Showing papers on "Asymptotic distribution published in 1973"
TL;DR: In this paper, the authors consider density estimates of the usual type generated by a weight function and obtain limit theorems for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity.
Abstract: We consider density estimates of the usual type generated by a weight function Limt theorems are obtained for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity Using these results we study the behavior of tests of goodness-of-fit and confidence regions based on these statistics In particular, we obtain a procedure which uniformly improves the chi-square goodness-of-fit test when the number of observations and cells is large and yet remains insensitive to the estimation of nuisance parameters A new limit theorem for the maximum absolute value of a type of nonstationary Gaussian process is also proved
703 citations
TL;DR: A simple theorem on generating functions is proved which can be used to establish the asymptotic normality of an(k) as a function of k and local limit theorems are turned to in order to obtain asymPTotic formulas for an( k).
365 citations
TL;DR: In this article, it is shown that in order for the trimmed mean to be asymptotically normal, it is necessary and sufficient that the sample be trimmed at sample percentiles such that the corresponding population percentiles are uniquely defined.
Abstract: In this paper it is shown that in order for the trimmed mean to be asymptotically normal, it is necessary and sufficient that the sample be trimmed at sample percentiles such that the corresponding population percentiles are uniquely defined (The sufficiency of this condition is well known) In addition, the (non-normal) limiting distribution of the trimmed mean when this condition is not satisfied is derived, and it is shown that in some situations the use of the trimmed mean may lead to severely biased inferences Some possible remedies are briefly discussed, including the use of "smoothly" trimmed means
246 citations
96 citations
TL;DR: In this paper, an Edgeworth-type expansion for the distribution of a minimum contrast estimator, and expansions suitable for the computation of critical regions of prescribed error (type one) as well as confidence intervals of prescribed confidence coefficient are presented.
Abstract: This paper contains an Edgeworth-type expansion for the distribution of a minimum contrast estimator, and expansions suitable for the computation of critical regions of prescribed error (type one) as well as confidence intervals of prescribed confidence coefficient. Furthermore, it is shown that, for one-sided alternatives, the test based on the maximum likelihood estimator as well as the test based on the derivative of the log-likelihood function is uniformly most powerful up to a term of order $O(n^{-1})$. Finally, an estimator is proposed which is median unbiased up to an error of order $O(n^{-1})$ and which is--within the class of all estimators with this property--maximally concentrated about the true parameter up to a term of order $O(n^{-1})$. The results of this paper refer to real parameters and to families of probability measures which are "continuous" in some appropriate sense (which excludes the common discrete distributions).
81 citations
73 citations
TL;DR: The theory of rank tests has been extended to include purely discrete random variables under the null hypothesis of randomness (including the two-sample and $k$-sample problems) and under contiguous alternatives for the two methods of assigning scores known as the average scores method and the randomized ranks method as mentioned in this paper.
Abstract: The theory of rank tests has been developed primarily for continuous random variables. Recently the asymptotic theory of linear rank tests has been extended to include purely discrete random variables under the null hypothesis of randomness (including the two-sample and $k$-sample problems) and under contiguous alternatives, for the two methods of assigning scores known as the average scores method and the randomized ranks method. In this paper the theory of rank tests is developed with no assumptions concerning the continuous or discrete nature of the underlying distribution function. Conditional rank tests, given the vector of ties, are shown to be similar, and the locally most powerful conditional rank test is given. The asymptotic distribution of linear rank statistics is given under the null hypotheses of randomness and symmetry (which includes the one-sample problem), and under contiguous alternatives. Three methods of assigning scores, the average scores, midranks, and randomized ranks methods, are discussed and briefly compared.
70 citations
TL;DR: In this paper, confidence intervals based on the maximum likelihood estimators for the location and scale parameters of the Double Exponential distribution are given by determining the distribution of the pivotal quantities ( − θ)/ and /σ.
Abstract: Confidence intervals based on the maximum likelihood estimators are given for the location and scale parameters of the Double Exponential distribution. These intervals are obtained by determining the distribution of the pivotal quantities ( – θ)/ and /σ. Exact distributions are determined for n = 3 and n = .5, and approximate distributions are provided for larger n. The asymptotic distributions are also given and the accluacy of these approximations are investigated. The powers of the associated tests of hypotheses are given and tolerance limits for the population are also provided. Some possible applications are indicated.
70 citations
TL;DR: In this paper, the authors generalized the hypothesis that very long-tailed data come from a symmetric stable distribution and showed that convolutions of these mixtures can be very slow to converge to the limiting stable distribution or even to assume the asymptotic behavior predicted by theory for large numbers of convolutions.
Abstract: The hypothesis that very long-tailed data come from a symmetric stable distribution can be generalized, namely, that the observations are generated by a mixture of a normal distribution and another symmetric stable distribution. The difficulty of statistical distinction of the two cases is computed as a function of the various parameters involved. Also, computations show that convolutions of these mixtures can be very slow to converge to the limiting stable distribution or even to assume the asymptotic behavior predicted by theory for large numbers of convolutions.
