Showing papers on "U-statistic published in 1993"

Consistency of the Generalized Bootstrap for Degenerate $U$-Statistics

Abstract: A generalized bootstrap version is defined for degenerate $U$-statistics. Our main result shows that the (conditional) distribution function of the bootstrapped degenerate $U$-statistic provides a consistent estimator for the unknown distribution function of the degenerate $U$-statistic under consideration. For the proof we rely on a rank statistic approach.

Unbiased Estimators and Their Applications: Volume 1: Univariate Case

Abstract: Will reading habit influence your life? Many say yes. Reading unbiased estimators and their applications vol 1 univariate case is a good habit; you can develop this habit to be such interesting way. Yeah, reading habit will not only make you have any favourite activity. It will be one of guidance of your life. When reading has become a habit, you will not make it as disturbing activities or as boring activity. You can gain many benefits and importances of reading.

Asymptotic Distributions of Weighted $U$-Statistics of Degree 2

Abstract: The limiting distribution of weighted $U$-statistics of degree 2 is found for a wide class of weights, including uniform weights. Nonnormal limits can occur for both degenerate and nondegenerate kernels. A compact expression is given for the cumulants of the distribution. Incomplete and randomly weighted $U$-statistics are also analyzed.

Addendum to “simple constructions of almost k ‐wise independent random variables”

Abstract: The constructions presented in the above paper use a finite field which is either GF(2") or GF( p) for some prime p. The constructions are presented assuming that one has a representation of the field (i.e., an irreducible polynomial of degree m or the prime p , respectively). Such representations could be found, with overwhelmingly high probability, in probabilistic polynomial-time (in m or I P I , respectively). The paper contained some remarks indicating how to achieve this goal using only a linear number of unbiased coin tosses. However, in retrospect we feel that some more details should be given. For uniformity of exposition, we denote by m the logarithm (to base 2) of the size of the required field. The field representations in both cases can be encoded by strings of length m. Furthermore, in both cases about a fraction of all m-bit long strings are valid representations, and one can efficiently determine whether a string is a valid representation. Hence, selecting a valid representation can be done by selecting candidates at random until a valid one is found. As indicated in the paper, to save on randomness, we use an efficient sampling which in turn uses a construction of a sequence of pairwise independent variables, each uniformly distributed in (0, l}". The problem which arises is that the standard constructions of such pairwise independent sequences use a field of similar cardinality (i.e., with at least 2"' elements), and hence we need a representation for this field, which brings us to a circular argument. The solution is to use the known pairwise independent constructions in a slightly less straightforward manner. Specifically, suppose we need to generate a t-long sequence of pairwise independent m-bit strings (e.g., in the above application t = O(rn)). The idea is to 1 m

$U$-Statistics of Random-Size Samples and Limit Theorems for Systems of Markovian Particles with Non-Poisson Initial Distributions

Abstract: : Limiting distributions of square-integrable infinite order U- statistics were first studied by Dynkin and Mandelbaum (1983) and Mandelbaum and Taqqu (1984). We extend their results to the case of non-Poisson random sample size. Multiple integrals of non-Gaussian generalized fields are constructed to identify the limiting distributions. An invariance principle is also established. We use these results to study the limiting distribution of the amount of charge left in some set by an infinite system of signed Markovian particles when the initial particle density goes to infinity. By selecting the initial particle distribution, we determine the limiting distribution of charge, constructing different non-Gaussian generalized random fields, including Laplace, alpha-stable, and their multiple integrals.

Unbiased estimating equations derived from statistics that are functions of a parameter

Abstract: SUMMARY The formation of a combination of the observed random variables and a parameter of interest fr having a distribution depending only on tf provides one approach to the analysis of problems with many nuisance parameters. The condition for the resulting formal maximum likelihood estimating equation for f to be unbiased is derived. Applications include a regression model for exponentially distributed random variables with errors in the explanatory variables.

Unbiased inestimability of the larger of two parameters

Abstract: Blumenthal and Cohen (1968c) and Dhariyal, Sharma and Krishnamoorthy (1985) considered the question of the existence of unbiased estimators of the larger (smaller) of the two parameters. In this paper, we first show the non-existence of such an unbiased estimator in case of two double exponential populations with unknown locations. We also give a general inadmissibility result and apply it to the uniform distribution

Asymptotic efficiency of the arithmetic-mean estimator of the unknown mean of a homogeneous random field

Abstract: We consider the estimation of the unknown mean of a homogeneous random field from observations on a system of homothetically expanding regions. We examine the asymptotic behavior of the variance of the arithmetic-mean estimator. The arithmetic-mean estimator is shown to be asymptotically efficient in the class of linear estimators.

A General Class of Unbiased Estimators of a Parameter

Abstract: In this note an attempt has been made to develop a class of unbiased estimators for an unknown parameter. Several estimators can be shown as a direct result of the proposed class.

Simulation Testing of Unbiasedness of Variance Estimators

Abstract: In this article I address the evaluation of estimators of variance for parameter estimates. Given an unbiased estimator X of a parameter θ, and an estimator V of the variance of X, how does one test (via simulation) whether V is an unbiased estimator of the variance of X? The derivation of the test statistic illustrates the need for care in substituting consistent estimators for unknown parameters.

THE MINIMUM VARIANCE UNBIASED ESTIMATOR AND BEST TEST OF THE PROCESS CAPABILITY LNDEX C_p

Abstract: In this paper, the best unbiased estimator and the best test of the (C_p index are deduced by using the theory of statistical inference, under the assumption that the process measurements arise from a normal, an exponential or a Gamma distribution.

Estimation of Partial Regression Coefficient

Abstract: A sampling scheme providing unbiased partial regression coefficient has been proposed. The proposed sampling scheme is not only unbiased but also superior to simple random sampling and that due to Singh and Bathla (1990) for estimation of partial regression coefficient.