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In a recent article I. J. Good has used Bayesian methods to conclude that there is strong evidence in favor of the validity of Bode's law. Here we reanalyze the observed planetary distances using non-Bayesian significance testing procedures, and reach the opposite conclusion. The three purposes of this article are (1) to reexamine the validity of Bode's law; (2) to develop some insight into the more general problem of testing whether or not an observed sequence of numbers follows some simple rule; and (3) to discuss the philosophical basis of Fisherian significance testing. It is not intended to provide an example of disagreement between Bayesian and non-Bayesian methods. Rather the two analyses are seen to be similar in execution, the different conclusions arising from different choices of the statistical models. [ABSTRACT FROM AUTHOR]

A measure of variation for categorical data is discussed. We develop an analysis of variance for a one-way table, where the response variable is categorical. The data can be viewed alternatively as falling in a two-dimensional contingency table with one margin fixed. Components of variation are derived, and their properties are investigated under a common multinomial model. Using these components~ we propose a measure of the variation in the response variable explained by the grouping variable. A test statistic is constructed on the basis of these properties, and its asymptotic behavior under the null hypothesis of independence is studied. Empirical sampling results confirming the asymptotic behavior and investigating power are included. [ABSTRACT FROM AUTHOR]

In this article we study the exact finite sample distribution function of a t-statistic that can be used to test hypotheses and construct confidence intervals for structural coefficients in simultaneous equations models. It is shown that the exact distribution function is equal to Student's t-distribution when one of its parameters is zero and converges to Student's t-distribution function as another of its parameters increases without bound. Computations of the exact distribution function and its first three moments indicate the error that might result if Student's t-distribution function is used as an approximation to the exact distribution function. [ABSTRACT FROM AUTHOR]

A step-by-step method is presented herein far partitioning a certain kind of hypothesis H about the m-way contingency table into (a) a series of hypotheses about marginal tables formed from the m-way table by ignoring one or more of table's m dimensions; and (b) a hypothesis about independence, conditional independence, or conditional equiprobability in the m-way table. This step-by-step method facilitates both the testing of H and the calculation of F, the estimated expected frequencies in the m-way table under FI. The method introduced herein for calculating F is easier to apply than the usual iterative-scaling method in many cases. [ABSTRACT FROM AUTHOR]

The derivation is presented of the asymptotic distribution of the number N of observations required to terminate the classical sequential probability ratio test (SPRT) with bounds a, b (a > 0 > b) of a simple parametric hypothesis H: theta = theta[sub 1] against a simple parametric alternative K: theta = theta[sub 1] + change(change > 0) when change tends to zero. it is shown that if the common probability density function (p.d.f.) of the basic random variables belongs to the Darmois-Koopman exponential class and if the true value of theta[sub 0] of theta is different from theta[sub 1], then as change arrow right 0, N change tends in probability to a constant A > 0 which is a specified multiple of a if theta[sub 0] > theta[sub 1] and of -b if theta[sub 0] < theta[sub 1]. Also there exists B > 0 which is finite and depends on a if theta[sub 0] < theta[sub 1] and on -b if theta[sub 0] < theta[sub 1] such that (N change - A)/B square root of change tends to N(0, 1) as change arrow right 0. Finally, sigma few numerical examples are worked to illustrate the magnitudes of A and B. [ABSTRACT FROM AUTHOR]

As is true of any statistical application, testing econometric statistical hypotheses involves constructing critical regions with prescribed probabilities of Type I error and determining probabilities of Type II error under alternative hypotheses. Such constructions and determinations presuppose knowledge of exact statistical distribution functions of econometric estimators and test statistics either directly, or indirectly, as in the case where a statistical distribution function that approximates the exact distribution with known margin of error is used. If this requisite information is not available, econometric statistical inference remains guesswork. Partly because of the complicated nature of systems of simultaneous econometric structural equations and the large numbers of structural constants required to characterize distribution functions of econometric statistics, however, only a few exact marginal distributions of econometric estimators and test statistics have been extracted so far. This article is intended as a contribution to the growing literature of econometric distribution theory and to assist the efforts of econometric statisticians in providing a rational basis for econometric statistical inference. [ABSTRACT FROM AUTHOR]

