Showing papers on "U-statistic published in 1981"
TL;DR: The traditional method for estimating or predicting linear combinations of the fixed effects and realized values of the random effects in mixed linear models is first to estimate the variance components and then to proceed as if the estimated values of variance components were the true values as mentioned in this paper.
Abstract: The traditional method for estimating or predicting linear combinations of the fixed effects and realized values of the random effects in mixed linear models is first to estimate the variance components and then to proceed as if the estimated values of the variance components were the true values. This two-stage procedure gives unbiased estimators or predictors of the linear combinations provided the data vector is symmetrically distributed about its expected value and provided the variance component estimators are translation-invariant and are even functions of the data vector. The standard procedures for estimating the variance components yield even, translation-invariant estimators.
168 citations
TL;DR: In this article, the first two moments of the maximum likelihood estimator and minimum variance unbiased estimator of P(X > Y) were obtained, when μ is known, say 1.
Abstract: Let X and Y be independent exponentially distributed random variables having parameters λ and μ respectively. Sharp boundsfor the first two moments of the maximum likelihood estimator and minimum variance unbiased estimator of P(X > Y) are obtained, when μ is known, say 1. When μ is unknown, sharp bounds for the first two moments of the maximum likelihood estimator of p(X > Y) are obtained and a lower bound for the variance of the minimum variance unbiased estimator is also obtained.
62 citations
TL;DR: In this article, the existence and asymptotic behavior of the uniformly minimum variance unbiased estimator of the unconditional probability is investigated under the condition that the species are uniformly distributed.
Abstract: We search a population by selecting one member at a time with replacement and observing the species of each selected member. We are interested in predicting the conditional probability of discovering a new species in the next selection after $n$ observations. The existence and asymptotic behavior of the uniformly minimum variance unbiased estimator of the unconditional probability is investigated under the condition that the species are uniformly distributed. We also compare the performance of the estimator as a predictor of the conditional probability with that of a linear unbiased predictor.
43 citations
TL;DR: In this article, the uniformly mimimum variance unbiased estimators and their variances from independent samples of lognormal distributions are concisely expressed using the hypergeometric functions, and their variance distributions are expressed in terms of hypergeometrical functions.
Abstract: The uniformly mimimum variance unbiased estimators and their variances from independent samples of lognormal distributions are concisely expressed using the hypergeometric functions
21 citations
TL;DR: In this paper, a general linear model for a column vector $y$ of data having $E(y) = X \alpha$ and $\operatorname{Var}(y), = \sigma^2H', where $\alpha$ is a vector of unknown parameters and $X and $H$ are given matrices that are possibly deficient in rank.
Abstract: Consider a general linear model for a column vector $y$ of data having $E(y) = X \alpha$ and $\operatorname{Var}(y) = \sigma^2H$, where $\alpha$ is a vector of unknown parameters and $X$ and $H$ are given matrices that are possibly deficient in rank. Let $b = Ty$, where $T$ is any matrix of maximum rank such that $TH = \phi$. The estimation of a linear function of $\alpha$ by functions of the form $c + a'y$, where $c$ and $a$ are permitted to depend on $b$, is investigated. Allowing $c$ and $a$ to depend on $b$ expands the class of unbiased estimators in a nontrivial way; however, it does not add to the class of linear functions of $\alpha$ that are estimable. Any minimum-variance unbiased estimator is identically [for $y$ in the column space of $(X, H)$] equal to the estimator that has minimum variance among strictly linear unbiased estimators.
15 citations
TL;DR: In this paper, Ghosh and DasGupta derived the rate of convergence of the coverage probability for estimating the mean of a U-statistic through Sproule's(1969, 1974) procedure.
