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1.
Abstract. For probability distributions on ? q, a detailed study of the breakdown properties of some multivariate M‐functionals related to Tyler's [Ann. Statist. 15 (1987) 234] ‘distribution‐free’ M‐functional of scatter is given. These include a symmetrized version of Tyler's M‐functional of scatter, and the multivariate t M‐functionals of location and scatter. It is shown that for ‘smooth’ distributions, the (contamination) breakdown point of Tyler's M‐functional of scatter and of its symmetrized version are 1/q and , respectively. For the multivariate t M‐functional which arises from the maximum likelihood estimate for the parameters of an elliptical t distribution on ν ≥ 1 degrees of freedom the breakdown point at smooth distributions is 1/( q + ν). Breakdown points are also obtained for general distributions, including empirical distributions. Finally, the sources of breakdown are investigated. It turns out that breakdown can only be caused by contaminating distributions that are concentrated near low‐dimensional subspaces.  相似文献   

2.
We investigate the asymptotic behavior of the probability density function (pdf) and the cumulative distribution function (cdf) of Student's t-distribution with ν > 0 degrees of freedom (t ν for short) for ν tending to infinity when the argument x = x ν of the pdf (cdf) depends on ν and tends to ± ∞ (?∞). To this end, we consider the ratio of the pdf's (cdf's) of the t ν- and the standard normal distribution. Depending on the choice of the argument x ν, the pdf-ratio (cdf-ratio) tends to 1, a fixed value greater than 1, or to ∞. As a byproduct, we obtain a result for Mill' ratio when x ν → ?∞.  相似文献   

3.
Wolfgang Wagner 《Statistics》2013,47(3):449-456
Let X1, X2, … be i.i.d.r.v. and write (X1+…Xn?An)/Bn?Fn, where Bn >0.AnER1, n≥1. It is known that solely one–sided asymptotic assumptions imposed on Fn imply Fn0. In the present note we show that stronger one–sided assumptions lead even to the existence of EX1 3 so that the BERRY-ESSEEN inequalities hold true.  相似文献   

4.
This article considers an empirical Bayes testing problem for the guarantee lifetime in the two-parameter exponential distributions with non identical components. We study a method of constructing empirical Bayes tests under a class of unknown prior distributions for the sequence of the component testing problems. The asymptotic optimality of the sequence of empirical Bayes tests is studied. Under certain regularity conditions on the prior distributions, it is shown that the sequence of the constructed empirical Bayes tests is asymptotically optimal, and the associated sequence of regrets converges to zero at a rate O(n? 1 + 1/[2(r + α) + 1]) for some integer r ? 0 and 0 ? α ? 1 depending on the unknown prior distributions, where n is the number of past data available when the (n + 1)st component testing problem is considered.  相似文献   

5.
We consider an inhomogeneous Poisson process X on [0, T]. The intensity function of X is supposed to be strictly positive and smooth on [0, T] except at the point θ, in which it has either a 0-type singularity (tends to 0 like |x| p , p∈(0, 1)), or an ∞-type singularity (tends to ∞ like |x| p , p∈(?1, 0)). We suppose that we know the shape of the intensity function, but not the location of the singularity. We consider the problem of estimation of this location (shift) parameter θ based on n observations of the process X. We study the Bayesian estimators and, in the case p>0, the maximum-likelihood estimator. We show that these estimators are consistent, their rate of convergence is n 1/(p+1), they have different limit distributions, and the Bayesian estimators are asymptotically efficient.  相似文献   

6.
This paper considers statistical inference for partially linear models Y = X ? β +ν(Z) +? when the linear covariate X is missing with missing probability π depending upon (Y, Z). We propose empirical likelihood‐based statistics to construct confidence regions for β and ν(z). The resulting empirical likelihood ratio statistics are shown to be asymptotically chi‐squared‐distributed. The finite‐sample performance of the proposed statistics is assessed by simulation experiments. The proposed methods are applied to a dataset from an AIDS clinical trial.  相似文献   

7.
A great deal of inference in statistics is based on making the approximation that a statistic is normally distributed. The error in doing so is generally O(n?1/2), where n is the sample size and can be considered when the distribution of the statistic is heavily biased or skewed. This note shows how one may reduce the error to O(n?(j+1)/2), where j is a given integer. The case considered is when the statistic is the mean of the sample values of a continuous distribution with a scale or location change after the sample has undergone an initial transformation, which may depend on an unknown parameter. The transformation corresponding to Fisher's score function yields an asymptotically efficient procedure.  相似文献   

8.
This paper deals with an empirical Bayes testing problem for the mean lifetimes of exponential distributions with unequal sample sizes. We study a method to construct empirical Bayes tests {δ* nl + 1,n } n = 1 for the sequence of the testing problems. The asymptotic optimality of {δ* nl + 1,n } n = 1 is studied. It is shown that the sequence of empirical Bayes tests {δ* nl + 1,n } n = 1 is asymptotically optimal, and its associated sequence of regrets converges to zero at a rate (ln n)4M?1/n, where M is an upper bound of sample sizes.  相似文献   