The principle of minimum discrimination information estimation is described and used to generate estimates for tests of hypotheses concerning various interactions and effects in the analysis of multidimensional contingency tables. All classical hypotheses for contingency tables can be generated by the use of this principle when certain marginals are considered as fixed. Analysis of information tables are given for a four-way contingency table. [ABSTRACT FROM AUTHOR]

Let t[sub I]= x[sub I]/s, I = 1, 2 be noncentral Student t-variates with noncentrality parameters delta[sub I], where x[sub 1], x[sub 2] and s are independent. The simultaneous distribution of t[sub 1] and t[sub 2] is studied and the power function of the studentized maximum and minimum modulus tests are tabulated for significance level 0.05, degrees of freedom 1, 2, 5, 10, 20, 50, infinity and delta[sub 1], delta[sub 2] = 0(1)5. Asymptotically exact approximations to these power functions are discussed. The marginal power of the maximum modulus test is also tabulated. [ABSTRACT FROM AUTHOR]

A test is scale and location invariant when the decision to accept or reject is not affected by multiplying each data point by the same positive constant or by adding the same constant to each data point. Two well-known statistics are shown to be test statistics of certain most powerful scale and location invariant tests: the ratio of range to standard deviation for normality versus uniformity, and the ratio of minimum deviation to standard deviation for normality versus exponentiality. The most powerful scale and origin invariant test of uniformity versus exponentiality is also considered. [ABSTRACT FROM AUTHOR]

Suppose that independent samples are available from two normal populations with means mu[sub 1] and mu[sub 2] and known variances. It is desired to test H: mu[sub 1] = 0. A possible preliminary test procedure in this case consists of first testing mu[sub 1] = mu[sub 2] and then testing H using either the sample mean from the first population or the pooled sample mean depending on the outcome of the test regarding equality of mu[sub 1] and mu[sub 2]. This procedure is shown to be biased. A power comparison of this procedure with the generalized likelihood ratio test of H indicates that the latter test is more powerful in a fairly extensive parametric region. It is noted that biasedness is frequently encountered in preliminary test procedures. In light of the example, it is recommended that power comparisons be made between such procedures and any available unbiased tests to ascertain whether extensive power loss regions exist or not. [ABSTRACT FROM AUTHOR]

A one-sample nonparametric test is proposed which sequentially examines the values of the Wilcoxon signed-rank statistic. The test rejects the null hypothesis of symmetry about zero if for any n between 1 and N the Wilcoxon signed-rank statistic for X[sub 1],..., X[sub n] differs from its null mean by |z|[sup alpha] N standard deviations; otherwise the null hypothesis is accepted. The critical constant |z|[sup alpha] N is determined by Monte Carlo sampling. [ABSTRACT FROM AUTHOR]

This article investigates empirical evidence on the structure of price expectations in the United States during the post-Korean War period. The study utilizes semiannual data which describe price expectations for six months and twelve months ahead. The objective is to use these two sets of data to test some of the well-known expectational hypotheses. Incidental to this we determine whether expectations satisfy the rationality hypothesis, and we briefly consider the accuracy of the predictions. [ABSTRACT FROM AUTHOR]

The powers of several one-sided Kolmogorov tests for a shift in the location of a normal distribution with known variance are computed and compared. An accurate approximation to the power of the ordinary Kolmogorov test for this problem is also given. [ABSTRACT FROM AUTHOR]

Under a reasonable hypothesis about the distribution of trade shipments among a group of K countries, or of non-self-directed acts among any group of K individuals, the likelihood of obtaining a given table of observed frequencies n[sub ij] is [Multiple line equation(s) cannot be represented in ASCII text] where n[sub I]=n[sub i1] + ... + n[sub iK] and m[sub I] = n[sub 1i] + ... + n[sub K I] are the total number of shipments made and received, respectively, by country I, N is the total number of shipments altogether, and Pi and Q[sub I] are the theoretical tendencies of country I to ship and to receive shipment respectively and satisfy SIGMA P[sub I] = SIGMA Q[sub I] = 1. In the attempt to test the correctness of the hypothesis, a critical problem is that of finding the maximum-likelihood estimates of the P's and Q's, that is, the values making L as large as possible. We give all solutions and show that, for the usual table of observations, the solution is unique and amounts to finding the value of a for which K-2 -R[sub 1] ... - R[sub K] is 0 - [Multiple line equation(s) cannot be represented in ASCII text] [ABSTRACT FROM AUTHOR]

