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The article presents the author's comments on an article "Optimum Sample Size for a Problem in Choosing the Population With the Largest Mean," by statistician Paul N. Somerville, published in the June 1970 issue of "Journal of the American Statistical Association." In the article Somerville described a procedure for selecting the population with the largest mean from normal populations with unknown means and a common known variance. The author claims that his procedure gives a unique minimax solution and that the minimax is a reasonable solution to the problem. According to the author, Somerville has not made it clear that his procedure gives only the local minimax solution. This makes practical applications of his procedure severely limited for it is difficult to think of a situation in which an experimenter would like to determine a local minimax value.

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 statistical approach to model a papermaking process and to control paper basis weight is presented. The process behavior is described by a mathematical model which take into consideration both the input-output dynamics ami disturbances coming into the system. The model building procedure was based on a three-step approach of identification, estimation and diagnostic checking. The disturbance model is accounted for by a discrete time series model, whereas the dynamic model is a discrete transfer function model. The simplicity of the modeling process lies in the fact that only the inputs-output data are necessary-without recourse to a complicated analysis of the physical system, itself. An optimal control equation was arrived at whereby the basis weight of the paper was to be controlled at a specified target value. Confirmation runs were conducted to check the effectiveness of the control equation. [ABSTRACT FROM AUTHOR]

The paper describes a double-sampling scheme for controlling the mean of a normal population. Results are presented which show considerable improvement over certain single-sampling schemes when judged by the criterion of average production run length. [ABSTRACT FROM AUTHOR]

This paper compares four major schemes used for forecasting mean demand to be used as input into an inventory model so that 'optimum' stockage levels can be obtained. The inventory model is the classical order up to S, infinite horizon model with carry-over from period to period and complete back-ordering. Maximum likelihood, exponential smoothing, standard Bayes and adaptive Bayes schemes are used and results, via Monte Carlo simulation, are obtained on the average costs per period for (1) stationary demand, (2) long-term trend and (3) 'shock' changes in mean demand. [ABSTRACT FROM AUTHOR]

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

The article discusses moments of the distribution of sample size, N in a sequential probability ratio test (SPRT). The present paper provides variance, the third and the fourth moments of N. The details are worked out in five common applications of the SPRIT. The relation of the variance of N to the truncation of a SPRT is discussed is also discussed in the paper. Scholar A. Wald indicated in passing how one can obtain the moments of N, but the only published literature where the author encountered a general expression for the variance of N. However, their expression is incorrect. Using scholar J. Wolfowitz's results, which they do, or differentiating Wald's, fundamental identity twice one gets provided. In many practical applications of the SPRT, μ and moments in an equation derived are differentiable functions of a real-valued parameter. The limiting expressions for the moments can then be determined by standard methods of mathematical analysis. However, for the third and fourth moments the actual technique may involve an excessive amount of arithmetic.

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. [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.

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.

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]

This paper discusses a procedure for finding robust; estimators of the location parameter of symmetric unimodal distributions. The estimators are based on robust rank tests and the methods used are applicable to other one parameter problems. To every density function there corresponds an asymptotically most powerful rank test (a.m.p.r.t.). For a set F of density functions the maximin rank test, R, maximizes the minimum limiting Pitman's efficiency of R relative to the a.m.p.r.t. for each member of F. This maximin test, R, can be used to construct estimators according to the proposal of Hodges and Lehman; it generates another estimator T in the following manner. If the test based on R is the a.m.p.r.t. for samples from a density function g(x - theta), then the estimator T will be the best linear unbiased estimate (b.l.u.e.) of the location parameter for samples from g(x). Unfortunately, the estimator T is not necessarily consistent for all members of F. A class of rank tests which generate linear combinations of a few order statistics is introduced and a simple estimator using the 33 1/3rd, 50th and 66 2/3rd percentiles is proposed. The relationship of the present paper to the work of Huber is discussed and it is shown that the b.l.u.e. corresponding to his least favorable distribution is the trimmed mean. [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]

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]

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.

