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A general method is developed for testing the goodness of fit of a continuous distribution against unspecified alternatives when the null hypothesis specifies only the functional form of the distribution and leaves some or all of its parameters unspecified. The exponential and normal distributions are treated in detail. For these distributions tables of percentile points of the test statistics are provided. Power comparisons of our tests with those developed by Lilliefors are also given. All the numerical results involved are derived using Monte Carlo methods.

Neyman & Pearson's (1933) conception of the power of a test of a statistical hypothesis, Ho, was developed, in the first instance as a means of guiding the choice between alternative tests. This, it was shown, could be done by comparing the effectiveness of the tests in discriminating between Ho and a set of admissible alternative hypotheses regarded as most relevant to the question under test. Where there is no doubt about the most appropriate test and no sequential scheme of sampling is possible, the power function may play a useful part in indicating, before the data are collected, how large the samples should be to avoid an inconclusive result. If this procedure is to be easily applied, a ready means must be available of calculating the power of the test for a given significance level and sample size. The tables of the power function of the t-test (Neyman & Tokarska, 1936) and Tang's Tables (1938) applicable in the Analysis of Variance, are examples of such aids. The present paper aims at providing in simple, if approximate, form a means of determining the power function of the test for the difference between two proportions. The test may be briefly outlined as follows. In two 'infinite' populations the proportions of individuals possessing a character A are pl(A) and p2(A) respectively. Random samples of m and n are drawn from the two populations and the result is represented in a 2 x 2 table, thus

SUMMARY The point of change in mean in a sequence of normal random variables can be estimated from a cumulative sum test scheme. The asymptotic distribution of this estimate and associated test statistics are derived and numerical results given. The relation to likelihood inference is emphasized. Asymptotic results are compared with empirical sequential results, and some practical implications are discussed. The cumulative sum scheme for detecting distributional change in a sequence of random variables is a well-known technique in quality control, dating from the paper of Page (1954) to the recent expository account by van Dobben de Bruyn (1968). Throughout the literature on cumulative sum schemes the emphasis is placed on tests of departure from initial conditions. The purpose of this paper is to examine a secondary aspect: estimation of the index T in a sequence {xt}, where the departure from initial conditions has taken place. The work is closely related to an earlier paper by Hinkley (1970), in which maximum likelihood estimation and inference were discussed. We consider specifically sequences of normal random variables x1, ..., xT, say, where initially the mean 00 and the variance o2 are known. A cumulative sum, cusum, scheme is used to detect possible change in mean from 00, and for simplicity suppose that it is a one-sided scheme for detecting decrease in mean. Then the procedure is to compute the cumulative sums t

On account of the unique property of samples from a normal population that the ratio n+1 (x-Fb)/V(n + 1)/s (where jt is the population mean, x = x1/(n + 1) and ns2 =Z(Xi= 1 is the ratio of a normal deviate to a stochastically independent estimate of its variance, Student's t-test is a suitable test of significance for the mean of a normal population. However, in a variety of cases, it is necessary to test for the mean of a population which does not follow the Gaussian law. Efforts have, therefore, been made to see how far Student's distribution may be used for the purpose in non-normal populations. Due, mainly, to the analytical difficulties of the problem, no extensive theoretical discussion has yet been given. Thus, Pearson & Adyanthaya (1929), Rietz (1939) and Nair (1941) have given experimental treatments, while the theoretical discussions of some others (Rider, 1929; Perlo, 1933; Laderman, 1939) have dealt only with trivially small sample sizes. The papers by Bartlett (1935) and Geary (1936, 1947) give results true for any sample size, though they are based on certain assumptions and approximations. The present paper deals with the population considered by Geary in his 1936 paper, subject to the same approximations. The second contribution by Geary (in which is derived the t-distribution in samples from a population which departs more from normality than that considered in the 1936 paper) came to my notice too late to be made use of in the present work; but it is proposed to consider it later on. Geary (1936) has obtained the distribution of the ratio (x -It) (n + 1)/s in the case of an asymmetrical population, whose fourth and higher cumulants are zero, by neglecting squares and higher powers of the third cumulant. We know from this how far the probability of an error of the first kind (i.e. the probability of rejecting the null hypothesis when it is true) in such a population differs from that for a normal distribution, provided we may neglect the square of the standardized third cumulant yl. Here again, on account of analytical difficulties, it is not possible, except for very small sample sizes, to consider the effect of terms containing higher powers of yl. However, we can assume the result derived by Geary to be correct for very small values of yl, as also for large sample sizes-but in such cases the deviation from values of the normal theory is practically negligible. Even then, it is of interest to know whether, in using the usual tables of the t-test (based on the normal distribution), we are committing the greater error in the probability of an error of the first kind or in that of an error of the second kind. In the present paper are derived the values of the probability of an error of the second kind (and hence, of the power of the test) when the usual t-tables are used to define the critical region. It may be mentioned here that this problem is only a special case of a general investigation, on which the writer is engaged, into the effect, on statistical tests, of differences between the actual and the assumed distribution laws of the universe sampled. The solution of these problems is hampered by analytical difficulties in the derivation of the probability laws (and particularly of power functions), and the present case is one of the few in which a mathematical, though only approximate, solution has been found possible.

