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

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An Analysis of Variance Test for Normality (Complete Samples)

Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution—the part of the distribution representing large but rare events—and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov (KS) statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data, while in others the power law is ruled out.

/pdf/power-law-distributions-in-empirical-data-4dybihkltl.pdf

Power-Law Distributions in Empirical Data

http://www.people.vcu.edu/~dbandyop/BIOS625/Peduzzi1996.pdf

A simulation study of the number of events per variable in logistic regression analysis.

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

1 Compositional data: some challenging problems.- 1.1 Introduction.- 1.2 Geochemical compositions of rocks.- 1.3 Sediments at different depths.- 1.4 Ternary diagrams.- 1.5 Partial analyses and subcompositions.- 1.6 Supervisory behaviour.- 1.7 Household budget surveys.- 1.8 Steroid metabolite patterns in adults and children.- 1.9 Activity patterns of a statistician.- 1.10 Calibration of white-cell compositions.- 1.11 Fruit evaluation.- 1.12 Firework mixtures.- 1.13 Clam ecology.- 1.14 Bibliographic notes.- Problems.- 2 The simplex as sample space.- 2.1 Choice of sample space.- 2.2 Compositions and simplexes.- 2.3 Spaces, vectors, matrices.- 2.4 Bases and compositions.- 2.5 Subcompositions.- 2.6 Amalgamations.- 2.7 Partitions.- 2.8 Perturbations.- 2.9 Geometrical representations of compositional data.- 2.10 Bibliographic notes.- Problems.- 3 The special difficulties of compositional data analysis.- 3.1 Introduction.- 3.2 High dimensionality.- 3.3 Absence of an interpretable covariance structure.- 3.4 Difficulty of parametric modelling.- 3.5 The mixture variation difficulty.- 3.6 Bibliographic notes.- Problems.- 4 Covariance structure.- 4.1 Fundamentals.- 4.2 Specification of the covariance structure.- 4.3 The compositional variation array.- 4.4 Recovery of the compositional variation array from the crude mean vector and covariance matrix.- 4.5 Subcompositional analysis.- 4.6 Matrix specifications of covariance structures.- 4.7 Some important elementary matrices.- 4.8 Relationships between the matrix specifications.- 4.9 Estimated matrices for hongite compositions.- 4.10 Logratios and logcontrasts.- 4.11 Covariance structure of a basis.- 4.12 Commentary.- 4.13 Bibliographic notes.- Problems.- 5 Properties of matrix covariance specifications.- 5.1 Logratio notation.- 5.2 Logcontrast variances and covariances.- 5.3 Permutations.- 5.4 Properties of P and QP matrices.- 5.5 Permutation invariants involving ?.- 5.6 Covariance matrix inverses.- 5.7 Subcompositions.- 5.8 Equivalence of characteristics of ?, ?, ?.- 5.9 Logratio-uncorrelated compositions.- 5.10 Isotropic covariance structures.- 5.11 Bibliographic notes.- Problems.- 6 Logistic normal distributions on the simplex.- 6.1 Introduction.- 6.2 The additive logistic normal class.- 6.3 Density function.- 6.4 Moment properties.- 6.5 Composition of a lognormal basis.- 6.6 Class-preserving properties.- 6.7 Conditional subcompositional properties.- 6.8 Perturbation properties.- 6.9 A central limit theorem.- 6.10 A characterization by logcontrasts.- 6.11 Relationships with the Dirichlet class.- 6.12 Potential for statistical analysis.- 6.13 The multiplicative logistic normal class.- 6.14 Partitioned logistic normal classes.- 6.15 Some notation.- 6.16 Bibliographic notes.- Problems.- 7 Logratio analysis of compositions.- 7.1 Introduction.- 7.2 Estimation of ? and ?.- 7.3 Validation: tests of logistic normality.- 7.4 Hypothesis testing strategy and techniques.- 7.5 Testing hypotheses about ? and ?.- 7.6 Logratio linear modelling.- 7.7 Testing logratio linear hypotheses.- 7.8 Further aspects of logratio linear modelling.- 7.9 An application of logratio linear modelling.- 7.10 Predictive distributions, atypicality indices and outliers.- 7.11 Statistical discrimination.- 7.12 Conditional compositional modelling.- 7.13 Bibliographic notes.- Problems.- 8 Dimension-reducing techniques.- 8.1 Introduction.- 8.2 Crude principal component analysis.- 8.3 Logcontrast principal component analysis.- 8.4 Applications of logcontrast principal component analysis.- 8.5 Subcompositional analysis.- 8.6 Applications of subcompositional analysis.- 8.7 Canonical component analysis.- 8.8 Bibliographic notes.- Problems.- 9 Bases and compositions.- 9.1 Fundamentals.- 9.2 Covariance relationships.- 9.3 Principal and canonical component comparisons.- 9.4 Distributional relationships.- 9.5 Compositional invariance.- 9.6 An application to household budget analysis.- 9.7 An application to clinical biochemistry.- 9.8 Reappraisal of an early shape and size analysis.- 9.9 Bibliographic notes.- Problems.- 10 Subcompositions and partitions.- 10.1 Introduction.- 10.2 Complete subcompositional independence.- 10.3 Partitions of order 1.- 10.4 Ordered sequences of partitions.- 10.5 Caveat.- 10.6 Partitions of higher order.- 10.7 Bibliographic notes.- Problems.- 11 Irregular compositional data.- 11.1 Introduction.- 11.2 Modelling imprecision in compositions.- 11.3 Analysis of sources of imprecision.- 11.4 Imprecision and tests of independence.- 11.5 Rounded or trace zeros.- 11.6 Essential zeros.- 11.7 Missing components.- 11.8 Bibliographic notes.- Problems.- 12 Compositions in a covariate role.- 12.1 Introduction.- 12.2 Calibration.- 12.3 A before-and-after treatment problem.- 12.4 Experiments with mixtures.- 12.5 An application to firework mixtures.- 12.6 Classification from compositions.- 12.7 An application to geological classification.- 12.8 Bibliographic notes.- Problems.- 13 Further distributions on the simplex.- 13.1 Some generalizations of the Dirichlet class.- 13.2 Some generalizations of the logistic normal classes.- 13.3 Recapitulation.- 13.4 The Ad(?,B) class.- 13.5 Maximum likelihood estimation.- 13.6 Neutrality and partition independence.- 13.7 Subcompositional independence.- 13.8 A generalized lognormal gamma distribution with compositional in variance.- 13.9 Discussion.- 13.10 Bibliographic notes.- Problems.- 14 Miscellaneous problems.- 14.1 Introduction.- 14.2 Multi-way compositions.- 14.3 Multi-stage compositions.- 14.4 Multiple compositions.- 14.5 Kernel density estimation for compositional data.- 14.6 Compositional stochastic processes.- 14.7 Relation to Bayesian statistical analysis.- 14.8 Compositional and directional data.- Problems.- Appendices.- A Algebraic properties of elementary matrices.- B Bibliography.- C Computer software for compositional data analysis.- D Data sets.- Author index.

