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Comments on the paper 'The effect of increasing sample size on the precision of an estimator' published in the Journal of The American Statistician. Applicability of the sampling to an infinite theoretical population; Considerations in sampling from finite population; Dependence of the estimator on the design.

A description is given of a quick and simple technique for checking, by means of the usual χ² and Kolmogorov-Smirnov tests, the goodness of fit of Weibull's distribution to data relating to the fatigue of textile materials tested for abrasion, repeated extension, repeated bending, etc., or subjected to any type of action susceptible of being treated by means of such a distribution. [ABSTRACT FROM AUTHOR]

This paper presents the derivation of formulae for estimating fraction defective for a process from either continuous production or discrete lots. The estimators derived are the best linear unbiased estimators (BLUE). Also presented are the variances of the estimators. [ABSTRACT FROM AUTHOR]

Investigates the accuracy of comparative estimating. Examination of two different methods of time interval selection; Emphasis on planner estimation error; Implications for managerial decision making.

Two equations have been obtained using regression analysis techniques which can be applied to determine storage space requirements with acceptable accuracy and minimum effort. By defining all items by means of a common parameter--packing factor--it was possible to ignore the individual weights of each item, yet still load the transport and storage unit to some practical load limits. The method used in this paper is general and can be applied to similar problems with different boundary conditions. [ABSTRACT FROM AUTHOR]

In a previous paper [4] an estimate of the 4-variate product moment E[{x[sub I] - E(x[sub I])} {x[sub j] - E(x[sub j])} {x[sub k] - E(x[sub k])} {x[sub h] - E(x[sub h])}] was obtained. This estimate had a slight bias, as we pointed out. In this paper an unbiased estimate of the 4-variate product moment is obtained. Asymptotic results for the 3-variate and 4-variate product moments and their estimates are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, the simple estimate formed by the ratio of the sample means, the estimate obtained by the statistician M.H. Quenouille's method of bias reduction, the unbiased estimate of the statisticians L.A. Goodman and H.O. Hartley and the estimate proposed by H.O. Hartley are compared. These estimates are respectively denoted by t1, t2, t3 and t4. Mean square errors of these estimates are compared under J. Durbin's model. In practice g has often been found to lie between 0 and 2. In this paper expressions for the Mean Square Errors of these estimates are obtained for general values of n, g and h, and they are compared for finite values of n when g = 0, 1 and 2. When g = 0. The method of obtaining exact expressions for the Mean Square Errors (MSE's) of these estimates is similar to that of J.N.K. Rao and J.T. Webster. After a considerable amount of simplification, the following expressions for the biases and MSE's are obtained. When t=0, Durbin compared t1 with t2, and Rao compared t1 with t4.

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.

This paper describes a method of estimating the number of marriages during an inter-censal period among a group of never-married persons. Two alternative procedures are suggested: the first based on data on the never-married category and the second based on data on the ever-married category. The essential data needed are marital distributions by quinquennial age-groups from two successive censuses. The marital data from the 1951 and 1961 censuses of India are used to discuss the methodological problems involved in the estimation of marriage frequencies and in the construction of nuptiality tables based on them. Marriage and death are taken as the two attrition factors for decrement of single population. Gross and Net Attrition Tables for the single population of India are constructed for each sex. Section 5 describes various nuptiality measures derived from the Attrition Tables. It is suggested that the approach developed in this paper can be utilized for reliable marital and, subsequently, fertility projections. [ABSTRACT FROM AUTHOR]

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 autoregressive schemes. [ABSTRACT FROM AUTHOR]