Cronholm and Revusky [4] have proposed a rank test based on independent subexperiments for the comparison of a treatment effect with a control when the treatment is destructive but the control produces at most a transient effect. This article provides a thorough study of its performance under two sampling situations. With the aid of a computer, the exact small sample power of the test is evaluated for some important alternatives. A modification of the test is proposed for the situation where numerical measurements are available rather than merely the ranks within each stage. The modification is found to substantially improve the power. Both tests are shown to be unbiased. Finally, the Pitman asymptotic efficiencies are obtained and comparisons are made with the appropriate Wilcoxon tests. [ABSTRACT FROM AUTHOR]

The lack of memory property of the exponential distribution plays an important part in the branch of applied probability. This property assumes the information regarding the probability distribution. In the present paper we give a characteristic property of the exponential distribution based on the means of the conditional distributions. Considering a random variable T with finite mean and such that P(T > O) > 0, and denoting by y a positive number such that P(T > y) > 0, we show that T has an exponential distribution if and only if the mean of the conditional distribution, given T > y, exceeds the mean of the unconditional distribution by the quantity y for all y. Also given in the paper is a similar characterization for the geometric distribution. Since the information concerning the expected values is easily accessible, we expect these properties to be useful in dealing with the practical problems. [ABSTRACT FROM AUTHOR]

This article proposes a refinement of the usual approximation due to Satterthwaite for making a significance test when the test statistic involves a single mean square in the numerator and a linear combination of mean squares in the denominator, that linear combination involving known positive constants. The test procedure is outlined in detail, and its derivation given in a separate section. The development of the technique essentially involves determining, by successive approximations, a function which serves as the test criterion. This function is given through terms in the cubes of the reciprocals of the degrees of freedom involved in the problem, thereby giving a deviation from the advertised significance level which is smaller (in magnitude) than the deviation provided by the Satterthwaite procedure. An expression approximating the power of the test is also developed. [ABSTRACT FROM AUTHOR]

This article reports a Monte Carlo evaluation of the levels of significance of chi-square tests for the order of two- and three-state Markov chains. Results are consistent with asymptotic theory only when samples are quite large. [ABSTRACT FROM AUTHOR]

This article suggests a distribution-free signed rank test for the parallelism of two regression lines. Pitman efficiency comparisons of the signed rank test and a normal theory t-test are presented. Monte Carlo sampling is used to compare the signed rank test with a nonparametric test proposed by Potthoff [7]. [ABSTRACT FROM AUTHOR]

Exact non-central distributions of individual characteristic roots have been obtained in two and three roots cases in connection with tests of the hypothesis are covariance matrices of two p variant normal populations. Powers of tests using individual roots are tabulated for the test of this hypothesis against various one-sided simple alternatives and comparisons of powers made. In this section the noncentral cfd's of individual roots, associated power function tabulations and comparisons of powers for testing sum of variables against various simple hypotheses. Smallest root: The non-central cdf's of the smallest root for p =2, 3 are obtained from the corresponding non-central cdf's of the largest root by making a few changes which are the same as discussed. Median root: Similarly, the non-central cdf of the median root for p =3 can be obtained from that of the largest root by effecting a few changes. Tabulations of powers of individual roots for test of hypothesis given earlier have been done extensively and are presented powers for selected values of the parameters.

