Robust estimation in the linear model with asymmetric error distributions

In the linear model X^{n × 1} = C^{n × p}θ^{p × 1} + E^{n × 1}, Huber's theory of robust estimation of the regression vector θ^{p × 1} is adapted for two models for the partially specified common distribution F of the i.i.d. components of the error vector E^{n × 1}. In the first model considered, the restriction of F to a set [−a₀, b₀] is a standard normal distribution contaminated, with probability , by an unknown distribution symmetric about 0. In the second model, the restriction of F to [−a₀, b₀] is completely specified (and perhaps asymmetrical). In both models, the distribution of F outside the set [−a₀, b₀] is completely unspecified. For both models, consistent and asymptotically normal M-estimators of θ^{p × 1} are constructed, under mild regularity conditions on the sequence of design matrices {C^{n × p}}. Also, in both models, M-estimators are found which minimize the maximal mean-squared error. The optimal M-estimators have influence curves which vanish off compact sets.     

On the robust estimation in poisson processes with periodic intensities

Under some regularity conditions, it is well known that the maximum likelihood estimator (MLE) is asymptotically normal and efficient. However, if the observation is contaminated, the MLE is not always an appropriate estimator. In this paper, we treat M-estimators and study their asymptotic behavior. By choosing estimation equations, robust M-estimators are presented for phase parameters.     

Tukey's linear sensitivity and order statistics

Tukey (1965,Proc. Nat. Acad. Sci. U.S.A.,53, 127–134) introducedlinear sensitivity as a measure of informativeness in a collection of order statistics. Here we study its general properties and discuss how it is related to the best linear unbiased estimator, Fisher information measure, and asymptotic relative efficiency. Also, we obtain explicit and asymptotic expressions for the linear sensitivity of a collection of consecutive order statistics from a location or from a scale family, and discuss its role in the comparison ofL-estimators. We conclude our discussion with examples from uniform, exponential and normal populations.     

Asymptotic analysis of stochastic programs

In this paper we discuss a general approach to studying asymptotic properties of statistical estimators in stochastic programming. The approach is based on an extended delta method and appears to be particularly suitable for deriving asymptotics of the optimal value of stochastic programs. Asymptotic analysis of the optimal value will be presented in detail. Asymptotic properties of the corresponding optimal solutions are briefly discussed.     

Robustness Comparisons of Some Classes of Location Parameter Estimators

Asymptotic biases and variances of M-, L- and R-estimators of a location parameter are compared under ε-contamination of the known error distribution F ₀ by an unknown (and possibly asymmetric) distribution. For each ε-contamination neighborhood of F ₀, the corresponding M-, L- and R-estimators which are asymptotically efficient at the least informative distribution are compared under asymmetric ε-contamination. Three scale-invariant versions of the M-estimator are studied: (i) one using the interquartile range as a preliminary estimator of scale: (ii) another using the median absolute deviation as a preliminary estimator of scale; and (iii) simultaneous M-estimation of location and scale by Huber's Proposal 2. A question considered for each case is: when are the maximal asymptotic biases and variances under asymmetric ε-contamination attained by unit point mass contamination at ∞? Numerical results for the case of the ε-contaminated normal distribution show that the L-estimators have generally better performance (for small to moderate values of ε) than all three of the scale-invariant M-estimators studied.     

An efficient Fréchet differentiable high breakdown multivariate location and dispersion estimator

A good robust functional should, if possible, be efficient at the model, smooth, and have a high breakdown point. M-estimators can be made efficient and Fréchet differentiable by choosing appropriate ψ-functions but they have a breakdown point of at most 1/(p + 1) in p dimensions. On the other hand, the local smoothness of known high breakdown functionals has not been investigated. It is known that Rousseeuw's minimum volume ellipsoid estimator is not differentiable and that S-estimators based on smooth functions force a trade-off between efficiency and breakdown point. However, by using a two-step M-estimator based on the minimum volume ellipsoid we show that it is possible to obtain a highly efficient, Fréchet differentiable estimator whilst still retaining the breakdown point. This result is extended to smooth S-estimators.     

Asymptotic behavior of M-estimator and related random field for diffusion process

The M-estimate which maximizes a positive stochastic process Q is treated for multidimensional diffusion models. The convergence in distribution of the process of ratio of Q's after normalizing is proved. The asymptotic behavior of M-estimates is stated. We present the asymptotic variance in general cases and in estimation by misspecified models.     

Metrization of Epi-Convergence: An Application to the Strong Consistency of M-Estimators

Strong consistency in the class of M-estimators is examined here as an application of epi-convergence, a functional convergence which is particularly suited for the study of convergence of the functions' minimizing values and arguments. Starting from a 1988 paper by J. Dupaova and R. Wets, which contains a thorough account of the relations between consistency and epi-convergence, a quantitative approach of the same topic is pursued here. Epi-convergence is compared with two definitions introduced in 1980 by one of the authors. The results are merged in order to define a distance between lower semicontinuous functions that is compatible with epi-convergence and bounds the distance between the minimizing arguments. These results applied to the statistical problem allow the definition of a bound of the distance between the estimator and the parameter.     