In a recent paper in this Journal, Rothenberg, Fisher and Tilanus [1] discuss a class of estimators of tile location parameter of the Cauchy distribution, taking the form of the arithmetic average of a central subset, of the sample order statistics. They show that the average of roughly the middle quarter of the ordered sample has minimum asymptotic variance within this class, and that asymptotically it eliminates about. 36 per cent of the efficiency loss of the median (the most commonly used estimate,r) in comparison to the maximum likelihood estimator (m.l.e.). Of course both the m.l.e, and the best linear unbiased estimator based on the order statistics (BLUE) achieve full asymptotic efficiency in the Cramer-Rao sense and there can be no dispute about the relative merits of the three estimators asymptotically, or about the inferiority of the median (with asymptotic efficiency 8/pi[sup 2] * This character cannot be converted in ASCII text) 0.8 compared with about 0.88 for the estimator of Rothenberg et al.). In any practical situation however, we will be concerned with estimation from samples of finite size and asymptotic properties will not necessarily give any guidance here. We are essentially concerned with two points in assessing the relative merits of estimators in small samples, their case of application and "small-sample efficiency" which is conveniently measured as the ratio of the Cramer-Rao lower bound to the variance of the estimator. In this paper various estimators of the location of the Cauchy distribution arc compared in these two respects for samples of up to 20 observations. The small-sample properties of the m.l.e. have been extensively discussed elsewhere (Barnett [2]) and relevant results are summarized where necessary. The main purpose of the paper is to discuss general linear estimators based on the order statistics, and to assess their utility in the present context. Since this paper was prepared a further interesting 'quick estimator', b [ABSTRACT FROM AUTHOR]

Given N points independently drawn from the uniform distribution on (0, 1), let p[sub n], be the size of the smallest interval that contains n out of the N points; let n[sub p], be the largest number of points to be found in any subinterval of (0, 1) of length p. This paper uses a result of Karlin, McGregor, Barton and Mallows to determine the distribution of n[sub p] for p = 1/k, k an integer. The paper gives simple determinations for the expectations and variances of p[sub n], for all fixed n > (N + 1)/2, and of n[sub 1/2]. The distribution and expectation of n[sub p] are estimated and tabulated for the cases p = 0.1(0.1)0.9, N =2(1)10. [ABSTRACT FROM AUTHOR]

Recently Professors T. J. Rothenberg, F. M. Fisher, and C. B. Tilanus published a paper proposing the class of trimmed means as estimators of the location parameter of the Cauchy distribution [5]. They showed that the asymptotic sampling variance of the estimators in this class is essentially minimized by using the middle 24% of the sample order statistics. The corresponding estimate has an asymptotic relative efficiency to the best estimator for complete samples (A.R.E.) of .87796 as compared to an A.R.E. of .81057 for the sample median. In this paper a few "quick estimators" are considered as estimators for the location parameter of the Cauchy Distribution. A "quick estimate" is a linear combination (a weighted average) of one or more order statistics. Our goal is to find a simple estimator, i.e. an estimator based on only a few order statistics, which has an A.R.E. of at least 90%. We found an estimator based on five order statistics which is considerably better than the optimum trimmed mean (using the middle 24% of the sample order statistics) and much better than the sample median. The A.R.E. of the optimum censored estimate with censored fractiles .38 and .62 is also found, and a comparison between the trimmed, censored, and proposed estimators is made. [ABSTRACT FROM AUTHOR]

Given a population of N units, it is required to draw a sample of n distinct units in such a way that the probability for the ith unit to be in the sample is proportional to its 'size' x. From the alternative methods of achieving this we consider here only the so-called systematic method which, to the best of our knowledge, was first developed by W. G. Madow (1949): The units in the population are listed in a 'particular' order, their x, accumulated and a systematic selection of n elements from a 'random start' is then made on the accumulation. In a more recent paper (H. O. Hartley and J. N. K. Rao (1962) ) an asymptotic estimation theory (for large N) associated with this procedure was developed for the case when the order of the listed units is random. In this paper we draw attention to certain properties of Madow's estimator: We utilize the fact that with systematic sampling the total number of different samples is N (rather than ([This eq. cannot be change in char.]) as with completely random sampling). This simplification in the definition of the variance of the estimator in repeated sampling enables us to identify the exact variance of Madow's estimator with a 'between sample mean square' in a special analysis of variance (see section 4) and compare it with the variance of the pps estimator in sampling with replacement as well as in other sampling procedures. We also develop two approximate methods of variance estimation (see section 5). We pay particular attention to the case when the units are listed in the order of their size. With this particular arrangement our method can be described as 'systematic with random start' and the gain in precision that we accomplish has of course, analogues in systematic sampling with equal probabilities employing ratio estimators in which there is a relation between the ratio ri =yi/Xi and xi Compared with other methods the present procedure combines the advantage of ease of systematic sample selection with the availability of exact variance formulas for any n and N. Moreover, it usually leads to a more efficient estimate. Its shortcoming resides in the fact that the estimation of the variance is based on certain assumptions. [ABSTRACT FROM AUTHOR]