Abstract: Csenki(1980a) utilized the rate of convergence results of Landers and Rogge(1976b) for suitably normalized random means to obtain the rate of convergence of the coverage probability for Chow-Robbins'(1965) fixed-width confidence interval procedure. In this paper, we utilize the rate of convergence results of Ghosh and DasGupta(1980) for suitably normalized random U-statistics to derive the rate of convergence of the coverage probability for estimating the mean of a U-statistic through Sproule's(1969, 1974) procedure. We show that Csenki's(1980a) convergence rate can be achieved with a substantial economy on moment condition. We also propose a two-stage procedure for this pro-1970 AMS Classification: 60F05, 62L10, 62G10
15 citations
TL;DR: For the simultaneous estimation of p-means of Poisson variables, this paper presented estimators that have significantly smaller mean square errors in the region of a pre-specified set of values, and are conservative, in the sense that their mean errors are never worse than that of the minimum variance unbiased estimator, when p > 3.
Abstract: Estimators are presented for the simultaneous estimation of p means of Poisson variables. These have significantly smaller mean square errors of estimation in the region of a pre-specified set of values, and are conservative, in the sense that their mean errors are never worse than that of the minimum variance unbiased estimator, when p > 3.
14 citations
TL;DR: In this paper, simple approximation formulas for the mean squared errors of the minimum variance unbiased and the maximum likelihood estimators of reliability of a system with m independent components are provided. But the performance of the proposed approximation formulas is investigated.
Abstract: Suppose S is a system with m independent components such that the system functions if and only if at least k components operate. Assume that the component lifetimes are identically exponentially distributed. This work provides simple approximation formulas for the mean squared errors of the minimum variance unbiased and the maximum likelihood estimators of reliability of this system. The performance of the proposed approximation formulas is investigated.
12 citations
TL;DR: In this article, the authors considered a general class of almost unbiased ratio estimators of which Murthy and Nanjamma's is a particular case and derived an optimum in this class.
Abstract: Murthy and Nanjamma [4] studied the problem of construction of almost unbiased ratio estimators for any sampling design using the technique of interpenetrating subsamples. Subsequently, Rao [7], [8] has given a general method of constructing unbiased ratio estimators by considering linear combinations of the two simple estimators based on the ratio of means and the mean of ratios. However, it is difficult to choose an optimum weight (Rao [9]) which minimizes the variance of the combined estimator since the weights are random in certain cases. In this note, we consider a different method of combining these estimators and obtain a general class of almost unbiased ratio estimators of which Murthy and Nanjamma's is a particular case and derive an optimum in this class. The case of simple random sampling where a similar class of almost unbiased ratio estimators can be developed is briefly discussed. The results are illustrated by means of simple numerical examples.
12 citations
12 citations
TL;DR: In this paper, a modification of the usual power series distribution is proposed and a statistical treatment of data generated by this probability model is indicated, which belongs to a multiparameter discrete exponential family and standard theory is then applied to obtain complete sufficient statistics of the parameters, sampling distribution of these statistics, maximum likelihood and minimum variance unbiased estimators, uniformly most powerful unbiased tests, and uniformly most accurate confidence intervals.
Abstract: A modification of the usual power series distribution is proposed and a statistical treatment of data generated by this probability model is indicated. It is shown that this modified power series distribution belongs to a multiparameter discrete exponential family and standard theory is then applied to obtain complete sufficient statistics of the parameters, sampling distribution of these statistics, maximum likelihood and minimum variance unbiased estimators, uniformly most powerful unbiased tests, uniformly most accurate confidence intervals and expected cover tolerance regions.
TL;DR: In this paper, necessary and sufficient conditions for the simple least squares estimator to coincide with the best linear unbiased predictor were developed for a general linear model and are generalizations of the condition given by Watson (1972).
TL;DR: In this article, a comparative study of product estimators proposed by Robson [1957] and Murthy [1964] is presented, and it is seen that the Robson's estimator gives a better performance.
Abstract: The paper presents a comparative study of product estimators proposed byRobson [1957] andMurthy [1964]. It is seen that the Robson's estimator gives a better performance.
