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This paper deals with the asymptotic relative efficiency (A.R.E.) and the asymptotic power of certain tests. These tests are distribution-free goodness-of-fit tests based on the empirical distribution function. They are the one and two sided Kolmogorov-Smirnov tests and a new one based on the absolute values of the order statistics. The method of obtaining the A.R.E. is due to Quade [5] and is outlined in Section 1. In this paper we are concerned with the case of scalar alternatives. The asymptotic power for the Kolmogorov-Smirnov tests is obtained in Section 2. Section 3 deals with a new test which allows a more facile use of a one-sided statistic while utilizing the same information normally included in a two-sided statistic. In Section 4, three examples are considered to illustrate the superiority of this new test.

This paper proposes a multi-stage procedure for the selection of the best of several populations, allowing more observations to be made on the more successful treatments. To this end, a weighting function operates on the results accumulated up to any one stage to determine the proportion of observations to be taken from each population for that stage. The paper is restricted to consider only binomial populations. Simulation experiments utilizing five somewhat arbitrary weighting functions indicate that this method has a higher probability of selecting the best population than conventional equal sample size experiments.

The purpose of this paper is to present an easily applied test useful in determining whether or not a supplier's data have been truncated. The proposed test has the following desirable properties: (i) it is the uniformly most powerful rank test, (ii) it is asymptotically uniformly most powerful, (iii) power computations can easily be made for arbitrary sample sizes, formulas for such computations being given in the paper. Although formulated in the context of verifying a supplier's data, the test can be applied to other situations in which false representation of data in the form of truncation is important. Such is the case, for example, in reliability demonstrations or legal suits involving physical measurements.

This paper considers the problem of obtaining efficient estimates for the parameters of a system of M regression equations. The disturbance terms of this system are assumed to be related by both serial and contemporaneous correlation. Under the further assumption that the serial correlation is a first order autoregressive process, the paper develops an estimator that is consistent and has the same asymptotic normal distribution as the Aitken estimator which assumes the covariance matrix to be known. The paper concludes with a discussion of some alternative covariance specifications and points out certain difficulties with the standard single equation procedures for handling auto-regressive schemes.

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 payoff function and a lower estimate with a piecewise linear payoff function, for example.

This paper treats essentially the first derivative of an estimator viewed as functional and the ways in which it can be used to study local robustness properties. A theory of robust estimation “near” strict parametric models is briefly sketched and applied to some classical situations. Relations between von Mises functionals, the jackknife and U-statistics are indicated. A number of classical and new estimators are discussed, including trimmed and Winsorized means, Huber-estimators, and more generally maximum likelihood and M-estimators. Finally, a table with some numerical robustness properties is given.

A new way of constructing uniformly better estimators (in the sense of minimum variance) from two independent unbiased estimators is presented. The problem of weighting estimators, when the ratio of their variances is unknown, is not straightforward. Yet this problem is particularly important in the recovery of inter-block information in balanced incomplete blocks. The theorem given in this paper suggests a simple method of constructing “better” estimators. The theorem has application in the analysis of balanced incomplete block designs; an example is given.

In this paper we obtain a multivariate generalization of the inverted beta distribution. Properties of this distribution and its connection with the multivariate Student-t distribution are discussed. Two asymptotic formulae for approximating the related probability integral are developed and illustrated with numerical examples. Application of this distribution to a problem in Bayesian inference is given.

This paper identifies a type of highly correlated noise—systematic noise—that often arises in applications of linear regression to observations of spacecraft. An explicit formula is given for the inverse of the covariance matrix. The inverse matrix has a simple series expansion in e2, where 1/e is the amplitude of the systematic noise. When the leading term, the so-called filter matrix, is used in place of the exact inverse to estimate the regression parameters, the systematic noise is eliminated. The filter matrix is shown to be optimum among all matrices that eliminate systematic noise from the estimates. An example is discussed.

In this paper a brief review is given of some important developments in the theory of sampling from finite populations during the decade 1950–60. More specifically different selection procedures and estimators in sampling with varying probabilities and in ratio method of estimation, method of optimum stratification and assessment of non-sampling errors have been covered in this review together with some general techniques of improving upon certain types of estimators.