In this article, the author will talk about researcher R.A. Fisher's contributions to statistics. It is well-known that his work in genetics was of comparable status. It is largely represented by his book "The Genetical Theory of Natural Selection," though in his subsequent work his further association with ecological and experimental studies in evolutionary genetics, and his share in the development of studies in the human blood groups, might especially be recalled. Fisher's contributions to statistics began with a paper in 1912 advocating the method of maximum likelihood for fitting frequency curves, although the first paper of substance was his 1915 paper in the journal "Biometrika," on the sampling distribution of the correlation coefficient. Fisher tried to pose problems of analysis as the reduction and simplification of statistical data. He put forward his well-known concept of amount of information in estimation theory, such that information might be lost, but never gained, by analysis. His concept has been of great practical value, especially in large sample theory.

This paper develops two models, each of which is designed to estimate the probability of surviving from birth to selected exact ages of early childhood: namely ages two, three and five. The models are designed for use in areas with deficient registration systems. They require, as input, statistics which can be derived from retrospective data supplied by census or survey respondents. The first model, the age model, converts statistics on the proportion dead of children ever born to women in age groups 20-24, 25-29 and 30-34 into estimates of q2, q3 and q5. The second model, the marriage model, converts statistics on the proportion dead of children ever born to women of five-year marriage duration intervals into these estimates. The models can be used independently or simultaneously. These models were developed from data generated by a large number of empirical fertility and mortality schedules. Regression analysis was used to determine the parameter values of the relationships specified, and several sets of equations for estimating values of qa for a =2, 3 and 5 comprise the final product of the paper. It should be noted that the conceptual basis for the models was first derived by William Brass. The data generated for the regression analysis provided an opportunity to test the original Brass estimated model. We are able to report that the model performed well over the wide range of fertility and mortality conditions included in the test. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to examine the effect of a variety of misspecifications broadly defined to include both over- and underspecifications upon three econometric estimators (i.e., ordinary least squares (DLS) on the structural equations, two stage least squares (2SLS) and limited information single equation (LISE) maximum likelihood). The analysis which follows in this paper differs in several important respects from that of Cragg. First, our Monte Carlo experiments have been conducted with an economically relevant model following the suggestions of SMITH (1971) in earlier work. Second, our application of 2SLS and LISE assumes knowledge of all the pre-determined variables which are arguments in the 'true' unrestricted reduced form. In CRAGG'S (1968) paper he suggests that with such single equation estimators a misspecification involving only one equation will not alter these techniques' performance patterns for the other equations in the model (CRAGG, 1968, p. 65). However, if the misspecification calls for the omission of a pre-determined variable and the variable appears only in the equation in question, then the first stage instruments for the right hand side endogenous variables constructed with 2SLS and LISE will (a) not satisfy the properties for consistent estimators with this equation,[3] and (b) not provide efficient estimates for all equations of the model (see MCCARTHY, 1971). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to suggest a methodology for using both types of information. Accordingly, the resulting technique is a more efficient approach for estimating state and local government expenditure determinants. The technique is a generalized Aitken estimator for a system of unrelated regressions and was first introduced by ZELLNER (1962). The second problem with past research is the result of the inadequacy of our models for publicly-provided goods and services. While there is a voluminous literature relating to public goods and collective consumption in general, the decision process underlying public provision of goods and services has not been subjected to comprehensive modeling.[1] Therefore empirical analyses of expenditure patterns have been based on incompletely developed models. Our approach will be to suggest a model which is representative of the existing literature, sketch its theoretical foundation, and discuss the areas for future research. The present paper will not, however, attempt to develop a more complete model of the public decision process. [ABSTRACT FROM AUTHOR]

