests are found to perform quite well in a Monte Carlo experiment reported elsewhere. Copyright 1993 by The Econometric Society.

/pdf/tests-for-parameter-instability-and-structural-change-with-26amgrdoht.pdf

Tests for Parameter Instability and Structural Change with Unknown Change Point.

but since logging operations and the burning and reburning that followed this logging, aspen and white birch have taken over the role as dominants. Thirty-two islands and 5 mainland areas were chosen for the study. Three species of hibernators, Eutamias minimum, Tamias striatus and Zapus hudsonias were taken on the mainland with only the Zapus being found on the islands. All six of the non-hibernating species taken were found on the islands. These include Microtus pennsylvanicus, Synaptomys cooper, Clethrionomys gapperi, Peromyscus rncaniculatus, Blarina brevicauda and Sorex cinereus. The degree of occupancy changed from 4 out of 21 islands in 1950 to 21 of 24 islands in 1952. Along with this increase in the proportion of islands occupied there was an increase in the number of species found on most of the islands although the average number of species found on each occupied island did not change appreciably. This increase in islands occupied was correlated with an increase in the density of the mainland populations. The water seemed to be an efficient barrier to travel during the summer but the ice of winter became a highway for dispersal of the various nonhibernating species in the winter. Considerable difference was found in the vulnerability of the several species to trapping. The Perornyscus were readily caught. The Clethrionomys were not as easy to capture as the former but were still readily taken. The Microtus, however, posed a different problem. They were difficult to capture and on two islands of about 2.2 and 4 acres respectively we were not able to elim-inate them after 16 days of trapping on a 15-foot grid. The populations of small mammals fluctuate quite widely and the several populations appear to be somewhat independent of each other. The islands appeared to lag behind the mainland in the development of their populations.

https://www.jstor.org/stable/pdfplus/1931034.pdf

Distance to Nearest Neighbor as a Measure of Spatial Relationships in Populations

This paper proposes a residual-based Lagrange multiplier (LM) test for a null that the individual observed series are stationary around a deterministic level or around a deterministic trend against the alternative of a unit root in panel data. The tests which are asymptotically similar under the null, belong to the locally best invariant (LBI) test statistics. The asymptotic distributions of the statistics are derived under the null and are shown to be normally distributed. Finite sample sizes and powers are considered in a Monte Carlo experiment. The empirical sizes of the tests are close to the true size even in small samples. The testing procedure is easy to apply, including, to panel data models with fixed effects, individual deterministic trends and heterogeneous errors across cross-sections. It is also shown how to apply the tests to the more general case of serially correlated disturbance terms.

Testing for Stationarity in Heterogeneous Panel Data

The statistical problem treated is that of testing the hypothesis that $n$ independent, identically distributed random variables have a specified continuous distribution function $F(x)$. If $F_n(x)$ is the empirical cumulative distribution function and $\psi(t)$ is some nonnegative weight function $(0 \leqq t \leqq 1)$, we consider $n^{\frac{1}{2}} \sup_{-\infty<x<\infty} \{| F(x) - F_n(x) | \psi^\frac{1}{2}\lbrack F(x) \rbrack\}$ and $n\int^\infty_{-\infty}\lbrack F(x) - F_n(x) \rbrack^2 \psi\lbrack F(x)\rbrack dF(x).$ A general method for calculating the limiting distributions of these criteria is developed by reducing them to corresponding problems in stochastic processes, which in turn lead to more or less classical eigenvalue and boundary value problems for special classes of differential equations. For certain weight functions including $\psi = 1$ and $\psi = 1/\lbrack t(1 - t) \rbrack$ we give explicit limiting distributions. A table of the asymptotic distribution of the von Mises $\omega^2$ criterion is given.

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Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes

Some (large sample) significance points are tabulated for a distribution-free test of goodness of fit which was introduced earlier by the authors. The test, which uses the actual observations without grouping, is sensitive to discrepancies at the tails of the distribution rather than near the median. An illustration is given, using a numerical example used previously by Birnbaum in illustrating the Kolmogorov test.

https://ecsocman.hse.ru/data/133/008/1218/journal_of_the_american_statistical_association49.pdf

A Test of Goodness of Fit

An inventory pricing problem

An integral representation of the positive stable distributions is used to give a tabulation of the asymptotic test statistic for Watson's optimal invariant test for uniformity of distribution on a circle.

/pdf/on-the-asymptotic-distribution-of-watson-s-statistic-55o3i6fsm9.pdf

Donald Allan Darling

Papers

Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes

A Test of Goodness of Fit

An inventory pricing problem

On the Asymptotic Distribution of Watson's Statistic