66 citations
TL;DR: In this paper, the authors used implicit differentiation to find the partial derivatives of an arbitrary orthogonal rotation algorithm, and showed that the transformation matrix which produces the rotation is usually a function of the data.
Abstract: Beginning with the results of Girshick on the asymptotic distribution of principal component loadings and those of Lawley on the distribution of unrotated maximum likelihood factor loadings, the asymptotic distribution of the corresponding analytically rotated loadings is obtained. The principal difficulty is the fact that the transformation matrix which produces the rotation is usually itself a function of the data. The approach is to use implicit differentiation to find the partial derivatives of an arbitrary orthogonal rotation algorithm. Specific details are given for the orthomax algorithms and an example involving maximum likelihood estimation and varimax rotation is presented.
58 citations
18 Dec 1973
TL;DR: In this article, a window estimator for the principal part of the asymptotic variance of a Normal distribution to Studentize the mean is proposed. But this estimator does not consider the variance of the normal distribution itself.
Abstract: : Estimates of location based on rank statistics and score functions are receiving considerable attention in studies of 'robust' statistics. When using such estimates, one frequently would also like to have an estimate of the variance of a Normal distribution to Studentize the mean. The paper proposes a window estimator for the principal part of the asymptotic variance. (Modified author abstract)
TL;DR: In this article, the Reed-Muench and Dragstedt-Behrens estimators are shown to be estimators of the mean, not the median, tolerance and to be nearly equivalent to the Spearman-Karber estimator.
Abstract: SUMMARY Theoretical properties of the Spearman-Karber estimator of the mean tolerance in a quantal response bioassay are discussed, and its limiting distribution is studied under three different limiting processes. The Reed-Muench and Dragstedt-Behrens estimators are shown to be estimators of the mean, not the median, tolerance and to be nearly equivalent to the Spearman-Karber estimator. The limiting distributions of the Reed-Muench estimator are considered in detail.
TL;DR: In this paper, a procedure for estimating the zero class from a truncated Poisson sample is developed, and asymptotic normality of the estimator is proved so that a confidence interval for the missing zero class can be obtained.
Abstract: A procedure for estimating the zero class from a truncated Poisson sample is developed. Asymptotic normality of the estimator is proved so that a confidence interval for the missing zero class can be obtained. An example is given to illustrate the results obtained.
TL;DR: In this article, the convergence in probability of the stopping time of a procedure due to Robbins (1959) is shown to be normal, which is then converted to convergence in distribution by using a theorem of Wittenberg (1964).
Abstract: Two problems of sequential estimation, viz. the estimation of the mean of a normal distribution with unknown variance and the estimation of a binomial proportion are studied as the cost per observation tends to 0. For the first problem the asymptotic distribution of the stopping time of a procedure due to Robbins (1959) is shown to be normal. For the second problem the stopping time of a modification of Wald's (1951) procedure is asymptotically normal when the parameter is different from $\frac{1}{2}$. When the parameter is $\frac{1}{2}$, this stopping time does not enjoy asymptotic normality. The method employed is to first prove the convergence in probability of the stopping time which is then converted to convergence in distribution by using a theorem of Wittenberg (1964). This method also yields a new proof of a theorem of Siegmund (1968).
TL;DR: In this paper, the authors derived the asymptotic normality of linear combinations of functions of order statistics of non-i.i.d. rv's in the case of bounded scores.
Abstract: Any triangular array of row independent rv's having continuous df's can be transformed naturally so that the empirical and quantile processes of the resulting rv's are relatively compact. Moreover, convergence (to a necessarily normal process) takes place if and only if a simple covariance function converges pointwise. Using these results we derive the asymptotic normality of linear combinations of functions of order statistics of non-i.i.d. rv's in the case of bounded scores.
TL;DR: In this paper, it was shown that the asymptotic probability of the classical Kolmogorov-Smirnov statistic exceeding any positive real number provides an upper bound for the corresponding probability when the underlying random variables are not necessarily identically distributed.
Abstract: For a sequence of random variables forming an $m$-dependent stochastic process (not necessarily stationary), asymptotic distribution and other convergence properties of the extremum of certain functions of the empirical distribution are studied. In this context, it is shown that the asymptotic probability of the classical Kolmogorov-Smirnov statistic exceeding any positive real number provides an upper bound for the corresponding probability when the underlying random variables are not necessarily identically distributed. The theory is specifically applied to the study of the limiting distribution, strong convergence and convergence of the first moment of the strength of a bundle of parallel filaments (which is shown to be the extremum of a function of the empirical distribution).
TL;DR: In this article, the minimax estimators of a cumulative distribution function $F$ are obtained for four types of loss functions, and it is shown that the sample distribution function is minimax under one of these loss functions.
