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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.

Several writers have studied the problem of obtaining confidence bands for a straight line regression under the requirement that the bands be at a specified confidence level, (1 - α) say, for values of the independent variable restricted to a pre-specified closed interval. The bands studied by other writers have been either bands of equal width throughout the interval or trapezoidal bands. This paper studies confidence bands of the classical hyperbolic type under the restriction noted above. Relevant distribution results differ according to whether the pre-specified interval on the independent variable is symmetric or asymmetric about the mean of the independent variable values in the experiment. In the former case the problem under study, is shown to be equivalent to a problem studied by Halperin et al (JASA, Sept. 1967) for which distribution theory and tables are already available. The symmetrical case is generalized to obtain confidence bands in multiple linear regression at level (1 - α) for an ellipsoidal region on the independent variables centered at the point of means of the independent variable values used in the experiment. For straight line regression some numerical comparisons are made with bands of the other types mentioned above. These comparisons suggest that the bands proposed in this paper are uniformly superior to bands of equal width in the sense of having smaller average width; the proposed bands appear to be superior to trapezoidal bands in the same sense for intervals of practical interest, i.e. within or reasonably close to the range on the independent variables used in the experiment. [ABSTRACT FROM AUTHOR]

The impact of dependence between individual test statistics is currently among the most discussed topics in the multiple testing of high-dimensional data literature, especially since Benjamini and Hochberg (1995) introduced the false discovery rate (FDR). Many papers have first focused on the impact of dependence on the control of the FDR. Some more recent works have investigated approaches that account for common information shared by all the variables to stabilize the distribution of the error rates. Similarly, we propose to model this sharing of information by a factor analysis structure for the conditional variance of the test statistics. It is shown that the variance of the number of false discoveries increases along with the fraction of common variance. Test statistics for general linear contrasts are deduced, taking advantage of the common factor structure to reduce the variance of the error rates. A conditional FDR estimate is proposed and the overall performance of multiple testing procedure is shown to be markedly improved, regarding the nondiscovery rate, with respect to classical procedures. The present methodology is also assessed by comparison with leading multiple testing methods. [ABSTRACT FROM AUTHOR]

This paper compares the Pearson Chi-Square and Kolmogorov goodness-of-fit tests with respect to validity under the following conditions: (1) the N independent observations are tabulated and arranged into k mutually exclusive groups that are equally probable under the hypothesis to be tested; and (2) both N and k are "small"; i.e., not greater than 50. A random sampling experiment was performed, and the results show that in general for the conditions considered, the Pearson test is more valid than the Kolmogorov test. [ABSTRACT FROM AUTHOR]

This article comments on statistical decision theory. The term statistical decision theory appears to be a condensation of A. Wald's phrase the theory of statistical decision functions, which occurs, in the preface to his monograph as well as earlier in Wald. Wald viewed his theory as a codification and generalization of the theory of tests and confidence intervals already developed by Neyman, often in collaboration with E. Pearson. The vignette on hypothesis testing by Marden presents an excellent review of the various manifestations of hypothesis testing. It is hard to choose a favorite among the wonderful Neyman-Pearson papers on the foundations of testing and confidence intervals. According to the foregoing, the spirit of decision theory is pervasive in contemporary statistical research. Common manifestations include both mathematical and numerical attempts to check the frequentist performance of proposed procedures. This includes comparative investigations of level and power for hypothesis tests or of precision of proposed estimators as, for example, might occur in a Monte Carlo comparison of variances and biases. The vignette on Bayesian analysis by Berger describes several different approaches to Bayesian analysis. None of these is directly the pure frequentist approach, in which the prior is a given distribution with the same frequentist validity as the family of distributions.

