Distribution-free Statistics Research Articles

Testing for ordered alternatives by combining independent distribution-free block statistics

A class of distribution-free tests for ordered alternatives in a block design is presented. On each block a distribution-free statistic is selected, and a weighted sum of these statistics is then formed. A procedure for selecting the weighting constants which maximize the asymptotic relative efficiency is presented. The improvement in the efficiency of the weighted sum of block statistics over the unweighted sum is considered and proves interesting. Some attention is also given to the idea of using different types of statistics on different blocks.

One-sample distribution-free test statisticV(n,k) sensitive to location differences

A location sensitive one—sample test V(n,n) similar to the Wilcoxon two—sample test was proposed by Riedwyl (1967) and stuided by Carnal and Riedwyl (1972). A generalization for grouped data was given by Maag, Streit and Drouilly (1973). In the present paper we discuss the application of the test for grouped data. We present a table of the significance limits and discuss the approximation by means of the normal distribution.

Some distribution-free rane-like statistics having the mann-whitney-wilcoxon null distribution

In this note we suggest a class of two-sample test statistics iich have, as their null distribution,the Mann-Whitney-Wilcoxon ill distribution. An interesting property of these statistics is lat many are not rank statistics; that is, they cannot be coumplited from, the ranks of the original observations. However, they %e still distribution-free when the two populations are identi-il. This class contains the Mann-Whitney-Wilcoxon test for the niality of location parameters of two distributions and a two-aiaple test for equality of spreads of two distributions recently ivestigated by Fligner and Killeen (1976)

Adaptive Distribution-free Regression Methods and their Applications

We consider the model Y = βx + Z, where the random variable Z has a continuoustype distribution that can be badly skewed, contaminated, or censored. To test the hypothesis H 0 : β = β0, we use the distribution-free statistic K(β0) = Σc(Q i )a(R i ), where c(·) and a(·) are increasing score functions and Q i and R i are the respective ranks of x i and y i – β0 x i . The score functions c(·) and a(·) can be adapted or chosen after observing the data without destroying the distribution-free nature of the test. A Monte Carlo study is presented which illustrates the excellent performance of an adaptive test when a wide range of distributions is considered for the residuals. Interval and point estimates of sβ can be found by employing the “inverse” of the testing procedure. These results are used to find estimates of the percentile lines. Two examples are given which involve lifetimes of electric motor insulation and grade point averages of beginning university students, respectively.

A class of asymptotically distribution-free statistics similar to students's t

Birnbaum (1970) considered the problem of finding a probability inequality of the Tchebychev type for a statistic similar to Student’s t assuming only that the underlining distribution is symmetric and unimodal. Birnbaum and Vincze (1973) derived the limiting distribution of that statistic. In this paper a class of statistics related to, but, simpler than, Birnbaum’s is presented. For this class, a probability inequality is derived, and also the inequality of Birnbaum is sharpened. The limting distribution of the class is shown to be the same as that of Birnbaum’s. Extension to the case of nonsymmetric distributions is given.

A Class of Asymptotically Distribution-Free Statistics Similar to Student's t

A Class of Asymptotically Distribution-Free Statistics Similar to Student's t

Poisson process and distribution-free statistics

The well-known connection between the Poisson process and empirical c.d.f.'s is exploited from a new point of view. Distributions of functions of empirical c.d.f.'s for finite sample size n are explicitly described in some new examples, and new qualitative information is obtained for some classical examples.

Poisson process and distribution-free statistics

The well-known connection between the Poisson process and empirical c.d.f.'s is exploited from a new point of view. Distributions of functions of empirical c.d.f.'s for finite sample size n are explicitly described in some new examples, and new qualitative information is obtained for some classical examples.

Distribution-free statistical hypotheses testing for stochastic processes

This paper extends some well known distribution-free structural results in the univariate and multivariate hypothese testing to stochastic processes. The study of distribution-free tests is essentially the study of similar sets and test functions. It is found that (i) all distribution-free statistics are based on permuations through of the basic PITMAN functions; (ii) the optimal tests are based on the likelihoood function of the alternatives; (iii) that a generation of DARMOIS-PITMAN-KOOPMAN families. Furthermore, it is observed that for martingales and MARKOV processes asymptotically optimal tests can be constructed by employing various limit theorems.

