Showing papers on "U-statistic published in 1989"
TL;DR: In this paper, the density-weighted average derivative of a general regression function is estimated using nonparametric kernel estimators of the density of the regressors, based on sample analogues of the product moment representation of the average derivative.
Abstract: This paper gives a solution to the problem of estimating coefficients of index models, through the estimation of the density-weighted average derivative of a general regression function. The estimators, based on sample analogues of the product moment representation of the average derivative, are constructed using nonparametric kernel estimators of the density of the regressors. Asymptotic normality is established using extensions of classical U-statistic theorems, and asymptotic bias is reduced through use of a higher-order kernel
999 citations
TL;DR: The delete-1 jackknife is known to give inconsistent variance estimators for nonsmooth estimators such as the sample quantiles as mentioned in this paper, which can be rectified by using a more general jackknife with $d$, the number of observations deleted, depending on a smoothness measure of the point estimator.
Abstract: The delete-1 jackknife is known to give inconsistent variance estimators for nonsmooth estimators such as the sample quantiles. This well-known deficiency can be rectified by using a more general jackknife with $d$, the number of observations deleted, depending on a smoothness measure of the point estimator. Our general theory explains why jackknife works or fails. It also shows that (i) for "sufficiently smooth" estimators, the jackknife variance estimators with bounded $d$ are consistent and asymptotically unbiased and (ii) for "nonsmooth" estimators, $d$ has to go to infinity at a rate explicitly determined by a smoothness measure to ensure consistency and asymptotic unbiasedness. Improved results are obtained for several classes of estimators. In particular, for the sample $p$-quantiles, the jackknife variance estimators with $d$ satisfying $n^{1/2}/d \rightarrow 0$ and $n - d \rightarrow \infty$ are consistent and asymptotically unbiased.
295 citations
TL;DR: In this article, the iterative least-squares procedure for estimating the parameters in a general multilevel random coefficients linear model can be modified to produce unbiased estimates of the random parameters.
Abstract: SUMMARY It is shown that the iterative least-squares procedure for estimating the parameters in a general multilevel random coefficients linear model can be modified to produce unbiased estimates of the random parameters. In the multivariate normal case these are equivalent to restricted maximum likelihood estimates.
156 citations
TL;DR: In this article, the authors compared the accuracy of the median unbiased estimator with that of the maximum likelihood estimator for a logistic regression model with two binary covariates, and showed that the former estimator is uniformly more accurate than the latter for small to moderately large sample sizes and a broad range of parameter values.
Abstract: This article compares the accuracy of the median unbiased estimator with that of the maximum likelihood estimator for a logistic regression model with two binary covariates. The former estimator is shown to be uniformly more accurate than the latter for small to moderately large sample sizes and a broad range of parameter values. In view of the recently developed efficient algorithms for generating exact distributions of sufficient statistics in binary-data problems, these results call for a serious consideration of median unbiased estimation as an alternative to maximum likelihood estimation, especially when the sample size is not large, or when the data structure is sparse.
97 citations
TL;DR: In this article, conditionally unbiased estimators of the selected mean given the ordering of the set of sample means based on the first stage sample are obtained. But, for several distributions such as the normal, with unknown mean, and binomial, no conditionally-unbiased estimators exist based on a one stage sample.
88 citations
TL;DR: In this article, the authors show that a similar result holds for the class of all stable unbiased state estimators, and this result is believed to have major implications for the design of robust estimators to deal with non-standard noise sources, modelling errors, etc.
77 citations
TL;DR: In this article, the authors studied the convergence rate of the central limit theorem for functions of independent random variables and applied it to the problem of estimating the probability of a function of independent variables.
Abstract: The rate of convergence in the central limit theorem for functions of independent random variables is studied in a unifying approach. The basic result sharpens and extends a theorem of van Zwet. Applications to $U$-, $L$- and $R$-statistics are also given, improving or extending the results of Helmers and van Zwet, Helmers and Huskova, Does and van Es and Helmers.
69 citations
TL;DR: In this article, the Breslow-Crowley bound for the difference between the empirical cumulative hazard and the Kaplan-Meier cumulative hazard estimators of the survival function was used to derive a Berry-Esseen bound for U-statistics.
Abstract: The Berry-Esseen bound for U-statistics, established by Helmers and Van Zwet (1982), is combined with the Breslow-Crowley (1974) bounds for the difference between the empirical cumulative hazard and the Kaplan-Meier cumulative hazard estimators of the survival function to derive a Berry-Esseen bound for the Kaplan-Meier estimator. We show that there exists an absolute quantity K such that the absolute difference between the standardized distribution function of Kaplan-Meier estimator at a fixed time point t and the standard normal cumulative distribution function is bounded above by where S(·) is the survival function and σ1is defined in Lemma 1.
