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In this paper a method is developed for estimating the parameters in the multivariate normal distribution in which the missing observations are not restricted to follow certain patterns as in most previous papers. The large sample properties of the estimators are discussed. Equivalence with maximum likelihood estimators has been established for a subclass of problems. The results of some simulation studies are provided to support the theoretical development. [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 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]

In experimental designs with randomized blocks, estimates of the inter-block variance often have few degrees of freedom since they depend only on the block averages (the inter-block information). These degrees of freedom are further reduced by the number of parameters appearing in the inter-block information. However, some parameters also appear in the intra-block information, and thus allow two independent estimators. The difference of these two estimators provides additional information about the inter-block variance. This paper shows how this extra information may be recovered to improve tests of hypotheses concerning inter-block parameters. [ABSTRACT FROM AUTHOR]

This paper uses a family of bivariate extreme-value distributions to estimate the probability of a large exceedance of a random variable given that a certain other random variable not independent of the first has exceeded a certain value. A simple method of reasonably good efficiency is given for estimating a bivariate extreme-value distribution from independent bivariate samples. The method is used to analyze the performance of a spacecraft command receiver which has an indication of data quality so that commands likely to be in error can be rejected. [ABSTRACT FROM AUTHOR]

This paper deals with certain aspects of variance-component estimation for the unbalanced two-way completely-random classification where the numbers of observations in the subclasses are treated as random variables not necessarily independent of some of the random effects of the model. General results are given on the expectations of two commonly-used estimators of the vector of variance components. Numerical approximations are presented for these expectations for one sub-family of the family of all possible joint distributions of the subclass numbers and the random effects. [ABSTRACT FROM AUTHOR]

The least squares estimator of a regression coefficient beta is vulnerable to gross errors and the associated confidence interval is, in addition, sensitive to non-normality of the parent distribution. In this paper, a simple and robust (point as well as interval) estimator of beta based on Kendall's [6] rank correlation tau is studied. The point estimator is the median of the set of slopes (Y[sub I]--Y[sub I])/(t[sub I]--t[sub I]) joining pairs of points with t[sub 1] is not equal to t[sub 1], and is unbiased. The confidence interval is also determined by two order statistics of this set of slopes. Various properties of these estimators are studied and compared with those of the least squares and some other nonparametric estimators. [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]

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]

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 presents three upper bounds for the variance of an estimator, based on observations selected with varying probabilities from a finite population, the elements of which are ranked with respect to the Y values. Accordingly, the usefulness of these bounds relates to the pre-enumeration analysis where one may well know the intended probabilities and joint probabilities corresponding to the sampling scheme but does not know the Y values. If, however, one can make a conservative guess at the largest Y value, one can use these bounds. Some examples are included to illustrate the theory. [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]

Formulas for the variances of weighted least squares estimators calculated with estimated weights based on equal replicate numbers are derived in this paper. Results are obtained for the two cases of multivariate and univariate error distributions. [ABSTRACT FROM AUTHOR]

It is claimed that the reasons for using matrices of derivatives, in appropriate situations, are as compelling as those for using matrices. This paper provides basic material for such use. Different types of matrix derivatives are defined and illustrated. Simple and easy techniques are then derived and are shown to be applicable to a considerable collection of matrix functions. Applications are made to such problems as establishing matrix integrals from scalar ones, determining maximum likelihood estimates for complex likelihood functions, optimizing matrix functions when there are matrices of side conditions, and evaluating the Jacobians of certain classes of transformations. The emphasis is on simplicity of derivation and on breadth of application. [ABSTRACT FROM AUTHOR]

In this paper we review the literature on the problem of handling multivariate data with observations missing on some or all of the variables under study. We examine the ways that statisticians have devised to estimate means, variances, correlations and linear regression functions from such data and refer to specific computer programs for carrying out the estimation. We show how the estimation problems can be simplified if the missing data follows certain patterns. Finally, we outline the statistical properties of the various estimators. [ABSTRACT FROM AUTHOR]

This paper presents a procedure for estimating the components in an economic time series assuming a constant seasonal component and a trend which may be estimated locally by a polynomial. The procedure is to estimate the trend and seasonal components for each consecutive two-year time span of quarterly data using conventional least-square techniques. Thus, for each quarter in the series, except the first seven and last seven, there are eight estimates for each of these components. A weighted average of these estimates is obtained using weights such that the variance of the average is a minimum. This procedure is used to adjust male unemployment in the United States for seasonal variation, and the results compare favorably with the official adjustment of the Bureau of Labor Statistics. [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]