On a Distribution Representing Sentence-Length in Written Prose

SUMMARY A new model for representing sentence-length distributions is suggested in equation (8) which is a special case of equation (2), with parameter y = -2 known a priori. Eight known sentence-length frequency counts taken from English, Greek and Latin prose were all satisfactorily described by distribution (8). For these eight fits, the average probability P(X2) was 0 50. A ninth observed distribution, taken from a Latin text of unknown authorship failed the x2 test applied to the fit of the data to the model in equation (8). This corroborates Yule's (1939) conclusion that it is highly unlikely that de Gerson could have written De Imitatione Christi. It is further conjectured that the last-mentioned observed frequency distribution could be well represented by the more general model in equation (2), with a parameter y much smaller than -2 THE first substantial investigation on sentence-length as a statistical tool to be used in deciding disputed authorship was published by Yule in 1939. Simple statistical indices such as the average number of words per sentence and the standard deviation of sentence-lengths were employed. Yule did not suggest a particular mathematical distribution model. Later (Yule, 1944) he explored word-frequency of an author in addition to sentence-length. Although Yule mentions in that book the negative binomial, he discards this distribution model as totally inadequate for representation of word frequencies and sentence-lengths. Williams (1940, 1970) suggests and uses the lognormal distribution as a model for sentence-length. To verify lognormality, Williams plots the observed cumulative percentage frequencies of sentence-lengths on log-probability paper in the hope that these plots will approach a straight line. No x2 tests are given for any of Williams's examples. Wake (1957), who discusses sentence-lengths in works of Greek authors, also makes use of the lognormal distribution by superimposing the observed histograms of the logarithms of sentence-lengths over the expected normal distributions. No x2 tests are given. The authorship of Greek prose is again investigated by Morton (1965) who works with distribution-free statistics such as the mean, the median, the quartiles and the deciles. Mosteller and Wallace (1963), in their study of the authorship of the Federalist papers, came to the conclusion that the mean and standard deviation of sentence- length was of no help in solving disputed authorship. In their particular research Mosteller and Wallace found the mean and standard deviations of sentence-length to be virtually identical for Madison and Hamilton. It can be shown, however, that two

A distribution-free test for symmetry in hierarchical data

Distribution-free tests for permutational symmetry of correlated variables have been offered recently in the literature. However, situations arise when all the planned measurements are available only on some subjects, while others drop out at various stages of the experiment. The present paper offers a distribution-free statistic for testing permutational symmetry of correlated variables in such hierarchical data, and develops a large-sample chi-square approximation. Explicit expressions for the test statistic have been provided for some special cases.

On the Goodness-of-Fit Problem for Continuous Symmetric Distributions

Abstract This article considers the problem of testing a completely specified continuous symmetric distribution against alternatives which are also symmetric about the same point. The symmetry is utilized in obtaining a new distribution-free statistic of the Kolomogorov-Smirnov type which can be used to halve the width of the Kolomogorov-Smirnov confidence band for the unknown distribution function.

Completeness theorems for characterizing distribution-free statistics

Completeness theorems for characterizing distribution-free statistics

Application of distribution-free statistics to the structural analysis of slow neutron cross section and resonance parameter data

The discovery of narrow intermediate structure and its explanation in terms of shape isomerism has encouraged the search for structure in the energy dependence of low-energy neutron cross-section and resonance parameter data. It is shown how the Wald and Wolfowitz distribution-free statistic can be applied to test for the energy-dependent structure of resonance parameter data in the presence of the well-known fluctuations which these parameters undergo from resonance to resonance. The statistic can also be used on a sequence of serial correlation coefficients and thereby test the assumed energy dependence of cross-section data. Illustrative examples refer to the neutron widths of 232Th, the fission and neutron widths of 234U and the serial correlation coefficients derived from the fission cross sections of 239Pu and 235U.

Analysis of remanent systematic error effects on circuit closures in geodetic spirit levelling

This paper is a summary of an unpublished paper and it attempts to give a complete solution of the problem posed by the presence of systematic errors in precise spirit levelling, using mathematical statistics. Distribution-free statistics, which do not require the precise knowledge of the distribution of the underlying populations, are used and to this end, a value of the expected systematic error in any given section is isolated from the total discrepancy of the independent mutual levelling operations thus facilitating a plausible least square adjustment of the geodetic network. The problem of weighting is considered in relation to the rank test-statistics. Complete mathematical derivations are not attempted in this summary.