IN FERRYING operations of valuable items (e.g., aircraft spare parts needed in a theater of operations during a war), the number of items to be carried in each aircraft is frequently under the control of the planner. By using more aircraft he can make the loading per aircraft smaller and hence possibly reduce the probability of large losses at the risk of increasing the probability of small losses. (The small loading also increases the actual ferrying cost.) Because of the value of the items it may be a good investment to buy this insurance. In making his decision as to how many items to load per aircraft, the planner therefore needs to know, as a function of the loading, the probability of losing any given number of items. Sometimes the mean and variance may give sufficient information, without knowledge of the more laboriously calculated probabilities. Usually however, the distributions are not even approximately symmterical nor even unimodal, so that an adequate approximation using only the mean and variance is not possible. In this paper expressions are given for the mean and the variance of the number of items lost, for the probability of losing a given number of items and for the probability generating functions. The moment generating functions may be found from the probability generating functions by substituting ek for 4. For models 1, 3, and 4 several formulas, in addition to the probability generating function and the general formula (m.4) are given for the probabilities of losing exactly a items.' These special formulas are useful for certain values of a. Apparently this problem has not been previously considered in this form in the statistical or operational research literature. We denote by k the number of flights to be used in the ferrying operation. No assumptions are required as to how many aircraft are used in the operations; there may be k aircraft, each making one trip, or fewer aircraft may be used, some or all making more than one trip. Each aircraft carries r valuable items, so that a total of N = kr items are ferried. Each item is assumed to be equipped with a life raft or some other means of preventing it from sinking in case the aircraft

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.

This paper presents the characteristics of the more important types of apportionment formulas. From the standpoint of effects of grants upon program or service aided, formulas are examined with reference to two criteria or conditions. A form of algebraic examination of formulas is presented which permits determination of characteristics of apportionment methods without numerical computations. This is done in terms of amount of grant, amount of local contribution, total program supported, rate of local effort, and relationship between central and local funds.

A large class of useful multivariate recursive time series models and estimation methods has appeared in the engineering literature. Despite the interest and utility which this recursive work has when viewed as an extension of regression analysis, little of it has reached statisticians working in regression. To overcome this we (a) present the relevant random-β regression theory as a natural extension of conventional fixed-β regression theory and (b) derive the optimal recursive estimators in terms of the extended regression theory for a typical form of the recursive model. This also opens the way for further developments in recursive estimation, which are more tractable in the regression approach and will be presented in future papers.

A method is proposed to eliminate trends from a sample from a non-stationary system of linear stochastic difference equations. The auto-covariance 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.

The problem of estimating the slope of a polynomial regression at a fixed point of the experimental region such that (a) the variance of the least-square estimate of the slope at the fixed point is a minimum and (b) the average variance of the least-square estimate of the slope is a minimum is discussed in this paper. In general these designs can be obtained using Kiefer-Wolfowitz [5] characterization of c-optimal designs, Federov [2] characterization of L-optimal designs, and Studden's [10] generalization of the Elfving Theorem [1]. After presenting a brief review of these characterization theorems, specific illustrations for the quadratic and cubic regressions are presented in detail.

In this paper the problem of choosing among point estimators on the basis of their small sample properties is discussed from the sampling point of view. The indeterminacy of most Monte Carlo studies is analysed and resolved within the framework of statistical decision theory. A first order autoregressive model is worked through in detail both for its own sake and to illustrate how a complete Monte Carlo study might be done.

This paper is a review of techniques for obtaining the treatment combinations that comprise a fraction of a factorial arrangement. Several procedures for constructing fractional replicate plans, which in their original form appear to be different, are presented in a manner which illustrates their similarities. The techniques discussed include orthogonal and non-orthogonal plans for both symmetrical and asymmetrical factorial experiments. The plans developed range from those that permit estimation of main effects only, to those that permit estimation of main effects and all two-factor interaction effects.

The estimation of the parameters of dosage response curves by the standard probit method is an iterative process beginning with approximations to the parameters and using one or more cycles of computations to “improve” these estimates until they converge. The present paper gives a table and method for computing the maximum likelihood solution which converges more rapidly than the standard probit method. A procedure is presented for obtaining more accurate initial approximations, and the problem of the bias of the maximum likelihood estimates in small samples is considered.

In a large scale survey, where a non-self-weighting design is used, the work at the tabulation stage becomes time consuming and costly due to the large number of multipliers (inflation factors) involved in obtaining the estimates. In this paper a technique is developed to substitute the multipliers by a very small number of multipliers called “randomized rounded-off multipliers” by a suitable randomizing process, thereby reducing the work at the tabulation stage. A suitable procedure has been suggested for determining the values of the randomized rounded-off multipliers which would minimize the increase in the variance of the estimator.

This paper carries on work in [2] and [4] on the asymptotic effects of small specification error in simultaneous equation estimation. The specification errors considered are inherent in the approximate nature of econometric models. It is shown that for any parameter and small enough specification error, the probability limits of two-stage least squares and limited-information maximum-likelihood, while different, lie on the same side of the true parameter value, the difference between them being of the second order of smalls relative to the effects of the specification error on either one of them. A method of consistently estimating the sign of the difference of the effects of specification error on the two estimators is given and a procedure suggested for choosing an estimator to reduce those effects. Two relatively simple examples are worked out.

This paper discusses the problems of slope-estimation and of linear extrapolation when the precision of the Y-measurements changes with different values of x. Tables are presented which show, for each of these problems, the optimum placement of x-values and the optimum distribution of the N observations for selected relationships between σ and x. The optimum placements are seen to be the same for the two problems, but for the latter, the optimum distributions depend on the nearness or remoteness of the extrapolation.