For Bernoulli random variables sequential tests of the simple hypotheses p = p[SUB o] vs. p = p[sub 1] are considered. In particular a comparison is made between the performance of sequential probability ratio tests (SPRTs) and curtailed single sampling plans (CSSPs), when both tests have the same error probabilities. In [1] it is shown that the CSSP has a strong optimality property. Nevertheless, if one admits randomized SPRTs, the latter are better. Numerical examples are considered in detail for p[sub o] = 1/2 and the CSSP with n = 2. The method of finding the randomized SPRTs with the correct error probabilities is indicated. These rules have positive probability of deciding without taking any observations. [ABSTRACT FROM AUTHOR]

In this article a procedure is given for constructing confidence bands of various forms for linear regression models. Application of the procedure yields, as special cases, hyperbolic, straight line segment, uniform width and trapezoidal confidence bands. [ABSTRACT FROM AUTHOR]

There is not much in the literature on tolerance limits for discrete distributions, since generally one will find it difficult to derive explicit expressions for these limits. Distribution-free tolerance limits for discrete cases were discussed. If the form of the distribution function is known, one should try to determine the formula of the tolerance limits for the particular family of distributions to which it belongs. Otherwise, an important information is not utilized. In applications for sampling inspection reliability, inventory control, the observed random variables are very often assumed to have a binomial, negative-binomial or Poisson distribution. In these cases one can obtain explicit expressions for tolerance limits, which have desirable optimum properties. In the present article we derive uniformly most accurate upper tolerance limits for families of distributions with the monotone likelihood ratio property. For these families there exist uniformly most accurate upper confidence limits. Furthermore, the distribution functions which belong to monotone likelihood ratio families are stochastically ordered. Thus, monotone relationship exists in these cases between tolerance limits and confidence limits. This relationship is utilized to derive the formulae of uniformly most accurate tolerance limits. The general theory is presented in Section 2. Section 3 is devoted to uniformly most accurate upper confidence limits.

This study is concerned with the extension of the Bradley-Terry model for paired comparisons to situations which allow an expression of no preference. A new model is developed and its performance compared with a model proposed by Rao and Kupper. The maximum likelihood estimates of the parameters are found using an iterative procedure which, under a weak assumption, converges monotonically to the solution of the likelihood equations. It is noted that for a balanced paired comparison experiment the ranking obtained from the maximum likelihood estimates agrees with that obtained from a scoring system which allots two points for a win, one for a tie and zero for a loss. The likelihood ratio test of the hypothesis of equal preferences is shown to have the same asymptotic efficiency as that for the Rao-Kupper model. Two examples are presented, one of which introduces a set of data for an unbalanced paired comparison experiment. Initial applications of the test of goodness of fit suggest that the proposed model yields a reasonable representation of actual experimentation. [ABSTRACT FROM AUTHOR]

A sample of size N is taken from an equiprobable t-category multinomial distribution. The precise values of Pearson's X[sup 2] and the likelihood ratio statistic A were computed for t = 3(1)6, N = 3(1)12; t = 6(1)14, N = 6(1)2t; t = 15(1)18, N = 6(1)28. The logarithms of the tail areas were smoothed and fitted by means of quadratic expressions. Much information is given concerning the accuracy of the chi-squared fits to the distributions of X[sup 2] and A. A table of -log[sub 10] P(x[sup 2] > a) is given, for a = 1(1)50, and d.f. = 1(1)20. The distribution of the total number of repeats within ceils is also discussed. [ABSTRACT FROM AUTHOR]

Natural test statistics for the hypothesis that an equation is overidentified have been developed by Anderson and Rubin and by Basmann. If the disturbances are jointly normal, serially uncorrelated, and small, both the above overidentification test statistics have the Fisher variance-ratio distribution asymptotically as the variance of the error terms gets small. This gives an analytic explanation of Monte Carlo results of Basmann. The results given apply to linear models in which predetermined variables are exogenous. [ABSTRACT FROM AUTHOR]

Cochran's Q test for differences between related-sample percentages or proportions has generally been incorrectly presented in secondary sources. The most common mistake results from failure to recognize that rows containing only 1's or only O's, i.e., only successes or only failures, do not affect the value of Q. The F test, however, is affected by such rows. The probabilities from the x[sup 2] and F approximations are compared with the exact probabilities in three sets of data. A rule of thumb, based on extensive study of the distribution of Q in small samples, is given as an aid in judging when the x[sup 2] approximation is satisfactory for practical purposes. [ABSTRACT FROM AUTHOR]