43 citations
TL;DR: In this paper, the authors give conditions for a law of the iterated logarithm for U-statistics which have a kernel with an estimator substituted into it, and also show convergence results for adaptive M-estimators and for cross-validation assessment statistics.
Abstract: Substituting an estimator in a statistic will often affect its limiting distribution. Sukhatme (1958), Randles (1982), and Pierce (1982) all consider the changes, if any, in the statistic's limiting normal distribution. This paper gives conditions for a law of the iterated logarithm for U-statistics which have a kernel with an estimator substituted into it. It also gives conditions for both strong and weak convergence. Applications of the theory are illustrated by constructing a sequential test for scale differences with power one. The theory also produces convergence results for adaptive M-estimators and for cross-validation assessment statistics. In addition, it is shown how to extend LIL results to a broad class of statistics with estimators substituted into them by use of the differential. In particular, a law of the iterated logarithm is described for adaptive L-statistics and is illustrated by an example of de Wet and van Wyk (1979).
35 citations
TL;DR: In this article, the central limit theorem for sums of dependent random variables indexed by a graph is applied to obtain asymptotic normality for the number of local maxima of a random function on certain graphs and for edges having the same color at both endpoints in randomly colored graphs.
Abstract: Petrovskaya and Leontovich (1982) proved a central limit theorem for sums of dependent random variables indexed by a graph. We apply this theorem to obtain asymptotic normality for the number of local maxima of a random function on certain graphs and for the number of edges having the same color at both endpoints in randomly colored graphs. We briefly motivate these problems, and conclude with a simple proof of the asymptotic normality of certain U-statistics. CENTRAL LIMIT THEOREM; DEPENDENT VARIABLES; MAXIMA OF RANDOM FUNCTIONS; RANDOM COLORINGS; U-STATISTICS; NEURAL NETWORKS
34 citations
TL;DR: In this article, the weak convergences of U- and V-statistics were established by Yoshihara (1976, Z. Warsch. Verw. Gebiete 35 237-252) for stationary absolutely regular processes.
TL;DR: In this paper, the authors consider a particular class of point estimators obtained by inverting suitable prediction functions and show that the estimator obtained from a best prediction function is optimal in a sense defined by Godambe and Durbin (1960) in the context of unbiased estimating equations.
Abstract: Changes in calibration curves from one time to the next, caused by drift, often require measuring devices to be recalibrated at frequent intervals. In such situations the usual practice is to estimate the unknown values of test samples using only data from the corresponding calibration period. Under a random coefficient regression model for the different calibration curves, however, it can be shown that it is more efficient to combine the data from all calibration periods to estimate the unknowns. We consider a particular class of point estimators obtained by inverting suitable prediction functions and show that the estimator obtained from a best prediction function is optimal in a sense defined by Godambe (1960) and Durbin (1960) in the context of unbiased estimating equations. We also compare the smallsample performance of this estimator with the usual estimator using the Pitman closeness criterion.
TL;DR: In this paper, the problem of minimum variance unbiased estimation of a U-estimateable function of two unknown truncation parameters based on independent random samples from two one-truncation parameter families was considered.
Abstract: Summary
We consider the problem of minimum variance unbiased estimation of a U-estimable function of two unknown truncation parameters based on independent random samples from two one-truncation parameter families. In particular, we obtain the UMVU estimator of the probability that Y > X.
TL;DR: In this article, the asymptotic behavior of the U -statistic and the one-sample rank order statistic for nonstationary absolutely regular processes was studied and some applications of these results for Markov processes as well as ARMA processes.
01 Dec 1989
TL;DR: In this paper, a linear errors-in-variables model (EV Model) with known or approximately known measurement error variance is used for constructing new approximately unbiased estimators for the slope coefficient in the EV model.
Abstract: This article presents a first direct application of finite sample distribution theory. The relevance of analytical finite sample research is exemplified in the framework of a simple linear errors-in-variables model (EV Model) with known or approximately known measurement error variance. Analytical results derived byRichardson/Wu (1970) are applied for constructing new approximately unbiased estimators for the slope coefficient in the EV model. The new estimators are compared with the biased least squares estimator and with asymptotic theory based corrected least squares estimators.
TL;DR: In this article, the uniformly minimum variance unbiased (best) estimator and a strongly consistent, asymptotically normal, unbiased estimator of each of Gini index and Yntema-Pietra index of lognormal distribution are derived.