We consider three problems in high-dimensional linear mixed models. Without any assumptions on the design for the fixed effects, we construct asymptotic statistics for testing whether a collection of random effects is zero, derive an asymptotic confidence interval for a single random effect at the parametric rate n , and propose an empirical Bayes estimator for a part of the mean vector in ANOVA type models that performs asymptotically as well as the oracle Bayes estimator. We support our theoretical results with numerical simulations and provide comparisons with oracle estimators. The procedures developed are applied to the Trends in International Mathematics and Sciences Study (TIMSS) data. for this article are available online. [ABSTRACT FROM AUTHOR]

This paper puts forth the view that diversity is an average property of a community and identifies that property as species rarity. An intrinsic diversity ordering of communities is defined and is shown to be equivalent to stochastic ordering. Also, the sensitivity of an index to rare species is developed, culminating in a crossing-point theorem and a response theory to perturbations. Diversity decompositions, analogous to the analysis of variance, are discussed for two-way classifications and mixtures. The paper concludes with a brief survey of genetic diversity, linguistic diversity, industrial concentration, and income inequality. [ABSTRACT FROM AUTHOR]

The article presents the author's comments on paper by researcher D. Basu related to randomization analysis of experimental data. Basu writes entertainingly, perhaps, but not informatively. Basu's paper discusses prerandomization, postrandomization, and unrecorded randomization. This discussion is irrelevant. But it is useful, perhaps, to make a remark. It also discusses the sufficiency principle. As Basu has written, this is a data-reduction principle. Basu also discusses researcher R.A. Fisher randomization test. It is obvious that the population in a randomization test of a randomized experiment is "the product of the statistician's imagination." With respect to Basu's writing on "the physical act of randomization," the author believes Basu is merely plain wrong. The paper also describes randomized pair trial. In his paper, Basu gives a hypothetical interchange of a statistician and a scientist and the author. The author suggests that this serves no useful purpose. The author finds the lack of knowledge that underlies Basu's thesis rather surprising, incongruous, and deplorable.

The paper reviews and extends a Bayesian full information analysis of the simultaneous equations model, based upon an extended natural conjugate prior density. The extended prior density belongs to a closed family and is compatible with the independent specification for each equation of a marginal prior density in the multivariate Student form. The paper establishes properties of the resulting posterior density, of some conditional and marginal densities, and of the posterior densities in the special case of seemingly unrelated regression equations. [ABSTRACT FROM AUTHOR]

A major concern in large-scale surveys is the problem of subpopulation estimation (domain estimation). This paper presents a study of four estimators for estimating domain totals. The domain considered in the study is an area type of domain, that is, a domain consisting of a combination of a certain number of area units belonging to different strata This paper uses some actual data and some fictitious data to compare variances and mean square errors of the four estimators. [ABSTRACT FROM AUTHOR]

Bayesian techniques are widely used in these days for simultaneous estimation of several parameters in compound decision problems. Often, however, the main objective is to produce an ensemble of parameter estimates whose histogram is in some sense close to the histogram of population parameters. This is for example the situation in subgroup analysis, where the problem is not only to estimate the different components of a parameter vector, but also to identify the parameters that are above, and the others that are below a certain specified cutoff point. We have proposed in this paper Bayes estimates in a very general context that meet this need. These estimates are obtained by matching the first two moments of the histogram of the estimates, and the posterior expectations of the first two moments of the histogram of the parameters, and minimizing, subject to these conditions, the posterior expectation of the Euclidean distance between the estimates and the parameters. Several applications of the main result are provided in the normal and other models. Also, the results are applied to an actual data set. [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]

Experimental design considerations have been virtually ignored by economists who have conducted computer simulation experiments with models of economic systems. The objective of this paper is to spell out in detail the relationship between existing experimental design techniques and techniques of data analysis and the design of simulation experiments with economic systems. We begin by defining the problem of experimental design as applied to computer simulation experiments. With the aid of an example model, we explore several techniques of data analysis and a number of specific experimental design problems. Although this paper is oriented towards the design of computer simulation experiments in economics, the techniques which are discussed are of a general nature and should be applicable to the design of simulation experiments in other disciplines. [ABSTRACT FROM AUTHOR]