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In a recent paper, Tiao and Guttman (1967) discussed the use of adjusted residuals to analyze the behaviour in moderate to large samples of the premium and protection of Anscombe's rule (herein designated as the A[sup (k)]-rule) when sampling is from the N(μ, σ²) distribution μ is to be estimated, and where k outlying observations are suspected of being spurious (k = 1 and 2). A discussion of two other rules, Semi-Winsorization (S[sup (1)]-rule) arid Winsorization (W[sup (1)]-rule), is given in Guttman and Smith (1969, 1971). This paper investigates the behaviour, for moderate to large η, of the A[sup (k)], S[sup (k)] and W[sup (k)rules, k = 1 and 2. To do this, we define rules based on adjusted residuals, which we shall denote as the A[sub k], S[sub k] and W[sub k] rules. Expressions for the premium and protection of the S[sub k] and W[sub k] rules are derived, and contrasted with these characteristics of the A[sub k] rule, obtained by Tiao and Guttman (1967). Some discussion of the case when σ² is unknown is also included. Here we assume that there is an independent estimate of σ², and we use rules with a different type of adjusted residual. [ABSTRACT FROM AUTHOR]

A two-stage procedure using shrinkage techniques is used for estimation in the simple linear regression model. A direct estimation of the predicted response as well as estimation of the parameters in the model are considered in this paper. Throughout the paper, 'a priori' values of the parameters are assumed to be available in the form of prior estimates or realistic guessed values based on the experimenter's knowledge and past experience. Some real problem examples are given to illustrate the use of this procedure. This technique may have some application to response surface analysis. [ABSTRACT FROM AUTHOR]

Suppose that f(t; 0) is the density function of the time to failure random variable of a device under an environment defined by a vector of generalized stresses S. It is desired to predict. θ[sub u], the value of θ at some low values of the stress, say S[sub u]. The elements of θ are re-parameterized as a function of S, such that θ[sub j] = f[sub j]( S; a[sub j]), where f[sub j] are known functions and the a[sub j] are unknown. Life tests are conducted at k different accelerated values of S, S[sub i] (i = l, 2, ··· k), and estimates of the elements of θ, θ[sub ji], are obtained from item censored samples. These estimates are then used to obtain point and interval estimates of a[sub j], using either the method of least squares, or the method of maximum likelihood. Having obtained the estimates of a[sub j], an inference about θ[sub u] can be made. Some problems in obtaining the least squares estimates of a[sub i] are pointed out in this paper. To illustrate this, an unrealistic but simple case is treated and discussed in this paper. It is assumed that f(t; 0) = λ exp (-λt), and that λ[sub i] = BY[sub i], where λ[sub i] is the hazard rate under an environment V[sub i], and B is unknown. A weighted least squares estimator for B is obtained, and it is shown that asymptotically this estimator converges in distribution to that of a unit normal. The maximum likelihood estimator for B is shown to have an inverted gamma distribution. Properties of the estimator of &lambda[sub u] are discussed. [ABSTRACT FROM AUTHOR]

A principal objective of this paper is to discuss a class of biased linear estimators employing generalized inverses. A second objective is to establish a unifying perspective. The paper exhibits theoretical properties shared by generalized inverse estimators, ridge estimators, and corresponding nonlinear estimation procedures. From this perspective it becomes clear why all these methods work so well in practical estimation from nonorthogonal data. [ABSTRACT FROM AUTHOR]

The estimates of Koch [1967a] have the undesirable property that they may change in value if the same constant is added to each of the observations. In this paper, an alternative procedure based on the same general, principles is developed and applied to a variety of models. As before, the estimators obtained are unbiased and consistent. They are also reasonably easy to compute. Finally, in the case of balanced experiments, they coincide with those obtained from the analysis of variance. On the other hand, their structure is more complex than that of the estimators considered in the previous paper. In particular, the derivation of their covariance matrix is much more complicated, and hence no attempt has been made here to study its properties. [ABSTRACT FROM AUTHOR]