Abstract: Here we derive the uniformly minimum variance unbiased (best) estimator and a strongly consistent, asymptotically normal, unbiased estimator of each of Gini index and Yntema-Pietra index of lognormal distribution . These estimators are in terms of generalized hypergeometric functions 1F2. Further, the variances of these estimators and the best estimators of variances of best estimators are found out. They are in terms of Kempe de Feriet's hypergeometric functions.
TL;DR: It is concluded that the indices based on information measures of degree [beta] = 2 allow us to construct unbiased estimates of their population values.
TL;DR: In this article, the problem of uniform minimum variance unbiased (UMVU) estimation of a U-estimateable function based on two independent random samples which are Type II censored and come from two one-truncation parameter families was considered.
Abstract: We consider the problem of uniform minimum variance unbiased (UMVU) estimation of a U-estimable function based on two independent random samples which are Type II censored and come from two one-truncation parameter families. In particular, we obtain the UMVU estimator of the probability that Y > X, and confidence interval for
TL;DR: In this paper, it was shown that a natural continuation of the example leads to an estimating equation for α that produces the maximum likelihood estimator, which is closely related to Fisher's motivation of maximum likelihood.
Abstract: C. R. Rao (1973, problem 5.11) constructed an example where a minimum variance unbiased estimator exists for the parameter function (1 – α)2 but no uniformly minimum variance unbiased estimator exists for α itself. The example was further discussed by Romano and Siegel (1986, example 9.6) who exhibited the family of unbiased estimators for α; the nonexistence of a uniformly minimum variance unbiased estimator comes from the fact that for a fixed value α0 of α the unbiased estimator with smallest variance depends on α0 and hence no unbiased estimator has uniformly smallest variance. In this note we show that a natural continuation of the example leads to an estimating equation for α that produces the maximum likelihood estimator. A second example, taken from Fisher (1958), exhibits a similar property. This example is both simpler and less artificial than the first, and we have found it very useful for teaching purposes. The general result is closely related to Fisher's motivation of maximum likeli...
TL;DR: In this paper, the authors consider two different models and present the minimum variance unbiased estimators of the expected failure rate of the revised software at any time of testing t, based on the data generated up to that point.
Abstract: As the formal methods of proving correctness of a computer program are still very inadequate, in practice when a new piece of software is developed and all obvious errors are removed, it is tested with different (random) inputs in order to detect the remaining errors and assess its quality. We suppose that whenever the program fails the error causing the failure can be detected and removed correctly. Thus, the quality of the software increases as testing goes on. In this paper, we consider two different models and present the minimum variance unbiased estimators of the expected failure rate of the revised software at any time of testing t, based on the data generated up to that point.
TL;DR: For a continuous distribution with a certain symmetry, several U-statistics of degree 2 are asymptotically as efficient as the invariant U-Statistics which are UMVU estimators of estimable parameters.
Abstract: For a continuous distribution with a certain symmetry, several U-statistics of degree 2 are asymptotically as efficient as the invariant U-statistics which are UMVU estimators of estimable parameters. To see the difference between two the statistics, we evaluate the limiting risk deficiency of the Ustatistic with respect to the invariant U-statistic, which is also equal to the coefficient of the reciprocal of the sample size in the ratio of their variances. For example, Gini's mean difference is asymptotically efficient for a continuous distributionwhich is symmetric with respect to a point on R1 . Its limiting risk deficiency is about 1.12 for a normal distribution.
TL;DR: In this article, the bias of the sample mean-of-the-ratios (MOR) statistic was analyzed and a new unbiased estimator alternative to the Hartley-Ross' (1954) based on the same statistics was proposed.
Abstract: We are offering two expressions of the bias of the sample mean-of-the-ratios and a new unbiased estimator alternative to the Hartley-Ross' (1954) based on the same statistics. From here, it is possible to build some operationally better estimators based on the mean-of-the-ratios statistic.
TL;DR: In this paper, a general class of almost unbiased estimators for the ratio and product of population means of two charactess is proposed, and the estimators are shown to be almost unbiased.
Abstract: In this paper we propose a general class of almost unbiased estimators for the ratio and product of population means of two charactess.
TL;DR: In this paper, the authors consider uniformly minimum variance unbiased estimation of estimable parameters when observations come from a certain class of discrete distributions with support depending on the parameter, and apply it to the special case where all parametric functions are estimable.
Abstract: This article considers uniformly minimum variance unbiased estimation of estimable parameters when observations come from a certain class of discrete distributions with support depending on the parameter. An example of Stigler (1972) comes as a special case, where all parametric functions are estimable and have uniformly minimum variance unbiased estimators.