The use of simulation, as a technique for attacking difficult problems, has increased greatly with the availability of the digital computer. This is illustrated by the large number of references in Shubik's (1960) bibliography[sup 2] and in the large number of studies published since then. Simulation is essentially an extension of a technique known as empirical sampling, or distribution sampling, which has been used in the field of statistics for many years. The limitations of the technique, which are well known to statisticians, are apparently not as well known, or at least not as well recognized, by those using simulation today. The first part of this paper contains an historical survey of distribution sampling as used by statisticians. The material was originally prepared in 1953 and is reproduced here in slightly revised form to bring this history to the attention of present day simulators in order that the lessons that can be learned from this part can more readily be incorporated in the development of methodology today. The second part of this paper discusses the relevance of empirical sampling to the present day state of the art of simulation. The technique of generating random members, developed for empirical sampling can be applied directly to simulation. However in other aspects simulation is more difficult than empirical sampling and here the theory of distribution sampling does not have much to offer. The difficulties are due to lack of independence among time series, non-stationarity of the time series, and the large number of parameters. [ABSTRACT FROM AUTHOR]

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]

This paper considers the development of a cost based industrial quality control system which functions both as an acceptance sampling plan and as a means for determining an assignable cause of quality variation in the manufacturing process. The manufacturing facility studied normally produces items at a relatively constant proportion defective. The facility is subject, however, to random failures in time which cause the proportion defective to shift upwards. The objective of the paper is to develop a sampling plan which minimizes the expected total annual cost of quality control by determining the sample size, acceptance number, and time period between successive samples. The optimization technique employed is a combination of a multivariate search technique and a partial enumeration. [ABSTRACT FROM AUTHOR]

This paper gives tables of the exact operating characteristic (O. C.) function of the exact average sampler number (A. S. N.) function, and of the probability of termination at any step for certain one-sided truncated sequential tests of the mean of a normal distribution when the variance is known. By looking at the smallness of the probability that a sequential test terminates after the equivalent fixed sample test it is observed that truncated sequential tests are more economical over a wider range of possible values of the mean than previously recognized. The O. C. and A. S. N. tables presented here may be used for approximate two-sided truncated sequential tests in the same manner as in [7]. Also, the scope of this paper is contrasted with the work of other authors. Aroian's direct numerical methods are used [2]. These methods emphasize the use of digital computers ire calculate the distributions of those test statistics required for computation of the operating characteristic (O. C.) function and the average sample number (A. S. N.) function. Such distributions are often convolutions involving sums of sequentially truncated random variables. The direct methods allow unlimited flexibility in the selection of test regions, and part of the discussion in this paper is more general than required for the specific tables calculated. [ABSTRACT FROM AUTHOR]

Copyright of INFOR is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of INFOR is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

This paper presents a procedure for matching double sampling plans to given single sampling plans. The procedure assumes sampling from a normal distribution with known standard deviation and assumes that lot average is the property of interest. It is applicable, for example, to sampling of bulk material. Double sampling plans with the "best" average sample size are discussed. An example shows how to use the procedure to find approximately matching double and single sampling plaits for sampling from a binomial distribution. [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]

A simple, unbiased estimator, based on a censored sample, has been proposed by Bain [1] for the scale parameter of the Extreme-value distribution. This estimator was shown to have high efficiency and to be approximately distributed as a chi-square variable if substantial censoring occurs. Further small sample and asymptotic properties of this estimator are considered in this paper. The estimator is modified so that it is more applicable to the complete sample case and a close chi-square approximation is established for all eases. The estimator is also shown to be related to the maximum likelihood estimator. [ABSTRACT FROM AUTHOR]

A method is described for assigning an observation to one of two normal populations with differing, unknown means and differing, unknown variances. The classification procedure rests on the likelihood ratio, which, for a given observation, is a function of four unknown parameters. Sample information is used to obtain a confidence region for these parameters. From this confidence region, a conservative confidence interval for the likelihood ratio is derived. Interpreting the likelihood ratio as a measure of the odds in favor of each population as the source of the observation, the interval can be used, in an obvious manner, for classification purposes. Simulation techniques are employed to examine the conservative nature of the interval Finally, as an illustration of the method, the results are applied to the determination of authorship of the disputed Federalist papers. [ABSTRACT FROM AUTHOR]

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 stepping region is presented which simplifies the construction of the Bayes regions. Several sampling plans are given and some of their properties are obtained, [ABSTRACT FROM AUTHOR]