Some distribution-free statistics and their application to the selection problem

The definition of the randomized rank-sum statistics introduced by Bell and Doksum [2] is extended to the case ofk populations, and properties of the statistics are developed which generalize some of the results of Bell and Doksum [2]. Both the randomized rank-sum and the nonrandomized rank-sum (heretofore called ‘scores’) statistics are employed in Gupta’s [9] subset approach to thek population selection problem. A useful small-sample property of the randomized rank-sum procedure is demonstrated. The asymptotic relative efficiency (ARE) of the procedures based on the two types of rank-sum statistics is shown to be equal to unity, and the ARE of either relative to the procedure based on sample means is shown to be the same as that of the ‘scores’ procedure relative to the means procedure in thek-sample testing problem. The three procedures are shown to have the same asymptotic performance characteristic when sample size ratios equal the asymptotic relative efficiencies. It is further shown that an ‘indifference zone’ procedure based on the randomized rank-sum statistics and the ‘indifference zone’ ‘scores’ procedure are asymptotically equally efficient.

A survey on distribution-free statistics based on distances between distribution functions

A survey on distribution-free statistics based on distances between distribution functions

Distribution-Free Tests of Independence

The object of this article is to characterize the family of all distribution-free (DF) tests of independence, and those subfamilies which are optimal for specified alternative classes. It is found (Theorem 3.2) that each DF statistic is a function of a Pitman (or permutation) statistic; and (Theorem 3.4) that the rank statistics are those whose distributions depend appropriately on the maximal invariant [Lehmann (1959), p. 227]. For parametric alternatives the MP (most powerful) [Lehmann and Stein (1949)] and the locally MP tests are found to be Pitman tests based on the likelihood function (Lemma 4.1 and Theorem 4.1); while the corresponding optimal rank tests are analogous (Lemmas 4.2, 4.3) to those in the 2-sample case [e.g. Capon (1961)], and are closely related to those of Bhuchongkul (1964). In Section 5 randomized statistics similar to those of [4] are shown (Corollary 5.2) to be asymptotically equivalent to the optimal nonrandomized statistic for specified parametric alternatives. For one reasonable nonparametric class of alternatives one proves (Theorem 6.1) that the normal scores test is minimax; while for the other class an unexpected statistic (Theorem 6.2) is minimax. Finally, in Section 7 the ideas of monotone tests developed by Chapman (1958) and others are extended, and analogous results (Theorem 7.1) are obtained for minimum power.

"Optimal" One-Sample Distribution-Free Tests and Their Two-Sample Extensions

The object of this paper is the development of a theory of optimal one-sample goodness-of-fit tests and of optimal two-sample randomized distribution-free (DF) statistics analogous to the well-known results of Hoeffding (1951), Terry (1952), Lehmann (1953), (1959), Chernoff and Savage (1958), Capon (1961) and others for two-sample nonrandomized rank statistics. For $Y_1, \cdots, Y_n$ a random sample from a population with continuous distribution function $G$, one tests in the one-sample case $H_0 : G = F$ vs. $H_1 : G \neq F$, where $F$ is some known continuous distribution function. From the Neyman-Pearson lemma, distribution-free tests that are most powerful (MP) for any $H$ vs. $K$ satisfying $KH^{-1} = GF^{-1}$, are obtained. From these MP distribution-free tests, one can on paralleling the derivations ([14], [25], [17], [18], [7]) for locally MP tests in the two-sample case obtain locally MP tests in the one-sample case. Further, it is found that the class of alternatives, for which a critical region of the form $\lbrack\sum J\lbrack F(y_i)\rbrack > c\rbrack$ is locally MP, is the class of $G$'s that consists of "contaminated" Koopman-Pitman distributions as given in Section 5. Randomized versions of the two-sample MP and locally MP rank statistics are considered and shown to be asymptotically equivalent to the locally MP rank statistics.

Goodness Criteria for Two-Sample Distribution-Free Tests

Some of the concepts and results of Chapman [3] for one-sample distribution-free tests are extended to the two-sample problem. Chapman's lead will be followed since his work gives a goodness criterion for comparing distribution-free tests, for finite sample sizes, over a large class of one-sided alternatives. Since all two-sample distribution-free statistics known to the authors satisfy Scheffe's [10] boundary condition, and all, except those of Pitman, also are strongly distribution-free (SDF) and therefore [1] are rank statistics, consideration may reasonably be restricted to rank tests. Such tests with the additional property of monotonicity are unbiased, partially ordered, and assume maximum and minimum powers for certain reasonable alternatives. Some of the maximum powers of the Mann-Whitney-Wilcoxon, Fisher-Yates, van der Waerden, Doksum, Savage, Epstein-Rosenbaum, and Cramer-von Mises statistics are tabulated. (For definitions and references, see Section 6.)