SUMMARY The comparison of sensitivities is considered in detail for a random effects model in which the different methods of measurement are applied to the same experimental units. An exact distribution is obtained and tabulated and approximations to it considered. Other applications of the results are outlined. The problem of comparing two or more methods of measuring the effects of experimental treatments arises in various fields of research. Cochran (1943) discussed this problem in considerable detail under the assumption that analysis of variance techniques were applicable. He suggested that a comparison of the sensitivities of two scales of measurement, or two experimental techniques, should depend both on the magnitudes of treatment effects and the experimental errors associated with the measurement scales, or experimental techniques. A method of measurement for which the differences between treatment effects show up well, and which, at the same time, has a relatively small error variance, will be considered a sensitive method. In this paper we distinguish between three types of situation in experimentation: (i) Type 1. The different methods of measurement are applied to the same experimental units. This can be done whenever not more than one method affects the experimental units. The results of the methods will be correlated. (ii) Type 11. The different methods of measurement are applied to separate subsamples of the same experimental units. Here only the 'treatment effects' for the methods will be correlated. (iii) Type 111. The different methods of measurement are applied to independent experimental units. Schumann & Bradley (1957, 1959) considered the type III situation. Following Cochran's line of thought, they effected a comparison of the sensitivities of two experiments through a comparison of two variance ratios. Their work was restricted to independent experiments and they were mainly concerned with similar experiments in which the variance ratios compared had identical degrees of freedom. They calculated critical values of the ratios of two independent F's to be used in hypothesis tests to compare sensitivities. Schumann & Bradley assumed that analysis of variance techniques were applicable and considered both model I (fixed effects model) and model II (random effects model) of the analysis of variance. Dar (1962) derived a normal approximation to the Schumann & Bradley density function for the ratio of two F's. He showed that the logarithm of the ratio was approximately normally distributed for large degrees of freedom. He also extended the theory to obtain an approximate chi-square test for the comparison of the sensitivities of several

SUMMARY The problem of detecting a single outlier in a linear regression model is considered. Results of Srikantan (1961), David & Paulson (1965) and Doornbos (1966) are unified and extended. The various cases considered are: (i) known variance, (ii) external studentization, and (iii) pooled studentization. One-sided and two-sided test statistics for detecting a single outlier are proposed. These statistics are maxima of suitably standardized or studentized weighted residuals. Bonferroni and other inequalities are used to obtain upper and lower limits for the true upper percentage points of the proposed statistics. Some appropriate measures of performance are introduced and studied.

