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The article presents information on the book, Foundations of Statistical Inference, edited by V. P. Godambe and D. A. Sprott. This book contains the Proceedings of a Symposium sponsored by the Rene Descartes Foundation and held at the University of Waterloo during ten days of April 1970. Rarely does one find the state of an art and the state of its artists revealed to such an extent under one cover. The book's twenty-eight papers and related discussions provide a comprehensive view of the world of statistical inference. J. Neyman presented the first paper in the Proceedings, on the Foundations of Behavioristic Statistics. It contains some nice historical notes, including a fascinating description of the development of sequential sampling and hypothesis tests having power equal to unity. Several papers, including those by Irwin D. J. Bross, I. J. Good, E. T. Jaynes, Oscar Kempthorne, H. E. Kyburg Junior and Herman Rubin deal with either philosophical issues or general criticisms of the foundations. The philosophy is typically quite loose and the criticisms do not lead to positive suggestions for replacing existing paradigms by new ones. The rest of the book deals mainly with mathematical-logical foundations, particularly with the status of Fraser's structured model for inference, aspects of the subjectivist Bayesian model, the status of likelihood methods and the special problems of sampling finite populations.

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In this paper a test method for time study developed from a theoretical basis of sequential analysis of mathematical statistics is presented. This method admits a statistical decision on the regions which contain a certain time ratio.
In the course of the test process, the instantaneous observations of the activities made at the work places are assigned by small integers which yield a simple sequential statistic. The observations need not be recorded. The sample size is, on the average, considerably smaller than that of comparable studies by normal work sampling.
The design and application of sequential work sampling tests are described with regard to the estimation method of work sampling and their most important features are discussed. [ABSTRACT FROM AUTHOR]

The 1967 Committee on Publications, chaired by David L. Wallace, found that many American Statistical Association members desired more review and survey papers. These have been hard to come by and so as the author's last act before leaving office, decided to provide a short survey paper on some related ideas in a field where nearly all of the statisticians sometimes work — that of contingency tables. These ideas are largely available in literature and yet they have not often been put together, though statisticians I. J. Good's monograph and Leo Goodman's many papers form good sources. But the author's paper is not intended as a review of the literature, only as a survey of one set of ideas about estimation in the analysis of contingency tables. The author fears that the first act of most social scientists upon seeing a contingency table is to compute chi-square for it. Sometimes this process is enlightening, sometimes wasteful, but sometimes it does not go quite far enough. The author collected 500 samples of writing published about 1961.

The article presents information about Samuel S. Wilks, a great contributor to statistics. Wilks' behavior toward applications was peculiarly split, he encouraged his students to work on applications, not always an easy thing to do in a strongly theoretical mathematics department; he often told students about applications that he regarded as "neat" or "cute" or "clever"; the statistical colloquium he guided often had speakers on practical applications of mathematical statistics; he himself wrote some practical papers in statistics, but in the classroom he rarely discussed applications. Repeatedly, however, practical problems explicitly influenced both his own publications and those of his students and Wilks often used applications as motivation for the discussion of distribution theory, usually going well beyond the needs of the original problem. The papers to be discussed exhibit incompletely and fragmentarily a major influence on the work of the man and his students. Wilks watched the development of the statistical theory of order statistics closely. Indeed he wrote a masterful summary of the literature. The general area involves the study of the statistical properties of ordered measurements.

Various methods based on Student t variates have been suggested and used for obtaining simultaneous confidence intervals for several means, or for several contrasts among means. Determination of an overall confidence level for such intervals involves evaluating the probability mass of a multivariate t distribution over a hypercube centered at the origin, with sides paralleling the coordinate planes, or obtaining bounds for this probability mass. Since such distributions involve many nuisance parameters, an impossible number of tables would be necessary in order to make exact confidence intervals. In the virtual absence of tables, approximations and bounds become important. In this paper, an attempt has been made to investigate the adequacy of certain suggested approximations [2], [5], [8] by computing the exact distributions for some particular cases. These exact distributions have been compared with approximations. This paper is concerned with two-sided confidence intervals, rather than one-sided intervals. [ABSTRACT FROM AUTHOR]

Let X[sub i1], X[sub i2],..., X[sub iota n], I=1, 2, be a pair of random samples from populations which are normally distributed with means theta[sub iota], and common known variance tau[sup 2]. The problem is to estimate the function psi(theta[sub 1], theta[sub 2]) = maximum (theta[sub 1], theta[sub 2]). In this paper we consider five different estimators (or sets of estimators) for psi(theta[sub 1], theta[sub 2]) and evaluate their biases and mean square errors. The estimators are (I) psi(X[sub 1], X[sub 2]), where X[sub I] is the sample mean of the ith sample; (ii) the analogue of the Pitman estimator, i.e. the a posteriori expected value of psi(theta[sub 1], theta[sub 2]) when the generalized prior distribution is the uniform distribution on two dimensional space; (iii) a class of estimators which are generalized Bayes with respect to generalized priors which are products of uniform and normal priors; (iv) hybrid estimators, i.e. those which estimate by (X[sub 1] + X[sub 2])/2 when |X[sub 1] -X[sub 2]| is small, and estimate by psi(X[sub 1], X[sub 2]) when |X[sub 1] - X[sub 2]| is large; (v) maximum likelihood estimator. The bias and mean square errors for these estimators are tabled, graphed, and compared. Also the invariance properties of these estimators are discussed with a rationale for restricting to invariant estimators. [ABSTRACT FROM AUTHOR]