01 Jan 1981
TL;DR: In this paper, necessary and sufficient conditions for non-negativity of variance estimators are reviewed and some new sufficient conditions are also given, where necessary conditions for variance estimator nonnegativity are discussed.
Abstract: Results on necessary and sufficient conditions for non-negativity of variance estimators are briefly reviewed. Some new sufficient conditions are also given.
TL;DR: In this paper, a three-way contingency table with factors A, B, and C is considered, where factors A and B are independent for each given level of factor C. Inference procedures concerning the marginal means of the levels of factor A are studied.
Abstract: Consider a three-way contingency table with factors A, B, and C. Assume factors A and B are independent for each given level of factor C. Inference procedures concerning the marginal means of the levels of factor A are studied. Maximum likelihood estimators, unbiased estimators, and uniformly minimum variance unbiased estimators are obtained. Exact variance formulas for the estimators are obtained. Large sample theory is developed and used for testing and confidence intervals. A variety of sampling models and generalizations are studied, including the case in which the number of levels of factor C is infinite. An example concerned with vaccine evaluation is given.
TL;DR: In this paper, distributional properties for a statistic T*, which has previously been reported to have power properties as a test of normality as attractive as those of the sample kurtosis or perhaps slightly more attractive.
Abstract: Distributional properties are given for a statistic T*, which has previously been reported to have power properties as a test of normality as attractive as those of the sample kurtosis or perhaps slightly more attractive. Asymptotic results, the mean and variance under normality, the range of variation, and approximation of critical values for testing normality are obtained
TL;DR: In this paper, a relationship in the treatment of the lower and upper truncations considered in Beg (1980) is pointed out and the minimum variance unbiased estimator of P = Pr{Y < X] for the (upper) truncated exponential distribution is obtained.
Abstract: In this note a relationship in the treatment of the lower and upper truncations considered in Beg (1980) is pointed out and the minimum variance unbiased estimator of P = Pr{Y
TL;DR: For spherical particles randomly dispersed in the space of a specimen, the estimators of the parameters of the space structure from the measurements obtained from extraction replicas are given in this article.
Abstract: For spherical particles randomly dispersed in the space of a specimen the estimators of the parameters of the space structure from the measurements obtained from extraction replicas are given. First an arbitrary form of the probability density function f(x) of the diameter X and then the generalized RAYLEIGH and lognormal distributions of X are considered. Unbiased estimators of the space parameters and of parameters of these distributions are found. The variances of these estimators are given and unbiased estimators of these variances are determined.
01 Nov 1981
TL;DR: In this article, generalized Blomqvist correlation, a generalization of the double median test, is formulated as a new U statistic with a lower variance and several open questions are answered, and some examples are given.
Abstract: : Generalized Blomqvist Correlation, a generalization of the double median test, is first formulated as a new U statistic with a lower variance Several open questions are answered, and some examples are given (Author)
01 Jan 1981
TL;DR: In this article, normal random variables with EX{in1}=EY{inj}=Θ, Θ ∈ R, Var X{ini} = σ{in 1}{su2}, Var Y{ini]=σ{in 2}su2], σ {in 1}> 0, σ ''in 2''> 0.
Abstract: Let X{in1}, X{in2},….,X{inn}, Y{in1}, Y{in2},…, Y{inm} be inaependent normal random variables with EX{in1}=EY{inj}=Θ, Θ ∈ R, Var X{ini} = σ{in1}{su2}, Var Y{inj}=σ{in2}{su2}, σ{in1}> 0, σ{in2}> 0, i=1, …, n, j=1, …, m, n > 1, m > 1.
TL;DR: In this article, a characterization of minimum variance unbiased estimators in the general linear model with restrictions on parameter space is presented, where the authors consider the case where the estimators are restricted to a fixed number of parameters.
Abstract: (1981). A characterization of minimum variance unbiased estimators in the general linear model with restrictions on parameter space. Series Statistics: Vol. 12, No. 4, pp. 465-477.