Three different tests have been proposed for the Wilks-Lawley hypothesis of the homogeneity of group means in multivariate analysis (i.e. of the equality of np-variate normal means). These tests are: the likelihood ratio test (Wilks, 1932); the analysis of variance analogue (Lawley, 1938) and the generalized Mahalanobis distance statistic (Pillai, 1954)*. We will call these test statistics W, L and D, respectively; their expressions as functions of the observations are given in ? 1-2. below. Under normal theory these tests are asymptotically equivalent. That is, suitably scaled, they have the same asymptotic distribution both when the null and the alternative hypotheses are true. This extends to non-normal theory, on central limit theorem considerations, provided the homoscedasticity and independence assumptions are retained. They are of course completely equivalent for all sample sizes in the univariate case, and also in the two-sample case. There is little to choose between them in respect of ease of computation and whilst intuitively it may be felt that W would be more seriously affected by departures from normality in small samples, relatively little is known of the robustness of the tests. (Barton & David, 1962 have studied the randomization theory of D and their results are available for the investigation of the effects of nonnormality on the first kind of error entailed by its use.) The null hypothesis percentage points of W were tabulated by Pearson & Wilks (1933) and Pillai (1954), gives those of D to a high degree of approximation. Both tabulations are for the usual percentage points, taken over a small range of values of N and p in the case of Pearson & Wilks, and a wide range of N andp in the case of Pillai; here N is the total sample size. The upshot of these considerations is that the distinction between the tests rests almost entirely on the comparison of their powers in small samples. Such comparison is made in the present paper for selected values of the several parameters concerned. In particular, the calculations have been restricted to the bivariate case p = 2, although the methods have been designed with the aim of application when p has any value. The overall aim has been to develop simple approximate methods of evaluating powers, checking the accuracy of approximation by comparison with exact values at points where these are obtainable by means of the theorems of Kshirsagar (1961) or the Fisher-Hsu-Roy theorem (cf. Hsu, 1940

SUMMARY The known distribution of residuals from linear regression models may be used to construct exact tests of significance. New tests for the presence of one or more outliers are considered in detail. Applications of the theory to other tests are discussed. Exact results are worked out for the normal and exponential error distributions; formulae are given for other nonnormal cases. All statistical tests are based on some model specifying the form or structure of the response. Linear models are a large and important class of the models currently used. The statistical tests based on linear models fall generally into two categories: (i) tests within the model that are sensitive to departures from some hypothesis about the parameters of the model; and (ii) tests of the model that are sensitive to departures from the assumptions of the model regardless of the parameters within the model. Tests of the latter type are based on the normalized residual vector, or some function of it, which has a known marginal distribution independent of the parameters of the model. Tests within the model are made conditionally given this ancillary residual vector. However, the distinction between these tests, at least for normal models, exists more in theory thain in practice. Section 2 contains some preliminary definitions and results which lead, in ? 3, to the distribution of residuals in both normal and nonnormal cases. In ? 4 a class of significance tests based on the structure of the regression problem is proposed and in ? 5 it is shown that the common analysis of variance tests for normal models belong to this class. In ? 6 this theory is applied to the problem of testing for the presence of one or more outliers. Examples of the derived tests are given for normal and exponential cases. Finally, in ? 7 the relation to other tests for nonadditivity and nonnormality is discussed.

A great deal of use has undoubtedly been made of least squares regression methods in circumstances in which they are known to be inapplicable. In particular, they have often been employed for the analysis of time series and similar data in which successive observations are serially correlated. The resulting complications are well known and have recently been studied from the standpoint of the econometrician by Cochrane & Orcutt (1949). A basic assumption underlying the application of the least squares method is that the error terms in the regression model are independent. When this assumption—among others—is satisfied the procedure is valid whether or not the observations themselves are serially correlated. The problem of testing the errors for independence forms the subject of this paper and its successor. The present paper deals mainly with the theory on which the test is based, while the second paper describes the test procedures in detail and gives tables of bounds to the significance points of the test criterion adopted. We shall not be concerned in either paper with the question of what should be done if the test gives an unfavourable result.