In problems where guidance to the proximity of a point is required, the probable success of the mission often is described by the probability of hitting within a given radius of the point. In this paper is presented a method of determining the probability of a hit within a given circle when it is assumed that the guidance error is distributed according to a bivariate normal distribution. This problem can be solved readily if the random error variables are independently distributed along two orthogonal axes with equal standard deviations and if they are not biased relative to the target location. Since these assumptions often are not valid, a series solution to the general problem is provided with an analysis of the error introduced by considering only a finite number of the terms.

An economic single sampling plan for a variable having a single-sided specification limit whose direct measurement is very expensive or destructive is suggested in this paper. Assuming existence of at least one correlated auxiliary variable whose measurement is not so expensive and of a known dispersion matrix of these variables, the suggested plan is shown to have easy practical application. Further the choice of only one auxiliary inexpensive variable highly correlated with the main variable is shown to be optimum in most of the practical situations.

In this paper we review the derivation of the predictive density function for the normal multiple regression model, state and prove a general theorem on optimal point prediction, and show how the predictive density can be employed in the analysis of an illustrative investment problem. Then we derive the predictive density function for the multivariate normal regression model and indicate how it can be used in the analysis of several problems.

The problem considered in this paper is one of allocating a predetermined aggregate among its components. An example from econometrics is the allocation among various competing consumption items of a consumer's budget, the total size of which is usually taken as determined outside the allocation framework. Under the specification adopted, one may treat any choice of (m — 1) of the m components of the predetermined aggregate as stochastic, fitting the allocation equations for these (m — 1) variables by Aitken's principle. The parameters of the equation excluded from the estimation procedure may be found by differencing; the estimates obtained in this way are shown to be invariant under any choice of which (m — 1) equations are fitted. Further, this invariance property is distribution free. These results hold good whether the aggregate is simple or weighted. As an alternative interpretation, moreover, one may regard this problem simply as one of fitting m equations containing a common regressor und...

Weiss [5] and Freeman and Weiss [2] have shown how to construct sampling plans which at least approximately minimize the maximum expected sample size in the case of Bernoulli and Normal populations. In this paper we construct sequential procedures of this type for the testing of an exponential parameter. An invariance property of the stopping region is presented which simplifies the construction of the Bayes regions. Several sampling plans are given and some of their properties are obtained.

It is desired to set confidence limits for the probability of successful operation at least until time x 0 of a series system of k dissimilar components. The components follow exponential failure laws, and data are available on failure times of components of each type. Exact confidence limits for k = 2 have previously been given by Lentner and Buehler. The present paper deals with a large-sample approximation to the exact solution for arbitrary k.

In many situations approximation of a set of data by a polygonal curve is more advantageous than approximation by a polynomial. If the join points of the polygonal curve are known, the problem is quite simple. If, however, they are to be chosen in some expeditious fashion, considerable numerical difficulties can arise if the curve-fitting problem is approached directly. In this paper it is shown that dynamic programming offers a simple direct approach to the determination of an optimal fit.

Reliability data often include information that the failure event has not yet occurred for some items, while observations of complete lifetimes are available for other items. Bayesian analysis requires a conjugate posterior density function after any combination of complete and incomplete observations. This paper considers density functions having failure rate function consisting of a known function multiplied by an unknown scale factor. It is shown that a gamma family of priors is conjugate for both complete and incomplete experiments. A very flexible and convenient model results from the assumption of a piecewise constant failure rate function.

The author [1], [2] has given an algorithm for finding that stratified allocation which minimizes the posterior variance of the overall population mean subject to a budget constraint under a model in which a normal prior distribution and independent normal sampling distributions were assumed. The budget constraint assumed a variable per unit cost of observation. In the present paper these results are extended to cover the case where there are fixed costs, as well as variable costs, associated with sampling in the ith stratum. The resulting algorithm is noted to be applicable in finding the optimal allocation of sampling effort (with fixed and variable sampling costs) under a variety of distributional assumptions. An interpretation is also given to two and higher stage design questions when there is differential prior information regarding the first stage units.

In this paper we consider the allocation problem for multivariate stratified surveys. If the stratum variances for the different variates are not distributed in the same way Neyman allocation optimizing the measurement of one variate is of limited value. In our formulation we determine the allocation such that sample estimates meet stated levels of precision or tolerance at minimum cost. Solution of the allocation problem is shown to be a programming problem and an example is given to illustrate it. By obtaining the solution to one plan a sampler essentially obtains the solution to a whole series of plans. The problem of tolerance setting is then discussed. An emprical solution, based on practical rather than some over-riding theoretical consideration, to the problem is given. A set of coefficients which elucidate the cost implications of each of the tolerances are derived.