This paper reviews three basic methods of non-linear least squares estimation and the best known modifications to the popular Gauss-Newton procedure. With regard to the Gauss-Newton procedure the paper identifies the problem of poor parameterization and gives simple examples to show how it may arise. A simple example is also given to show that one popular method of handling constrained boundaries can lead to erroneous results. A modification, based on stepwise linear regression techniques, to handle both of these problems is presented and discussed and an example is given to illustrate its effectiveness. The required fundamentals of stepwise linear regression are reviewed. [ABSTRACT FROM AUTHOR]

This paper presents the results of a Monte Carlo study of the accuracy of an approximation to the power of the chi-square goodness of fit test with small but equal expected frequencies. Various combinations of sample size, number of groups, and alpha level are considered, and in most instances the actual power of the test is estimated to be less than the nominal power. The degree of accuracy appears to be more related to the size of the sample than to the size of the expected frequencies. The following rule of thumb is offered for obtaining crude estimates of the actual power from the nominal power for sample sizes from 10 to 50: The actual power of the test equals about eight-tenths of the nominal power. [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]

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. [ABSTRACT FROM AUTHOR]

A sampling procedure involving repeated application of double sampling for stratification on several successive occasions is considered. The formulae for the estimate of the mean on any current occasion and its variance are obtained. The determination of optimum allocation of available resources and the extension of the procedures for uni-stage sampling, to multi-stage sampling, are briefly indicated. The use of the design, suggested in this paper, is illustrated by its application to a recently conducted survey for estimation of coconut production in the State of Assam. [ABSTRACT FROM AUTHOR]

In this article, the author expresses his views on rotational and parallel traverses in the rapid integration of geographic areas. In an interesting report, James P.Latham has drawn attention to the possibilities of high-speed electronic scanning equipment in the rapid integration of irregular geographical areas. The basis of the traverse method is that the sum of intercepts along a traverse line is used to estimate the proportion of the total area occupied by the phenomena being intercepted. Results obtained from the simulated traverses confirm the findings of Latham's important paper on the general utility that one expects to find from rapid scanning of geographic patterns.

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 yon 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. [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]

This paper describes a double sampling scheme to estimate with at least a given precision the mean of a variate whose variance has two components one of which is unknown. Although the procedure is described almost entirely in terms of a particular problem (quality control of net weight), it may be applied in a more general situation (section 5). The consequences of using either the mean square error or the range to estimate variance are investigated. [ABSTRACT FROM AUTHOR]

In the process industries data must often be obtained under conditions of closedloop operation; that is, under conditions where feedback control is being applied. In the analysis of such data care is needed to properly take account of the manner of its generation. In particular, if standard open-looped procedures of model identification, estimation and diagnostic checking are applied to closed-loop data incorrect models may result and lack of fit not be detected. This paper discusses the identification, estimation and diagnostic checking of closedloop systems and illustrates the ideas on two real sets of data. [ABSTRACT FROM AUTHOR]

In this paper we have obtained the maximum likelihood estimate of the fraction defective under a fully-curtailed double sampling plan. The asymptotic variance of the maximum likelihood estimate is also obtained. The results are generalized to multiple sampling plan under full-curtailment. A simple device for executing a curtailed multiple sampling plan and the calculation of the ASN, the probability of acceptance therefrom is discussed. [ABSTRACT FROM AUTHOR]

This paper establishes a procedure for estimating the various spectral parameters in an echo type situation where both the signal is received as well as lagged and attenuated forms of the signal. The lag parameters are assumed frequency dependent and the basic assumption made is that these lags are sufficiently large to cause significant variation in the spectrum over the narrow trend of frequencies considered. The asymptotic properties of the estimates of these parameters are established and some simulations and an analysis of real data are considered. [ABSTRACT FROM AUTHOR]