In two previous papers, a model for multivariate paired comparisons was proposed and the associated methodology and large sample properties were developed. The present study considers the problem of relating the response pattern on overall quality to that on a specified set of attributes. A regression equation is derived for a joint distribution of responses to dichotomous items and is applied to the multivariate paired comparison model. A test of the significance of the responses to specified attributes in estimating the responses to overall quality is presented. (Author)

In the Neyman-Pearson theory of testing statistical hypotheses, the efficiency of a statistical test is to be judged by its power of detecting departures from the null hypothesis. Thus besides knowing the random sampling distribution of a given statistic T under this hypothesis, say Ho, it is also necessary to know the distribution of T under admissible hypotheses alternative to Ho. Hence the power function of the test is obtained. In the case of the wellknown tests using X2, t and F, the evaluation of their power functions involves the use of what have been called non-central distributions. For example, if we are applying the t-test to examine if a sample has come from a normal population with mean # = 0(HO), we know that under Ho, t has a 5 % chance of exceeding the 5 % point of its distribution. But in order to compute the power of the test we wish to know the chance that t exceeds this point when M has alternative values, not equal to zero. This chance is given by the non-central t-integral. This distribution has been studied by Fisher (1931), Neyman (1935), Neyman & Tokarska (1936) and Johnson & Welch (1939). In a similar way, the non-central X2and F-distributions arise in consideration of the power functions of the X2and variance-ratio tests. The power function may be used either to determine the extent of the departures from Ho in a given direction, which will be detected as significant (at a prescribed level) with a given probability, or it may be used to determine in advance the size of experiment necessary to ensure that a worth-while difference will be established as significant, if it exists. But apart from its value in this connexion, the study of non-central distributions is of considerable interest. The mathematical forms of these distributions of t, x2 and F have been long known, but their use without extensive tabling has not been easy. The present paper is therefore concerned with two lines of investigation: (a) The derivation of certain approximations to the probability integrals of (i) non-central X2, and (ii) the ratio of non-central X2 to an independent central X2, which we have termed noncentral F. These approximations, depending on tabled functions, permit easy calculation. (b) Discussion of the ways in which these distributions may be used in connexion with the power functions of statistical tests.

SUMMARY The smooth empirical Bayes approach is based on using past observations to estimate a specified type of approximation to the prior distribution. This approach is applied to problems of empirical Bayes hypothesis testing and to the compound decision problem. Problems involving composite hypotheses are considered which have, hitherto, received much less attention than the case of two simple hypotheses. The methods are shown to be asymptotically optimal and results of studies involving finite sample numbers are reported, giving some indication of the rate of approach to optimality in specific cases.

There are a number of variable selection procedures in current use in multiple regression analysis. In some of these procedures, a new variable is entered into regression when its contribution to the explanation of the response is larger than that of any other variable not yet entered. The new entrant is then tested by a standard F-test to see if its contribution is 'significant'. Because of the method of selecting the largest contributor, an F-test of this type cannot be theoretically correct but the procedure is used because the exact distributions are not known. This raises questions like 'Is one seriously in error in using the F-tests.' 'If so, can adjustments be made to the F-test to correct the difficulty.' The purpose of this paper is to discuss some new results which throw light on these questions. (Author)

SUMMARY In this paper the locally most powerful rank test for testing the symmetry axis of a symmetric circular distribution is derived. Then an efficiency expression for a class of linear rank tests is obtained. The results are applied to the class of vonMises distributions and efficiency results for the sign test and the Wilcoxon test are calculated. 1. INTRODIUCTION Circular distributions are encountered in many areas of scientific investigation. Perhaps the most important example is the distribution of phases of periodic phenomena, for example, in biology and physics. Another area of application is the analysis of directions, for example, in earth sciences, migration, etc. Several examples are given by Batschelet (1965). Assume that Xl, ..., Xn are independent observations from a random variable, which takes its values on the unit circle. In this paper we are concerned with the problem of finding nonparametric tests for the hypothesis that the underlying distribution is symmetric with respect to the horizontal axis against the alternative of a displaced or rotated centre of symmetry. We proceed as follows: First we obtain a locally most powerful rank test against rotation alternatives, ? 2. After studying the distribution of a class of related test statistics under the hypothesis as well as under alternatives, ? 3, we derive the efficiency of such test sequences with respect to parametric competitors and we show that a fully efficient nonparametric test exists, ? 4. Finally, we apply some of the results to the class of von Mises distributions, ? 5.