The article presents information on a nomogram for the "Student"-fisher t test. A nomogram is given for estimating the probability (P) for a given value of the "Student"-Fisher t test. W.S. Gosset, an employee of the Guiness brewing company in Dublin, published papers in 1908 in which he correctly solved three problems: the probable error of a mean, the distribution of the mean divided by its estimated standard deviation and the distribution of the estimated correlation coefficient between independent variates. Later "Student" and economist R.A. Fisher calculated tables of the relevant t distribution and Fisher gives a table of t and probabilities, corresponding to various degrees of freedom. Fisher and F. Yates, scholar provide in addition a column for P. It seemed that presentation of the P, degrees of freedom, t relationship in the form of a nomogram would be advantageous. It makes possible a fairly exact estimate of probabilities less than 0.0001 and makes it possible to get an estimate of P for any value of t from 1 to 65, instead merely of selected values.

A table is presented containing critical points for the non-central F with non-centrality of one-half for type one error .05, .10, .25 and .50 and numerator and denominator d.f. including 1-30, 40, 60, 120, 200,400, 1000. The critical points may be used to test the hypothesis that aset of exact linear restrictions imposed on the parameters of a linear model yield estimators that are better in generalized MSE than the unrestricted, o.l.s., estimators. Tabulation is discussed and some examples are presented on how the tables may be used. [ABSTRACT FROM AUTHOR]

A double sample test for a hypothesis concerning the mean mu of a normal random variable with unknown variance sigma[sup 2] is presented in this note. This test is an alternative to the usual single sample t test and is made by taking samples at two stages. After the first sample has been observed the hypothesis is rejected, accepted or a second sample is taken. This double sample test is a modification of a double sample test suggested by D. Owen. However, for the test being suggested here if the second sample is taken the information from the first sample is used in making the final decision. A table is included which is a tabulation of the rejection and acceptance points for various sample sizes and alpha = .05 and alpha = .01. Additional tables are included which give some power values and expected sample sizes for the new test and Owen's test. These power values are compared with power values of various single sample t tests. Conclusions about desirable sample sizes are presented for both the new test and Owen's test. [ABSTRACT FROM AUTHOR]

The principle of minimum discrimination information estimation is employed to obtain RBAN (Regular Best Asymtotically Normal) estimates of the cell frequencies of an r x r contingency table under hypotheses of either symmetry or marginal homogeneity. For the latter, a convergent iterative procedure is given to compute the estimates. The associated minimum discrimination information statistics are distributed asymptotically as chi[sup 2] under the null hypotheses. The procedures differ from those previously presented in the literature and permit of extension to multidimensional contingency tables. An example is given. [ABSTRACT FROM AUTHOR]

The problem considered here is a multivariate generalization of the classical Behrens-Fisher problem. Based on samples of size n[sub iota], i = 1,..., k, from k multivariate normal populations (of perhaps different dimension), a test statistic is proposed to test null hypotheses about linear relationships between components of the mean vectors of the k populations. Under the null hypothesis, the test statistic has a central F distribution and under alternatives, a non-central F distribution. For three different specializations of the above problem, the proposed test statistic reduces to tests proposed by Scheffe, Bennett, and Anderson respectively. [ABSTRACT FROM AUTHOR]

The subsample values of a statistic t are the values of t for subsets of the whole sample. Subsample values may be used as indicators of variability of t. For real valued statistics t, estimating a parameter theta, the subsample values are defined to form a set of typical values if the intervals between the ordered subsample values each include theta with equal probability. The typical value property holds exactly in some cases and approximately in others. [ABSTRACT FROM AUTHOR]

The size and power of Student's t-test are discussed under weaker than normal conditions. It is shown that assuming only a symmetry condition for the null hypothesis leads to effective bounds on the dispersion of the t-statistic. (The symmetry condition is weak enough to include all cases of independent but not necessarily identically distributed observations, each symmetric about the origin.) The connection between Student's test and the usual non-parametric tests is examined, as well as power considerations involving Winsorization and permutation tests. Simultaneous use of different one-sample tests is also discussed. [ABSTRACT FROM AUTHOR]