A growing interest has been witnessed in recent, years in multivariate discrete probability models. The multivariate logarithmic series distribution (LSD) is a multivariate analogue of the univariate LSD. This paper investigates the marginal and the conditional distributions of the multivariate LSD. It. records its moment properties and also provides some regression and correlation analysis. An interesting modal property of the distribution is discovered. The paper also discusses the estimation of the parameters by the methods of maximum likelihood and unbiased minimum variance. At the end is discussed in detail an application of the multivariate LSD to the field of population and community ecology. [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]

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]

We propose a new approach to the problem of high-dimensional multivariate ANOVA via bootstrapping max statistics that involve the differences of sample mean vectors. The proposed method proceeds via the construction of simultaneous confidence regions for the differences of population mean vectors. It is suited to simultaneously test the equality of several pairs of mean vectors of potentially more than two populations. By exploiting the variance decay property that is a natural feature in relevant applications, we are able to provide dimension-free and nearly parametric convergence rates for Gaussian approximation, bootstrap approximation, and the size of the test. We demonstrate the proposed approach with ANOVA problems for functional data and sparse count data. The proposed methodology is shown to work well in simulations and several real data applications. [ABSTRACT FROM AUTHOR]

Ordinary differential equation (ODE) is widely used in modeling biological and physical processes in science. In this article, we propose a new reproducing kernel-based approach for estimation and inference of ODE given noisy observations. We do not assume the functional forms in ODE to be known, or restrict them to be linear or additive, and we allow pairwise interactions. We perform sparse estimation to select individual functionals, and construct confidence intervals for the estimated signal trajectories. We establish the estimation optimality and selection consistency of kernel ODE under both the low-dimensional and high-dimensional settings, where the number of unknown functionals can be smaller or larger than the sample size. Our proposal builds upon the smoothing spline analysis of variance (SS-ANOVA) framework, but tackles several important problems that are not yet fully addressed, and thus extends the scope of existing SS-ANOVA as well. We demonstrate the efficacy of our method through numerous ODE examples. [ABSTRACT FROM AUTHOR]

The article presents the authors' comments on researcher D. Basu's paper related to randomization analysis of experimental data. Basu's paper on researcher R.A. Fisher's randomization test for experimental data (FRTED) is certainly entertaining. Although much of the paper is devoted to the thesis that Fisher changed his views on FRTED, apparently the primary point of the paper is to argue that FRTED is "not logically viable." Admittedly, FRTED is not the ultimate statistical weapon, even in randomized experiments, but calling it illogical is rather bizarre. Basu criticizes FRTED through two primary arguments. His first line of criticism follows from his attack on a nonparametric test labeled as "Fisher's randomization test." Basu's second line of criticism of FRTED takes the form of a discussion between a statistician and a scientist. The author sees nothing illogical about the FRTED, it is relevant for those rare situations when a purely confirmatory test of a priori sharp hypothesis is to be made using a priori defined statistic having an associated priori definition of extremeness. FRTED cannot adequately handle the full variety of real data problems that practicing statisticians face when drawing causal inferences, and for this reason it might be illogical to try to rely solely on it in practice.

The article presents the author's comments on researcher D. Basu's paper related to randomization analysis of experimental data. Basu has provided researchers with an interesting and provocative critique of significance tests related to randomized experiments. It does seem to be true that there is not a unified mathematical theory of significance tests developed by researcher R.A. Fisher. Nevertheless, it is important to point out a fallacy in Basu's criticism of nonunique significance level. After confessing to a "ruthless cross-examination" of the wrong topic, the non-Fisherian nonparametric tests, Basu suggests that Fisher's silence in 1956 may be used to condemn the randomization test. The empirical evidence confronting Fisher certainly suggested the necessity of randomization in most field experiments, if the standard methods of analysis were to be used. The final substantial issue of Basu's paper is that of the ancillarity of the design outcome. Technically Basu is quite correct, if the randomization has validated a parametric model, the design outcome is then ancillary by design. It would, however, be as well not to forget the purpose of an ancillary statistic, since other definitions.