SUMMARY Point estimates of, confidence intervals for, and hypothesis tests about the age, in generations, of a Galton-Watson process with known offspring distribution are given for the supercritical and critical cases. Extensions to other branching processes are discussed briefly.

SUMMARY Consider data arranged into k 2 x 2 contingency tables. The principal result is the derivation of a statistical test for making an inference on whether each of the k contingency tables has the same relative risk. The test is based on a conditional reference set and can be regarded as an extension of the Fisher-Irwin treatment of a single 2 x 2 contingency table. Both exact and asymptotic procedures are presented. The analysis of k 2 x 2 contingency tables is required in several contexts. The two principal ones are (i) the comparison of binary response random variables, i.e. random variables taking on the values zero or one, for two treatments, over a spectrum of different conditions or populations; and (ii) the comparison of the degree of association among two binary random variables over k different populations. Cochran (1954) has investigated this problem with respect to testing if the success probability for each of two treatments is the same for every contingency table. Cochran's recommendation is that the equality of the two success probabilities should be tested using the total number, summed over all tables, of successes for one of the treatments. Cochran considers the asymptotic distribution of the total number of successes, for one of the treatments, conditional on all marginals being fixed in every table. He recommends this technique whenever the difference between the two populations on a logistic or probit scale is nearly constant for each contingency table. The constant logistic difference is equivalent to the relative risk being equal for each table. Mantel & Haenlszel (1959), in an important paper discussing retrospective studies, have also proposed an asymptotic method for analysing several 2 x 2 contingency tables. Their worlk on this problem was evidently done independently of Cochran, for their method is exactly the same as Cochran's except for a modification dealing with the correction factor associated with a finite population. Birch (1964) and Cox (1966) clarified the problem by showing, that under the assumption of constant logistic differences for each table, same relative risk, the conditional distribution of the total number of successes, for one of the treatments, leads to a uniformly most powerful unbiased test. Birch and Cox also derived the exact probability distribution of this conditional random variable under the given model. In this paper, we investigate the more general situation where the difference between the logits in each table is not necessarily constant. Procedures are derived for making an inference with regard to the hypothesis of constant logistic differences. Both the exact and asymptotic distributions are derived for the null and nonnull cases. This problem has been discussed by several investigators. A constant logistic difference corresponds to no interaction between the treatments and the k populations. The case k = 2 corresponds to one in which Bartlett (1935) has derived both an exact and an asymptotic procedure. Norton (1945)

Several sequential analogs of Stein's two-stage test procedure are presented for testing hypotheses about the mean of a normal population with unknown variance and with specified error probabilities. When sequential experimentation is feasible, they provide alternatives to the sequential normal test (variance known) or the sequential t-test. If the variance is assumed known, the procedures may still be recommended since the added cost may be only a very few additional observations on the average, and the performance of the tests does not depend on the validity of any assumption about the variance. Unlike the t-test, these procedures do not require that the alternative hypothesis be specified in standard deviation units.

SUMMARY Randomization models for 2 x 2 tables in the matched and unmatched cases are proposed and, in terms of these models, the usual large sample statistical tests are shown to be conservative. The likelihood surface for the unmatched case is discussed. In the matched case, the model takes more explicit account of the effect of the matching criteria used in the trial than do the more familiar probability models. A simple analogue in the case of continuous measurements is indicated.

SUMMARY In an earlier paper the authors gave a general definition of residuals and derived some asymptotic properties. In the present paper the results are extended to study test statistics computed from residuals.

Parametric and nonparametric procedures for the prediction of a time series are discussed. In each case the increase in the mean squared error of prediction over its minimum level due to the use of estimated spectra is assessed. The fitting of simple parametric models as approximations is also discussed. (Author)

P. L. Hsu (1940) has shown that for any linear hypothesis the E2-test is the uniformly most powerful of all the tests whose power function depends on a certain function, A, of the population parameters. Two other tests of importance, namely, those associated with the multiple correlation coefficient and Hotelling's T2 (Hotelling, 1931), have the similar property of being uniformly more powerful than all other tests whose power functions depend on the respective functions of population parameters involved in the distributions of R and T2. It is the purpose of this paper to establish such an optimum property of these two tests. We shall consider them separately.