Consider the problem of testing the hypothesis H[sub 0]: theta Is an element of W[sub 0] against the alternative H[sub 1]: theta Is an element of W[sub 1] when a certain subset E[sub 0] subset W[sub 1] is given. This subset E[sub 0] is such that the risk of accepting Ho will be significant economically only if theta Is an element of E[sub 0]. For this problem a test obtained by generalising the equal probability test of Arrow (1960) is suggested. Applications of the test procedure to the problems of combining several independent tests and of comparing several populations with a standard are discussed. [ABSTRACT FROM AUTHOR]

In an earlier paper [5] the authors compared the power of the BLUS test with the probability of a correct decision of the Durbin-Watson bounds test. A method to compute the distribution of the Von Neumann ratio under the null hypothesis and under the alternative hypothesis was given. In the present paper the latter method is used to tabulate the BLUS-test statistic and to compute the exact significance points of the Durbin-Watson test for several examples. Powers of both tests are computed and compared. It appears that, for the cases considered, the power of the exact Durbin-Watson test exceeds that of the BLUS procedure, while the latter is greater than the probability of a correct decision in the Durbin-Watson bounds test. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to make the reader aware of the applicability and advantage of a particular mathematical model. The application is typified by an example and the advantage is via confidence intervals; that is, shorter confidence intervals are possible using the model than if one ignores it, providing the applicability is valid. It is also shown that an improved estimate can be obtained through use of the model. Let f(y|mu, sigma) be a normal density with mean mu and variance sigma[sup 2] and let g(mu|lambda, beta) be a normal density with mean lambda and variance beta[sup 2]. A sequence y[sub 1], y[sub 2],..., y[sub n+1] of independent observations from the mixture off and g can be considered as follows: An unobservable mu [sub i] is first drawn from g(mu|lambda, beta) and then y[sub i] which can be observed is drawn from f(y|mu[sub i], sigma). Confidence intervals on mu[sub n+1] are obtained which are based on the observations y[sub 1],..., y[sub n+1] and which are shorter than the standard interval based on y[sub n+1] only for any n. Shorter intervals are obtained for two cases: (i) lambda unknown, sigma, beta known; (ii) only sigma/beta = c known. [ABSTRACT FROM AUTHOR]

The article focuses on an inequality for a class of bivariate chi-square distributions. A recurrent problem in simultaneous inference is discussed in the article. While the joint distributions of test statistics may be known in form, these frequently depend on parameters, values of which are unknown. In such cases it is not possible to give simultaneous procedures, which hold exactly with specified probability; instead, it is expedient to give approximate procedures for which useful probability bounds can be established. Considerable effort has been expended in this direction, particularly in connection with simultaneous inference regarding means. The study is concerned with the construction of two-sided simultaneous confidence limits for variances in the bivariate case. In the p-variate normal case, researcher C.G. Khatri has given a lower bound to the simultaneous confidence coefficient associated with one-sided lower confidence limits and with additional assumptions regarding the covariance matrix, Khatri has similarly bounded the simultaneous confidence coefficient in the case of one-sided upper limits; however, two-sided limits was not examined.

Given a sample of n vector observations from a multivariate normal population, Anderson and Roy-Gnanadesikan have given for the variances a set of confidence intervals which are approximate in that a lower bound only is known for the joint confidence coefficient. In the present study, exact procedures are developed in terms of multivariate Chi-square distributions, and more general approximate procedures are given via Bonferroni's inequality. A numerical investigation suggests that the Bonferroni lower bound is fairly sharp for a variety of parameter values, and it always is superior to the Roy-Gnanadesikan procedure in the bivariate case examined. A lower bound in terms of independent statistics further is examined far a special class of one-sided intervals. [ABSTRACT FROM AUTHOR]

A new class of tests is presented for testing the hypothesis of no second order interaction in three-way contingency tables. This hypothesis is expressed first in the form of linear contrasts in logarithms of the parameters of a multinomial experiment and the test is based on the general theory of locally asymptotically most stringent tests. Specific test criteria are then given for testing no second order interaction in 2 X2 X 2, 2 x 2 x K and I x J x K contingency tables. Two examples are discussed. [ABSTRACT FROM AUTHOR]