This paper compares general and linear regression for the biextremal and Gumbel bivariate extreme models. The technique is based on the values of the correlation ratios and coefficients. The computations show that linear regression is a good approximation to the general one in both cases. [ABSTRACT FROM AUTHOR]

The minimum norm quadratic unbiased estimator type (MINQUE type) of estimates considered in this article are obtained by requiring identical values for the ratios of the a priori variances to the a priori error variance and letting this common value tend to infinity. The resulting estimates are invariant quadratic unbiased estimators with certain parametric and nonparametric optimality properties: assuming normally distributed random effects the efficiency of the proposed estimates to the minimum variance quadratic unbiased estimates (MIVQUE's) approaches unity when the true variance ratios are identical and tend to infinity. Assuming nonnormal effect distributions in the model with two variance components, the estimates are asymptotically efficient: in a sequence of designs where the number of classes and the number of observations on each class approach infinity, it is shown that the asymptotic variances of the estimates are equivalent to the theoretical minimum variances for invariant quadratic unbiased estimators. The result is interesting and useful since the usual analysis of variance (ANOVA) estimate of between-classes variance has strictly larger asymptotic variance for the unbalanced one-way model. Commonly considered estimates result from this procedure; the usual residual mean square (assuming that all non-error effects are fixed) is the resulting estimate of the error variance. In the one-way model the resulting estimates coincide with estimates considered by Thomas and Hultquist (1978), Burdick and Graybill (1984), Ahrens, Kleffe, and Tenzler (1981), and Kaplan (1982). In particular, the convergence of the MINQUE estimates was proved in the latter two papers in the context of the unbalanced one-way model. The procedure yields computationally convenient estimates in the general mixed ANOVA model. Computing formulas are... [ABSTRACT FROM AUTHOR]

In a graphical procedure for comparing k treatment means in a one-way ANOVA, one displays uncertainty intervals around the sample means and judges any pair to be significantly different if and only if their uncertainty intervals do not overlap. A graphical procedure is a Multiple Comparison Procedure (MCP) if and only if it controls the experimentwise error rate. In this paper we consider some new graphical MCP's for the unbalanced one-way ANOVA design. These procedures are based on different approximations to the Tukey-Kramer (TK) procedure (e.g., Kramer 1956). As such, they constitute alternatives to Gabriel (1978) (and its modification by Andrews, Snee, and Sarner 1980), which is based on approximating a less efficient MCP (the GT2 of Hochberg 1974). Two of the four procedures considered here are based on best and simple upper bounds to all the confidence-interval lengths of the TK method and hence must be conservative. The other two procedures are based on approximations (here too we have the best vs. the simple procedure), but simulations were used to find that their true experimentwise error rates are less than the nominal ones; that is, these procedures are still on the conservative side. The choice of a particular procedure will depend then on the relative importance of simplicity, efficiency, and the security of having a controlled experimentwise error rate. [ABSTRACT FROM AUTHOR]

White and Hultquist (1965) developed a method of combining finite fields mapped into a finite commutative ring to provide confounding plans for mixed factorial experiments where the numbers of the levels of factors have to be prime or the power of a prime. This paper extends their procedure to cover mixed factorials where the numbers of levels of factors need to be relatively prime, and not necessarily all prime, thus covering a wider range of mixed factorial experiments amenable to the traditional way of analysis of variance suggested by White and Hultquist. [ABSTRACT FROM AUTHOR]

This paper presents a piecewise regression method for continuous models containing max or min operators, or both. This method does not require knowledge of the zone in which a shift in regimes occurs. Moreover, it allows the application of analytical derivatives to maximize the likelihood function, which greatly simplifies the estimation of the model. The method proposed exhibits fast convergence and can be used for an arbitrary number of regimes and variables. [ABSTRACT FROM AUTHOR]

Let (X[sub I], Y[sub I])(I = 1, 2,..., n) be independent identically distributed as (X, Y). Then the rth ordered X variate is denoted by X[sub r:n] and the associated Y variate, the concomitant of the rth order statistic, by Y[r:.]. This paper considers statistics of the form n[sup -1] Sigma[sup n, sub I = 1] J(I/(n + 1)) Y[sub [I:n]] and more generally of the form n[sup -1] Sigma[sup n, sub I = 1] J(I/(n + 1))H(X[sub I:n], Y[sub [I:n]), where J is a bounded smooth function and may depend on n. Under certain regularity conditions, the asymptotic normality of these statistics is established. These statistics are used to construct consistent estimators of various conditional quantities, for example E(Y|X = x), P(Y is an element of A|X = x) and var(Y|X = x). [ABSTRACT FROM AUTHOR]