Some recent developments in two-sample tests for shift alternatives on a circle and in tests for uniformity of a circular distribution are related in this paper. An intuitive correspondence, supported by formal analysis, is found between the two classes of test statistics in form, asymptotic null distribution and asymptotic behaviour under alternatives and leads to a method for constructing and inferring the properties of specific two-sample tests from the mathematically simpler tests for uniformity. Several useful two-sample tests on the circle, some previously suggested, the others new, are systematically derived.

IN TESTING EQUALITY OF MEANS OF TWO Normal populations, Student's t-test is not generally applicable when the variances of the two populations may be unequal and their ratio unknown. A procedure is considered whereby a preliminary test is performed to ascertain whether or not the variances may be regarded as equal. On the basis of the outcome of this preliminary test, one proceeds to test for equality of means. Tables of size and power are included to indicate the behavior of the proposed tests. (Author)

SUMMARY This paper deals with some properties of three nonparametric tests which can be used for detecting a difference in location of two populations wlhen samples are censored on the occurrence of a given order statistic. The null distributions of the test statistics are presented together with some tables of critical values. Exact expressions are obtained for the powers of the tests under exponential and rectangular alternatives. Finally, the expected durations of the tests are compared. A number of tests which permit early termination of an experiment have been proposed for detecting a difference in the location of two populations. These tests have important applications in life-testing situations where testing time may be costly. The criterion for termination may be a preselected period of time from the start of the experiment or the occurrence of a given order statistic in one of the samples or the two samples combined. In this paper we consider three nonparametric tests which may be used to provide a one-sided test for difference in location of two populations. The common feature of the tests is that the experiment is terminated on or before the occurrence of a preselected order statistic in one of the samples. We first consider the distributions of the test statistics under the null hypothesis that the two samples have been drawn randomly from populations with identical cumulative distribution functions. Some tables of critical values are presented. Expressions are given for the test powers under the alternative of a translation in an exponential distribution and for changes in the location and scale parameters of a rectangular distribution. Finally some results are given for the expected durations of the tests.

Samples of n sub 1 and n sub 2 are taken from two populations of unit vectors or directions in two dimensions. A distribution-free test is here provided of the null hypothesis that the two populations have the same form. It could be useful for detecting a difference of mean directions in unimodal populations. (Author)

A mathematical study is made of the asymptotic efficiency of certain methods of estimation for the negative binomial and the Neyman Type A distributions. The efficiency of the methed report thod of moments for these distributions is found to be high for certain triangular shaped regions in parameter space and in these triangular shaped regions variances of the distributions are close to their respective means. It is concluded that the methods using three statistics are asymptotically more efficient than the ones using only two statistics. An appendix is provided in which the efficiency of the statistical data for the various methods are listed in a group of tables.

A statistical analysis is made by use of the null hypothesis test for random samples which have been drawn from a population with the continuous distribution function F(x). It is useful for distributions on a circle since its value does not depend on the arbitrary point chosen to begin cumulating the probability density and the sample points. Sampling experiments on bird flight are plotted by use of an IBM 650 computer.

Testing statistical hypothese , Testing statistical hypothese , کتابخانه دیجیتال جندی شاپور اهواز

A statistical test based on a sequence of observations is termed closed if an upper bound, say n, is placed on the length of the sequence. In the case of Bernoulli trials several such tests have been constructed by solving a related problem in the theory of random walk with an absorbing barrier (Spicer, 1962), by modifying statistical procedures based on an infinite sequence of observations (Wald, 1947; Armitage, 1957), or by empirical methods (Bross, 1952). As each of these procedures tends to be less powerful than the optimal test based on the entire series of n observations, a sequential version (Alling, 1963, ? 4.3) of the latter type of test is of interest and will be described in this note. The proposed test, as might be anticipated, tends to be less economical in the use of observations than the closed sequential procedures currently being employed.