The paper is concerned with testing an application of a theory of investment behavior based on the neo-classical theory of optimal capital accumulation. The demand for capital is determined to maximize net worth under the neo-classical conditions of production. With the assumption of a lag in the completion of investment projects, gross investment is a distributed lag function of period-to-period changes in desired capital stock. The investment process is divided into a series of separate stages, the first stage being a change in the demand for capital services, intermediate stages being for letting contracts, and so on, and the final stage the actual investment expenditure. To extend the theory to include anticipated investment expenditures, these are treated as a special kind of intermediate stage. The purposes of the paper are to test hypotheses about the internal structure of the investment process when viewed in this way, and to investigate whether forecasting actual investment expenditures by treating the investment process in this way can improve upon the method used by the OBE-SEC. [ABSTRACT FROM AUTHOR]

The article presents discussion of a paper by statistician I.J. Good. The author supposes that most statisticians approaching the problem of testing goodness of fit would have been very much less ambitious than Good has been. Most of the people, if spend some time looking at the first planet, Mercury and the last two, Neptune and Pluto, do not agree with Bode's law, whereas all the intermediate planets from Venus to Uranus seem to fit the law remarkably well. Those statisticians who were sufficiently knowledgeable would presumably meditate on the matters that Good has meditated on; but, having done this, most people would make a merely informal judgment and completely shirk the challenge that Good has met, to evaluate the weight of evidence. Good remarks that if one is going to test goodness of fit of a theory to some data, using either a Neyman-Pearson approach or a Bayesian approach, one have to frame explicit hypotheses. Over a good many years the author has, from time to time tried to think about the general problem of testing goodness of fit, and remain skeptical about whether goodness of fit can be adequately discussed either in Neyrnan-Pearson or in Bayesian terms because of this necessity to have, not only an explicit null hypothesis, but also a fully defined explicit alternative hypothesis or set of alternative hypotheses.

Results are given of an empirical sampling study of the sensitivities of nine statistical procedures for evaluating the normality of a complete sample. The nine statistics are W (Shapiro and Wilk, 1965), square root of b[sub I] (standard third moment), b[sub 2] (standard fourth moment), KS (Kolmogorov-Smirnov), CM (Cramer-Ven Mises), WCM (weighted CM), D (modified KS), CS (chi-squared) and u (Studentized range). Forty-five alternative distributions in twelve families and five sample sizes were studied. Results are included on the comparison of the statistical procedures in relation to groupings of the alternative distributions, on means and variances of the statistics under the various alternatives, on dependence of sensitivities on sample size, on approach to normality as measured by the W statistic within some classes of distribution, and on the effect of misspecification of parameters on the performance of the simple hypothesis test statistics. The general findings include: (I) The W statistic provides a generally superior omnibus measure of non-normality; (ii) the distance tests (KS, CM, WCM, D) are typically very insensitive; (iii) the u statistic is excellent against symmetric, especially short-tailed, distributions but has virtually no sensitivity to asymmetry; (iv) a combination of both square root of b[sub 1] and b[sub 2] usually provides a sensitive judgment but even their combined performance is usually dominated by W; (v) with sensitive procedures, good indication of extreme non-normality (e.g., the exponential distribution) can be... [ABSTRACT FROM AUTHOR]

The article discusses the analysis of cross-classified data. There are two different kinds of contingency tables, that is, truncated contingency tables in which the entries in some of the cells of the table are omitted from the analysis and the more usual contingency tables in which none of the cells of the table are omitted. In the usual non-truncated contingency table, one need to calculate statistician Karl Pearson's chi-square statistic for testing the null hypothesis of independence in the table, using the appropriate number of degrees of freedom. The concept of "quasi-independence" is useful in analyzing tables for which some of the cells are truncated and also in analyzing the more usual kinds of cross-classification tables since it leads to methods that focus attention in turn on various subsets of the entire table, making possible a more detailed analysis of the association between the row and column classifications in the table. For the non-truncated cross-classification table, if the rows and columns are not independent, it will often be of interest to estimate what are the kinds of interaction or association present in the table and what are their magnitudes.