This paper reports results from an empirical study of the ratio estimator for a finite population total. From each of six real populations, 1,000 simple random samples, 1,000 restricted random samples, and three nonrandom samples of size 32 are drawn. Performance of the ratio estimator and of five estimators of its variance is compared with theoretical results generated using (a) prediction (superpopulation) models and (b) probability sampling distributions. The results, presented graphically, show that theory based on prediction models can reveal relationships that are essential in making inferences, but that are concealed in probability sampling analyses. [ABSTRACT FROM AUTHOR]

Loglinear models are classified as direct or indirect depending on whether the maximum likelihood estimates of cell values are explicit functions of the sufficient statistics or not. For saturated (hence, direct) models, Goodman (1970) and Bishop, Fienberg, and Holland (1975) used the delta method to calculate the asymptotic variances of various u terms in the loglinear models. In the present paper, this approach has been generalized to direct unsaturated hierarchical loglinear models. General rules for determining closed form expressions for asymptotic variances in such situations are obtained; bounds for the asymptotic variances of a terms in indirect models are considered; and these rules are compared with other methods of producing asymptotic variances. [ABSTRACT FROM AUTHOR]

Most of the literature on survey sampling deals with a single stratifying variable. In this paper an attempt is made to study the effect of using two stratifying variables. We present an approximation to the variance of the study variable under the assumption of a linear regression on the two stratifying variables. This approximation depends only on the number of strata, the simultaneous density of the stratifying variables, and the correlations between the study variable and each of the stratifying variables. The results indicate that in many practical situations the gain from using two stratifying variables over one is nontrivial. [ABSTRACT FROM AUTHOR]

This paper suggests that user accessibility as well as speed and accuracy should be used as a criterion to evaluate algorithms for generating random variables. FORTRAN implementations of several of the best current normal algorithms are studied. Two of the algorithms were also implemented in assembler for making comparisons with ether studies and for assessing the cost of accessibility. A new mixing algorithm which combines features of the convenient and fast methods of Marsaglia et al. is introduced. [ABSTRACT FROM AUTHOR]

This paper deals with income distribution of capital gains and their effect on measures of income inequality for 1962. Aggregate accrued gains are allocated to various income classes by using microdata from the Survey of Financial Characteristics of Consumers. The results show that accrued gains are distributed more unevenly than money income. The Gini coefficient drops from 0.41 to 0.35 when accrued losses for 1962 are subtracted from income, but it increases to 0.43 when average gains for 1960-64 are added to income. Variance of natural logarithms, the other measure of inequality used, also shows a similar pattern. [ABSTRACT FROM AUTHOR]

Inferences concerning the variance ratios of a random linear model are important in genetics and other areas of scientific investigation. For example, in genetics, the heritability parameters are functions of the variance ratios of a random model. This paper derives simultaneous confidence intervals for all or certain subsets of the parameters of a balanced incomplete random model. This procedure is based on the mean squares of the analysis of variance, and the mean squares are linear functions of a set of minimal sufficient statistics for estimating the parameters of the model. The procedure is demonstrated for a layout of six treatments, ten blocks, and five observations per block. [ABSTRACT FROM AUTHOR]

This paper develops a method for handling a nonorthogonal analysis of covariance in an iterative manner using balanced analysis of variance residual and expectation operators. In essence, it extends previous work of the author for the nonorthoganal AOV problem to the nonorthogonal AOC problem. The iterative AOC method has the property of guaranteed convergence. In addition, under certain (convergent) conditions, successive approximations to the residual sum of squares for the AOC model are shown to be monotonically decreasing. This property is used to minimize iteration in hypothesis testing. [ABSTRACT FROM AUTHOR]