Abstract: In this paper the minimax estimators of a cumulative distribution function $F$ is obtained for four types of loss functions. The result is quite general in that no restrictions are imposed on the unknown $F$. Moreover, the estimates do not depend upon the weight function used in the definition of the loss functions. It is also shown that the sample distribution function is minimax under one of these types of loss functions.
TL;DR: In this paper, a Poisson limit is established for the probability that $k$ events occur and an asymptotic distribution for the number of upcrossings of a high level by certain stochastic processes is considered as an application.
Abstract: Certain mixing sequences of dependent `rare' events are considered and a Poisson limit is established for the probability that $k$ events occur. An asymptotic distribution for the number of upcrossings of a high level by certain stochastic processes is considered as an application.
TL;DR: In this article, terms and theorems of the theory of asymptotic distribution are used investigating numerical methods for the solution of certain integrodifferential equations, specially applied on the study of collisionless plasmas, which are described by the so called Vlasov-equation.
Abstract: In this paper terms and theorems of the theory of asymptotic distribution are used investigating numerical methods for the solution of certain integrodifferential equations. These simulation methods are specially applied on the study of collisionless plasmas, which are described by the so called Vlasov-equation. Interpretating the solution of such equations as the asymptotic distribution of suitably constructed point sequences, one gets a mathematical frame for these methods and succeeds in proving convergence.
TL;DR: In this paper, the authors studied the asymptotic behavior of eigenvalues of a pseudodifferential operator with certain conditions imposed on the symbol of the operator, with remainder.
Abstract: This paper is devoted to the study of asymptotic behavior of eigenvalues of a pseudodifferential operator in . With certain conditions imposed on the symbol of the operator, the asymptotics of the eigenvalues, with remainder, is obtained.Bibliography: 10 titles.
TL;DR: In this article, the scale parameter of an exponential distribution is reparameterized as a function of the stress according to the Arrhenius Re-action Rate Model and the location parameter is re-parameterised as a linear function of stress.
Abstract: The scale parameter of an exponential distribution is reparameterized as a function of the stress according to the Arrhenius Re-action Rate Model. The location parameter is reparameterized as a linear function of the stress. Maximum likelihood estimators of the parameters of the Arrhenius model and the weighted least squares estimators of the linear model are obtained using data from censored sample accelerated life tests. The asymptotic normality of the maximum likelihood estimators is ascertained by examining the shapes of their maximum relative likelihood functions. The asymptotic normality of the least squares estimators is ascertained by establishing certain criteria for convergence. An unbiased estimator for the mean life of the device at use condition stress μ u is obtained. Confidence limits for μ u , are difficult to obtain, and alternatively a plausibility interval for μ u is obtained.
TL;DR: In this article, asymptotic expansions of the non-null distributions of the likelihood ratio, Hotelling's and Pillai's statistics for multivariate linear hypothesis are given in terms of normal distribution function and its derivatives, assuming the matrix of noncentrality parameters is of the same order as the sample size.
Abstract: New asymptotic expansions of the non-null distributions of the likelihood ratio, Hotelling's and Pillai's statistics for multivariate linear hypothesis are given in terms of normal distribution function and its derivatives, assuming the matrix of noncentrality parameters is of the same order as the sample size.
TL;DR: In this article, a class of simple linear estimators of the standard deviation of the normal distribution is considered, based on the differences between the sums of the observations in the tails of the ordered sample.
Abstract: A class of simple linear estimators of the standard deviation of the normal distribution is considered. They are based on the differences between the sums of the observations in the tails of the ordered sample. For normal distributions they are asymptotically unbiased and possess good efficiency—even for small samples. Also for a wide range of non-normal distributions they are reasonably robust asymptotically.
TL;DR: In this article, empirical distributions of Z = (X-bar - ()/(N-1/2)) were obtained for several sample sizes to investigate the approach of the sampling distribution of Z to a normal distribution.
Abstract: From each of two dozen populations varying widely in shape and skewness, empirical distributions of Z = (X-bar - ()/(N-1/2 were obtained for several sample sizes to investigate the approach of the sampling distribution of Z to a normal distribution. For..
TL;DR: In this article, it was shown that under certain conditions, the maximum likelihood and the minimum variance unbiased estimators of a positive integral power of the natural parameter in an exponential family have the same asymptotic distribution.
Abstract: It is shown that under certain conditions, the maximum likelihood and the minimum variance unbiased estimators of a positive integral power of the natural parameter in an exponential family have the same asymptotic distribution.
01 Jan 1973
TL;DR: In this article, it was proved that there exist systems of generalized primes in which the asymptotic distribution of integers is N(x)= Ax+O(x log7 x) with A>0 and y C [0, 1] and in which Chebyshev inequalities lim inf "(x)log x 0 lim su 7r(x) log x < 00 X-cO x *CO x
Abstract: It is proved that there exist systems of generalized primes in which the asymptotic distribution of integers is N(x)= Ax+O(x log7 x) with A>0 and y C [0, 1) and in which the Chebyshev inequalities lim inf "(x)log x 0 lim su 7r(x)log x< 00 X-cO x *CO x