Several writers have studied the problem of obtaining confidence bands for a straight line regression under the requirement that the bands be at a specified confidence level, (1 - α) say, for values of the independent variable restricted to a pre-specified closed interval. The bands studied by other writers have been either bands of equal width throughout the interval or trapezoidal bands. This paper studies confidence bands of the classical hyperbolic type under the restriction noted above. Relevant distribution results differ according to whether the pre-specified interval on the independent variable is symmetric or asymmetric about the mean of the independent variable values in the experiment. In the former case the problem under study, is shown to be equivalent to a problem studied by Halperin et al (JASA, Sept. 1967) for which distribution theory and tables are already available. The symmetrical case is generalized to obtain confidence bands in multiple linear regression at level (1 - α) for an ellipsoidal region on the independent variables centered at the point of means of the independent variable values used in the experiment. For straight line regression some numerical comparisons are made with bands of the other types mentioned above. These comparisons suggest that the bands proposed in this paper are uniformly superior to bands of equal width in the sense of having smaller average width; the proposed bands appear to be superior to trapezoidal bands in the same sense for intervals of practical interest, i.e. within or reasonably close to the range on the independent variables used in the experiment. [ABSTRACT FROM AUTHOR]

The first part of this note is devoted to some of the simpler series representations of the doubly non-central t-distribution. In the second part, the computational considerations arising from the series representations are discussed. A table showing some of the results of the method is presented. [ABSTRACT FROM AUTHOR]

The topic of the shrinkage of an estimator for the mean of a normal distribution towards a point ([3] Thompson) is extended to shrinkage to an interval centered at μ = 0. An effort is made to keep the form of the modified estimator relatively simple, while obtaining more flexibility than is possessed by the sample mean x. The family of estimators advocated is: μ = x + k σ2x/4θ ln [(x - θ )2 + kσ 2x/ (x + &#x03B8)2 + k σ 2x. Curves of MSE/σ2x (where MSE is mean square error) versus |μ/σx| are given for a number of values of θ and k. A table is given by the use of which k and θ may be selected on the basis of the minimum value of MSE/σ2/x, the width of the interval for which MSE/σ2/x < 1, the maximum value of MSE/σ2/x, and the value of [μ/σx| for which MSE/σ2/x is a maximum. [ABSTRACT FROM AUTHOR]

This note gives a systematic method of computing the probability P[sub n](beta) of the event [Multiple line equation(s) cannot be represented in ASCII], where F[sub n] and F are the empirical and theoretical cumulative distribution functions and beta is a function on the interval (0, 1). The result is used to determine not only the significance level but also the exact power for an appropriate testing situation. When beta is a linear function, P[sub n](beta) may represent the probability distribution of the one-sided Kolmogorov-Smirnov statistic D[sup +, sub n]=sup[sub -Infinity < x < Infinity] [F[sub n(x) - F(x)]. Numerical examples are given in section 2, and they show that the method is extensively useful for practical applications. [ABSTRACT FROM AUTHOR]

Tests are developed concerning the mean of the three-parameter Weibull distribution with the variance known and with the variance unknown, and concerning the variance with the mean unknown. The shape parameter is considered known. These tests are compared with tests developed by Hatter and Dubey based on the usual statistics for the normal case. A two-sample test for difference in means and for difference in reliability is also discussed. For the most part the tests are simple yet provide improvements over existing methods. [ABSTRACT FROM AUTHOR]

Balls are drawn one after another from k cells C[sub 1], ..., C[sub k] according to the multinomial distribution. Suppose the ith cell initially contains N[sub I] balls, and sampling stops as soon as any of the k cells, say C[sub alpha], empties first. Let X[sub I] denote the number of balls taken from cell C[sub iota] (all I is not equal to alpha) at stopping time. The joint asymptotic (as N[sub I] arrow right Infinity) distribution of the X[sub iota] (I is not equal to alpha) is derived under the most general configuration of the multinomial cell probabilities p[sub 1], ..., p[sub k]. Conditions on p[sub iota] and N[sub I] are given under which the asymptotic distribution is shown to be either normal or truncated (restricted) normal. An application of the asymptotic distribution theory for N[sub iota] = N[sub 0] (I =1, ..., k) in setting up approximate tests and confidence intervals for the largest p[sub I] is also given. Under certain conditions on pi and N[sub I] it is shown that the asymptotic probability that C[sub I] empties first is equal to the probability content of a positive orthant under a multivariate normal distribution. [ABSTRACT FROM AUTHOR]