Data editing by a computer program can be regarded as a form of statistical quality control. In that perspective, costs and benefits of alternative programs can be compared to one another and, of course, to no control plan at all. This paper outlines one approach to cost-benefit comparison and utilizes a realistic example to suggest that computerized data editing is generally superior to clerical editing. It also warns against edit programs which provide for record rejections and off-line review. Most importantly, it urges that a systematic comparison of costs can assist significantly in edit planning. [ABSTRACT FROM AUTHOR]

Let X[sub 1], ... , X[sub m] and Y[sub 1], ... , Y[sub n] be independent random samples from populations having continuous d.f.'s psi((x-micro)/sigma) and psi((y-nu)/tau) respectively. The classical F-test of a hypothesis concerning angle = tau/sigma is known to be non-robust. This paper examines several robust alternative procedures and compares them on the basis of Pitman a.r.e and Monte Carlo studies of power functions. An approximate permutation test [13] and a "jackknife" procedure [9] are found to be most satisfactory; while a class of "rank-like" tests [10] are found to be "useful inefficient statistics" [ABSTRACT FROM AUTHOR]

This paper presents the exact sampling distributions of the ordinary and the two-stage least squares estimators of a structural parameter in a structural equation with two endogenous variables in a complete system of stochastic equations. The results show that the distributions of the two estimators are essentially similar to each other. It can also be seen that both distributions depend crucially upon the deviation of a regression coefficient of disturbance terms of two endogenous variables from a structural parameter, and that the first estimator possesses moments up to the order N-2, while the second possesses them up to the order K-1, where N is the sample size and K is the number of exogenous variables excluded from the equation to be estimated. The small sample properties of the estimators are investigated by numerical evaluations of the density functions. [ABSTRACT FROM AUTHOR]

Let X[sub (1)] is less than or equal to X[sub (2] is less than or equal to ... is less than or equal to X[sub (n)] be an ordered sample of a random variable X which has continuous probability distribution function F (x), and let F[sub n] (x) be the corresponding empirical distribution function. The following three statistics, introduced by A. Renyi, are considered: [Multiple line equation(s) cannot be represented in ASCII text] The paper presents table of exact probabilities for these statistics for finite sample sizes. The limiting distributions of these statistics for sample size n arrow right Infinity are discussed, and sample sizes are indicated for which these limiting distributions can be used instead of the exact distributions. Numerical examples for the use of the tables are presented, as well as applications to testing hypotheses on life distributions and to one-sided estimation of probability distribution functions from censored data. [ABSTRACT FROM AUTHOR]

The choice of a Dodge-type continuous sampling plan is usually based upon the requirement of a maximum limit on the expected fraction of accepted defective material (AOQL). This limit is contingent upon the process producing the material being in "a state of statistical control". White (1966) has provided a method by which a corresponding limit (UAOQL) can be obtained without the assumption that the process in statistical control. White assumed that all defective items were replaced by good items. This paper presents a method for calculating UAOQL's when defective items are removed but not replaced by good units. [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]

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]

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]

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

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]

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]

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]

In a linear regression model, when errors are autocorrelated, several asymptotically efficient estimators of parameters have been suggested in the literature. In this paper we study their small sample efficiency using Monte Carlo methods. While none of these estimators turns out to be distinctly superior to the others over the entire range of parameters, there is a definite gain in efficiency to be had from using some two-stage procedure in the presence of moderate high levels of serial correlation in the residuals and very little loss from using such methods when the true rho is small. Where computational costs are a consideration a mixed strategy of switching to a second stage only if the estimated rho is higher than some critical value is suggested and is shown to perform quite well over the whole parameter range. [ABSTRACT FROM AUTHOR]

Consider a clinical trial to compare two treatments where response is dichotomous and patients enter the trial sequentially. This paper investigates the conduct of such a trial where the "Play the Winner Rule" (PWR) is used to assign patients to the different therapies. The implementation of the PWR in a clinical trial tends to place more patients on the better treatment. Both theoretical and numerical investigations show that over a wide range of situations this rule leads to near optimum results when used in a two-stage manner. Furthermore, these results are insensitive to optimum sample size requirements. [ABSTRACT FROM AUTHOR]

The aim of this paper is two-fold: (1) To give a unified treatment of the theory of order statistics when the parent distribution is not necessarily continuous. (2) To assess the effects of grouping on the distribution of order statistics and to indicate the convenience, under suitable conditions, of using order statistics for the estimation of parameters from grouped data with or without censoring. [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]

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. [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]

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]