9.
The Kolmogorov-Smirnov (K–S) one-sided and two-sided tests of goodness of fit based on the test statistics D+ n D? n and Dn are equivalent to tests based on taking the cumulative probability of the i–th order statistic of a sample of size n to be (i–.5)/n. Modified test statistics C+ n, C? n and Cn are obtained by taking the cumulative probability to be i/(n+l). More generally, the cumula-tive probability may be taken to be (i?δ)/(n+l?2δ), as suggested by Blom (1958), where 0 less than or equal δ less than or equal .5. Critical values of the test statis-tics can be found by interpolating inversely in tables of the proba-bility integrals obtained by setting a=l/(n+l?2δ) in an expression given by Pyke (1959). Critical values for the D's (corresponding to δ=.5) have been tabulated to 5DP by Miller (1956) for n=1(1)100. The authors have made analogous tabulations for the C's (corresponding to δ=0) [previously tabulated by Durbin (1969) for n=1(1)60(2)100] and for the test statistics E+ n, E? n and En corresponding to δ f.3. They have also made a Monte Carlo comparison of the power of the modified tests with that of the K–S test for several hypothetical distributions. In a number of cases, the power of the modified tests is greater than that of the K–S test, especially when the standard deviation is greater under the alternative than under the null hypo-thesis.  相似文献   

10.
Let X1,X2,… be independent and identically distributed nonnegative random variables with mean μ, and let Sn = X1 + … + Xn. For each λ > 0 and each n ≥ 1, let An be the interval [λnY, ∞), where γ > 1 is a constant. The number of times that Sn is in An is denoted by N. As λ tends to zero, the asymtotic behavior of N is studied. Specifically under suitable conditions, the expectation of N is shown to be (μλ?1)β + o(λ?β/2 where β = 1/(γ-1) and the variance of N is shown to be (μλ?1)β(βμ1)2σ2 + o(λ) where σ2 is the variance of Xn.  相似文献   

11.
The least-absolute-deviation estimate of a monotone regression function on an interval has been studied in the literature. If the observation points become dense in the interval, the almost sure rate of convergence has been shown to be O(n1/4). Applying the techniques used by Brunk (1970, Nonparametric, Techniques in Statistical Inference. Cambridge Univ. Press), the asymptotic distribution of the l1 estimator at a point is obtained. If the underlying regression function has positive slope at the point, the rate of convergence is seen to be O(n1/3). Monotone percentile regression estimates are also considered.  相似文献   

12.
The uniformly asymptotical normality of frequency polygons for ψ-mixing samples is investigated under the given conditions. Moreover, the corresponding rate of convergence is also derived, which is nearly O(n? 1/6) for the given assumptions.  相似文献   

13.
Denote M n the largest of n independent STSD variables. This article shows the rate of convergence of (M n  ? b n )/a n to the extreme value distribution exp (?e ?x ) which is characterized by the supremum metric.  相似文献   

14.
Classes of higher-order kernels for estimation of a probability density are constructed by iterating the twicing procedure. Given a kernel K of order l, we build a family of kernels Km of orders l(m + 1) with the attractive property that their Fourier transforms are simply 1 — {1 —$(.)}m+1, where ? is the Fourier transform of K. These families of higher-order kernels are well suited when the fast Fourier transform is used to speed up the calculation of the kernel estimate or the least-squares cross-validation procedure for selection of the window width. We also compare the theoretical performance of the optimal polynomial-based kernels with that of the iterative twicing kernels constructed from some popular second-order kernels.  相似文献   

15.
Abstract. We consider the problem of efficiently estimating multivariate densities and their modes for moderate dimensions and an abundance of data. We propose polynomial histograms to solve this estimation problem. We present first‐ and second‐order polynomial histogram estimators for a general d‐dimensional setting. Our theoretical results include pointwise bias and variance of these estimators, their asymptotic mean integrated square error (AMISE), and optimal binwidth. The asymptotic performance of the first‐order estimator matches that of the kernel density estimator, while the second order has the faster rate of O(n?6/(d+6)). For a bivariate normal setting, we present explicit expressions for the AMISE constants which show the much larger binwidths of the second order estimator and hence also more efficient computations of multivariate densities. We apply polynomial histogram estimators to real data from biotechnology and find the number and location of modes in such data.  相似文献   

16.
A self-contained FORTRAN subroutine is provided which computes factors for Wald-Wolfowitz type tolerance limits allowing arbitrary combinations of sample size n and degrees of freedom ν. The exact calculations from our program reveal inadequacies of two existing approximations, especially when ν ? n. Numerous applications where νn ? 1 are cited; two of these are discussed and illustrated.  相似文献   

17.
18.
This article presents some structural properties of the inverse Gaussian distribution, together with several new characterizations based on constancy of regression of suitable functions on the sum of n independent identically distributed random variables. A decomposition of the statistic λσ (X?1i?X?1) into n - 1 independent chi-squared random variables, each with one degree of freedom, is given when n is of the form 2r.  相似文献   

19.
Suppose we have a random sample of size n from a multivariate distribution with finite moments, for which a parametric form is not available. We wish to obtain a confidence interval (CI) for the length of its mean. The usual method is to Studentize. The resulting CIs are not exact. The error in their nominal levels is ~n ?1/2 and ~n ?1 in the one-sided and two-sided cases. We show how to reduce these errors to ~n ?3/2 and ~n ?2.  相似文献   

20.
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