The Statistical Analysis of Compositional Data

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.

https://www.jstor.org/stable/pdfplus/2286009.pdf

EDF Statistics for Goodness of Fit and Some Comparisons

Goodness-of-Fit Techniques

"a comprehensive book, well presented and clearly written." The New Zealand Statistician "Chapters contain some theory but they are mainly directed toward application with some numerical illustrations which generally use simulated sets to be found in the AppendixThe level of exposition is generally clear and directthe book should prove useful to practising statisticians." Short Book Reviews (International Statistical Institute) "the authors have carried out the helpful task of bringing together into a useful reference volume a large amount of material on an interesting topic which has previously been scattered throughout the literature." Royal Statistical Society

Goodness-of-Fit-Techniques

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...

K-Sample Anderson–Darling Tests

SUMMARY This paper gives modifications of eleven statistics, usually used for goodness of fit, so as to dispense with the usual tables of percentage points. Some test situations are illustrated, and formulae given for calculating significance levels.

Michael A. Stephens

Papers

EDF Statistics for Goodness of Fit and Some Comparisons

Goodness-of-Fit Techniques

Goodness-of-Fit-Techniques

K-Sample Anderson–Darling Tests

Use of the Kolmogorov-Smirnov, Cramer-Von Mises and Related Statistics without Extensive Tables