Exact small sample behavior in two-way contingency tables is investigated for Pearson's chi-square statistic (X²), Light and Margolin's C statistic and its related R² measure of association, Kuliback's minimum discrimination information statistic (2&lcirc;), and Goodman and Kruskal's Lambda. R² is shown to be identical to Goodman and Kruskal's tb, leading to a test for independence based on 4. In small samples from a product of multinomials model, the null distribution of C is better approximated by a x² distribution than is the null distribution of X²; both are considerably better approximated by a x² distribution than is the null distribution of 2&lcirc;. It is proved for tables with two columns and any number of rows that if the column totals are equal, then X² ≤ 2&lcirc;; thus, X² is more conservative than 2&lcirc;. Hence, use of 2&lcirc; should be avoided in testing independence in tables with small samples. [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]

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. [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]

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]

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.

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]

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

The 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 gives numerical comparisons of the efficiency of Ordinary Least Squares (OLS) and Grouping Estimators in simple linear regression. The disturbances are assumed to have unequal variances, and an assumption is made about the form of this heteroscedasticity. It is shown that for some types of heteroscedasticity a Grouping Estimator can be more efficient than Ordinary Least Squares. [ABSTRACT FROM AUTHOR]

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

This paper deals with certain aspects of variance component estimation for the unbalanced one-way random classification where the number (N[sub I]) of observations in the ith class is treated as a random variable not necessarily independent of the class effect (A[sub iota]). It is assumed that in general P(N[sub I] = 0) > 0. The conditional expectations (given the number of observations in each class) of all estimators of the between variance component (sigma[sup 2, sub alpha]) belonging to a certain class of estimators are derived. A general expression is found for the expected value of that estimator of sigma[sup 2, sub alpha] yielded by analysis of variance of class means. The limit of this expression (as the number of classes arrow right Infinity) is given; and it is shown that, if the bivariate distribution function of A[sub I], N[sub I] belongs to a certain class of distribution functions, then this limit is less than sigma[sup 2, sub a]. Numerical approximations to the expected values of two estimators of sigma[sup 2, sub a] are presented for one subclass of such distribution functions. [ABSTRACT FROM AUTHOR]

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

In many linear regression problems, the values of the independent variable or variables may be subject to certain constraints. For example, the independent variables may necessarily be positive; as another example, the variables may not only all be positive but are powers of a single variable (e.g., polynomial regression on time). Previous writers considering the problem of obtaining confidence bands on a regression function for all values of the independent variable have not utilized such constraints; the usual basis for such bands has been the multiple comparison procedure of Scheffe which places no constraints at all upon the independent variables. Any procedure utilizing constraints will necessarily yield a uniform improvement over the method of Scheffe (assuming both methods are applicable) in the sense of yielding narrower bands for a given confidence probability. In the present paper a nontrivial lower bound is obtained for the confidence probability associated with a multiple comparison procedure appropriate to the case where it can be assumed that each independent variable must be of specified sign; this includes, as a subclass, polynomial regression on a non-negative independent variable. This result gives a basis for a multiple comparison procedure less conservative than that of Scheffe when both are applicable. Implementation of the procedure requires the percentage points of a heretofore untabulated distribution. Tables of percentage points of this distribution appropriate to linear combinations of two, three, or four parameters are presented. [ABSTRACT FROM AUTHOR]

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

This paper investigates the properties of the minimum variance unbiased (M.V.U) and maximum likelihood (M.L.) estimators of the reliability functions of systems composed of two subsystems connected in series. The study falls into two parts, one for the Poisson case and one for the exponential case. In each of these cases the situations are distinguished between, where the two subsystems are identical and situations subsystems are different. In the Poisson case under minimum variance unbiased estimators a system A is considered which is composed of two subsystems connected in series. Failure time points of the subsystem follow a Poisson process with intensity. An experiment is performed on n independent replicates of each of the considered subsystems over a period of length. Under the exponential case, a system A is considered, same as Poisson case which consists of two subsystems connected in series. Failure time points of the two subsystems follow a Poisson process. Independent observations are available on the interfailure time lengths; namely, the life-lengths of the subsystems.