This paper compares two point estimators of fraction defective of a normal distribution when both population parameters are unknown; the minimum variance unbiased estimator, &Fmacr;(χ), and the maximum likelihood estimator, &Fcirc;(χ). Using minimum mean squared error as a criterion, it is shown that the choice of estimator depends upon the true value of F(χ), and the sample size. In the domain .0005 ≤ F(χ) ≤ .50, the maximum likelihood estimator is generally superior even for small sample sizes, except for F(χ) less than about 0.01, or greater than 0.25. Furthermore, the bias in the m.l.e. is slight over much of the domain where this estimator has smaller mean squared error. As a practical solution to the estimation problem, it is suggested that the m.v.u.e. be calculated, and if this estimate is between 0.01 and 0.25, it should be replaced with the m.l.e. This combined estimator is shown to be nearly as efficient as the better of the m.v.u.e. and m.l.e. throughout the domain of F(χ). [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 proposes several methods of estimating parameters in stable distributions. All the methods involve sample characteristic functions. One of the methods which is based upon the method of moments is treated in some detail. Asymptotic normal distributions for the proposed moment estimators are provided. Moreover, all methods provide consistent estimators. The estimation problem is treated for both univariate and multivariate stable distributions. [ABSTRACT FROM AUTHOR]

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-beta regression theory as a natural extension of conventional fixed-beta 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. [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.

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]

The problem considered in this paper is how to combine estimators of the common mean from two samples corresponding to normal populations with different unknown variances. Attention is confined to the case where it is known that the variance of one specific population exceeds that of the other. Three classes of unbiased estimators are presented, one of which is based on a preliminary test of significance regarding the ratio of the population variances. The gain achieved by utilizing the knowledge that the ratio of variances exceeds one is investigated by comparing the efficiencies of these estimators with an estimator presented by Graybill and Deal [1] in which no restriction on the ratio of variances is present. [ABSTRACT FROM AUTHOR]

Maximum likelihood and other BAN estimators have been shown to possess certain optimal asymptotic properties in estimating the parameters of probability distributions satisfying specific regularity conditions. The subject of non-regular estimation is concerned with problems in which these conditions do not hold. In many such problems, classical-lower bounds on the variance of unbiased estimators, such as the Cramer-Rao bound, lead to the trivial result V(t) * (These characters cannot be converted in ASCII text) 0, where t is any unbiased estimator. A number of alternative bounds for application in the non-regular case have been derived. In this paper previous results of this type are reviewed and an additional bound is given. The specific applications of interest involve estimation of a location parameter. Applications of the bounds to the exponential, uniform and Pearson Type III distributions are investigated. [ABSTRACT FROM AUTHOR]

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 under an exact linear constraint relating this regressor to the m regressands of the system. [ABSTRACT FROM AUTHOR]

In this paper it is assumed that, using a sample survey, two factors are to be studied, comparisons between the "levels" of the factors are of greatest interest, and there is "interaction" between the factors. Attention is concentrated on situations in which only two levels of each factor are to be compared, but extensions to more complex surveys are discussed. Assuming independent sampling, optimal sample size allocations are obtained. Where these allocations require recourse to programming algorithms, approximate solutions are given. If independent sampling is not feasible, a double sampling procedure is suggested. To indicate how sub-sampling from the first phase sample is to be carried out, a sampling rule (possessing optimal conditional precision properties) is derived. Then, a procedure to determine the optimal first phase sample size is given. Finally, it is demonstrated that this double sampling procedure can be applied to estimation of the (finite) population mean when double sampling with stratification is used. [ABSTRACT FROM AUTHOR]

In this note we examine the bias and small sample efficiency of certain estimators for the parameters of a linear regression function when some observations are missing. The estimators studies in this paper were suggested by the large sample study reported in this issue of the Journal. We conclude that our asymptotically unbiased estimators of beta and mu[sub y|x], have negligible bias in sample sizes as small as n = 20, and that our asymptotically unbiased estimator of sigma[sup 2] may have an 8% bias when n = 20. The small sample and asymptotic efficiencies of these estimators are nearly the same for n = 60; when n = 20 the difference between these two efficiencies depends on the correlation coefficient rho and the pattern of missing observations. [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]