This paper deals with estimating the location and the scale parameters of the extreme value distribution when the sample size is very large, using sample quantiles. An estimator is given for the location parameter when the scale parameter is known, based on one or more (up to 15) order statistics. Also given is an estimator for the scale parameter when the location parameter is known, built on two order statistics. An iterative procedure is utilized to estimate the parameters when both are unknown, using two order statistics. The problem of estimating the percentage point is also dealt with, and a comparison with Lieblein's method is included. [ABSTRACT FROM AUTHOR]

The decision-theoretic approach to point estimation involves the choice of an estimate to minimize the expected loss associated with the estimate. The purpose of this paper is to indicate the influence of risk aversion on point estimates for classes of payoff functions including the piecewise linear and quadratic payoff functions. Increased risk aversion results in a point estimate closer to zero for a quadratic pay. off function and a lower estimate with a piecewise linear payoff function, for example. [ABSTRACT FROM AUTHOR]

A method is proposed to eliminate trends from a sample from a nonstationary system of linear stochastic difference equations. The autocovariance matrix and the spectral density matrix of the detrended component of the sample are derived. The latter matrix turns out to have the same form as the spectral density matrix for a stationary system when expressed in terms of the roots of the system. Since the parameters of the system are assumed known throughout the paper, the problem of statistical inference does not arise. [ABSTRACT FROM AUTHOR]

This paper develops a theoretical framework for the unrelated question randomized response technique suggested by Walt R. Simmons. The statistical efficiency of this technique is compared with the Warner technique under situations of both truthful and untruthful responses. Methods of allocating the total sample to each of two subsamples required by the unrelated question approach are developed. Recommendations are made concerning choices of values for those parameters which can be assigned at the discretion of the investigator. [ABSTRACT FROM AUTHOR]

When two correlation coefficients are calculated from a single sample, rather than from two samples, they are not statistically independent, and the usual methods for testing equality of the population correlation coefficients no longer apply. This paper considers the situation when the sample is from a multivariate normal distribution. Several possible large sample testing procedures are given, all based on Fisher's z-transformation. Power curves are given for each procedure and for seven values of the asymptotic correlation between the two sample correlation coefficients. [ABSTRACT FROM AUTHOR]

A progressive-censoring model arises from a life test of a sample of items in which one or more of the survivors may be removed from the test at the time of any failure. Such a model is often more realistic for actual failure data which must be analyzed by a statistician than one in which all survivors are assumed to be removed from test simultaneously. This paper deals with the situation in which the underlying failure-time distribution for the population sampled is the two-parameter Weibull distribution. The reliable life for the population is defined to be the 100(1-R) percent point of the failure-time distribution, where R is a specified population survival proportion, or reliability. An exact confidence bound on reliable life based on three observed ordered failure times is derived for this progressive-censoring model. The criterion used for selecting the order numbers of the three failure times upon which the bound is based depends upon computed values of the power function of the test associated with the bound. A table from which lower bounds can be obtained is given for R equal to .95, confidence level .90, sample size equal to 2, 3, . . . , 6, and all possible censorings. [ABSTRACT FROM AUTHOR]

This paper deals with proportional stratified sampling in the situation where the estimation variable X is difficult and expensive to observe, while the possibly erroneous stratification variable Y is easy and inexpensive to get at. The usually biased estimate [Multiple line equation(s) cannot be represented in ASCII text] is compared with the unbiased estimate [Multiple line equation(s) cannot be represented in ASCII text] where the P[sub I] are stratum weights and y[sub I] and x[sub I] are means of the units sampled from the I:th stratum. The two estimates are similar in that they utilize information from only those population units that make up the sample. While mu[sub a] is the more inexpensive estimate, mu[sub b] is usually preferable if one judges by the size of the mean square error, which, among other things, depends on the number of strata and the location of the stratum boundaries. In particular, the properties of mu[sub a] and mu[sub b] are discussed in connection with an example involving the bivariate normal distribution. [ABSTRACT FROM AUTHOR]

In this paper we examine how well the asymptotic null distribution of a two sample test due to Lehmann approximates the small sample distribution of the test, compare the validity of this Lehman test with the validity of the two sample t test under the null hypothesis of equal means, and compare the power of this Lehmannn test with the power of the t test. Our general conclusion is that experimenters will prefer to use the t test when the underlying distribution is the scale contaminated compound normal distribution and the sample sizes are less than thirty. [ABSTRACT FROM AUTHOR]

An algorithm for the determination of the economic design of X-charts based on Duncan's model is described in this paper. This algorithm consists of solving an implicit equation in design variables n (sample size) and k (control limit factor) and an explicit equation for h (sampling interval). The use of this algorithm not only yields the exact optimum but also provides valuable information so that the sensitivity of the optimum loss-cost (L*) can be evaluated. Loss-cost contours are used to discuss the nature of the loss-cost surface and the effect of the design variables. The effect of two parameters, the delay factor (e), and the average time for an assignable cause to occur (1/lambda), on the optimum design is evaluated. Numerical examples are used for illustrations. [ABSTRACT FROM AUTHOR]

In this paper, successive sampling procedures using multi-stage sampling design have been developed. It is found that it is generally advantageous to retain a fraction of the sample selected on previous occasions for improving the estimate of a mean on the most recent occasion. This type of partial replacement may sometimes be recommended for estimating a mean over all occasions. This is specially true if the experimenter is interested in not only obtaining an overall estimate for the entire period but also in separate estimates for each occasion. In a sampling enquiry repeated on three occasions it has been observed that for estimating the mean on the third occasion it would be preferable to repeat the same sample fraction from one occasion to the next, while for estimating the mean over all occasions the sample fraction repeated on the second occasion should not be repeated on the third but in its place a sub-sample from the sample selected afresh on the second occasion should be retained. [ABSTRACT FROM AUTHOR]

This is an expository paper on the fixed sample size indifference zone formulation of the ranking and selection problem. A particularly simple proof of a theorem which is useful in this formulation is given, and its use is illustrated. [ABSTRACT FROM AUTHOR]

The experiments to be discussed in this article was designed to investigate empirically the distribution of the sample version of the measure of association (G), &b.gamma;. Professors, William H. Kruskal and Leo A. Goodman have published an article giving interpretive motivation for the coefficient, and have developed the large sample theory for G. In the present research, sampling experiments were performed to test the adequacy of the large sample theory for 5X5 population cross classifications and samples of size ten, twenty-five and fifty. The numerical experiment described in this paper was based on a number of 25 cells multinomial populations, each considered as represented by the probabilities in a 5X5 population cross classification. Gamma is defined by Goodman and Kruskal in terms of probabilities. The 100 population cross classifications were chosen to be representative of practical situations in psychology. The method use to obtain them was to begin with 5X5 cross classification having complete association along the main diagonal and with a variety of marginal distributions.

The determination of sample size to meet certain probability requirements is a problem which often faces an experimenter when obtaining a confidence interval or testing a hypothesis. This paper includes some new graphs useful in selecting sample size and gives a reference to a new textbook which includes a number of other similar type graphs. The method of construction is explained in detail and examples are included. [ABSTRACT FROM AUTHOR]

Usually auxiliary information based on just one variate is used to improve the precision of estimators of population totals, means, etc. In this paper a method is proposed of using information on several variates to achieve higher precision. The technique of difference estimation is employed throughout. It is shown that the variances of difference estimators are comparable to those of ratio estimators. The results are extended to double sampling procedures and sampling over two occasions. [ABSTRACT FROM AUTHOR]

In the classification of the elements in a binomial population, there is in many cases a chance of misclassifying an item. When an item in a sample from a binomial population is misclassified, it causes a bias, independent of the standard error, in the estimate of the population parameter P. This paper analyzes this bias when each of two interviewers independently classify the items in a single sample. Upper and lower bounds for the bias are derived. [ABSTRACT FROM AUTHOR]

This paper is concerned with maximum likelihood estimation of a process (or lot) average proportion of defectives based on attribute samples that have been curtailed either with rejection of a lot on finding the kth defective or with acceptance on finding the Kth nondefective. The MLE, &pcirc;, based on inspections from a sequence of m lots is shown to be simply the ratio of the number of defectives found to the total number of items inspected, and the asymptotic variance of this estimator is shown to be approximately &pcirc;(1 — &pcirc;)/σ[sup m][sub 1] y[sub i], where σ[sup m][sub 1] y[sub i] is the total of all items inspected. [ABSTRACT FROM AUTHOR]

This paper presents two definitions of maximum cluster, that have sometimes been used to test for non-random clustering. We compare the power of the tests based on these statistics, and show that when the clustering interval is sufficiently small, one of the tests is more powerful for a wide class of alternative hypotheses. We show that this test is the generalized likelihood ratio test for an alternative hypothesis related to a particular type of non-random clustering. [ABSTRACT FROM AUTHOR]