Asymptotic Expansions for Closed Orbits in Homology Classes

In this paper, we study the behaviour of the counting function associated to the closed geodesics lying in a prescribed homology class on a compact negatively curved manifold. Our main result is an asymptotic expansion. We also obtain results in the wider context of periodic orbits of Anosov flows.     

Asymptotic Behavior of Loss Probability in GI/M/1/K Queue as K Tends to Infinity

We obtain an asymptotic behavior of the loss probability for the GI/M/1/K queue as K for cases of <1, >1 and =1.     

Asymptotic expansions for the conditional sojourn time distribution in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1-PS queue

We consider the M/M/1 queue with processor sharing. We study the conditional sojourn time distribution, conditioned on the customer’s service requirement, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. The asymptotic formulas relate to, and extend, some results of Morrison (SIAM J. Appl. Math. 45:152–167, [1985]) and Flatto (Ann. Appl. Probab. 7:382–409, [1997]). This work was partly supported by NSF grant DMS 05-03745.     

Asymptotic Results and a Markovian Approximation for the M(n)/M(n)/s+GI System

In this paper for the M(n)/M(n)/s+GI system, i.e. for a s-server queueing system where the calls in the queue may leave the system due to impatience, we present new asymptotic results for the intensities of calls leaving the system due to impatience and a Markovian system approximation where these results are applied. Furthermore, we present a new proof for the formulae of the conditional density of the virtual waiting time distributions, recently given by Movaghar for the less general M(n)/M/s+GI system. Also we obtain new explicit expressions for refined virtual waiting time characteristics as a byproduct.     

Asymptotic distributions of non-central studentized statistics

Let X1,...,Xn be independent and identically distributed random variables and Wn = Wn(X1,...,Xn) be an estimator of parameter θ.Denote Tn =(Wn - θ0)/sn,where sn2 is a variance estimator of Wn.In this paper a general result on the limiting distributions of the non-central studen-tized statistic Tn is given.Especially,when s2n is the jacknife estimate of variance,it is shown that the limit could be normal,a weighted χ2 distribution,a stable distribution,or a mixture of normal and stable distribution.Applicati...     

Analysis of customers’ impatience in queues with server vacations

Many models for customers impatience in queueing systems have been studied in the past; the source of impatience has always been taken to be either a long wait already experienced at a queue, or a long wait anticipated by a customer upon arrival. In this paper we consider systems with servers vacations where customers’ impatience is due to an absentee of servers upon arrival. Such a model, representing frequent behavior by waiting customers in service systems, has never been treated before in the literature. We present a comprehensive analysis of the single-server, M/M/1 and M/G/1 queues, as well as of the multi-server M/M/c queue, for both the multiple and the single-vacation cases, and obtain various closed-form results. In particular, we show that the proportion of customer abandonments under the single-vacation regime is smaller than that under the multiple-vacation discipline. This work was supported by the Euro-Ngi network of excellence.     

Asymptotic Analysis of the GI/M/ 1 /n Loss System as n Increases to Infinity

This paper provides the asymptotic analysis of the loss probability in the GI/M/1/n queueing system as n increases to infinity. The approach of this paper is alternative to that of the recent papers of Choi and Kim (2000) and Choi et al. (2000) and based on application of modern Tauberian theorems with remainder. This enables us to simplify the proofs of the results on asymptotic behavior of the loss probability of the abovementioned paper of Choi and Kim (2000) as well as to obtain some new results.     

On stable laws for estimating functions and derived estimators

Summary Stable laws forM-estimators, maximum likelihood and other estimators and obtained through parallel results for the estimating functions and relative compactness of some related estimating functional processes. Work supported by the Office of Naval Research, Contract No. N00014-83-K-0387.     

Nonparametric inference on the difference of location parameters of correlated variables from fragmentary samples

Summary In this paper, two types of robust estimators and approximate confidence intervals for the difference of location parameters of correlated random variables are proposed and investigated when some observations are missing. It is shown that the suggested estimators are consistent and asymptotically normally distributed. In addition, the proposed approximate confidence intervals are also shown to enjoy some nice asymptotic properties.     

Asymptotic behavior ofL-statistics for a large class of time series

In this paper we derive the asymptotic normality ofL-statistics with unbounded scores for a large class of time series. To handle the dependence structure, we use the concept ofm(n)-decomposability as an alternative to classical mixing concepts.Research supported by the Office of Naval Research Contract N00014-91-J-1020.Part of this work was done while the author was at the Department of Mathematics, KUN, Nijmegen, The Netherlands.     

Two-sample rank tests based on exceeding observations

Simple rank statistics are used to test that two samples come from the same distribution. Šidák’s E-test (Apl. Mat. 22 (1977), 166–175) is based on the number of observations from one sample that exceed all observations from the other sample. A similar test statistic is defined in Ann. Inst. Stat. Math. 52 (1970), 255–266. We study asymptotic behavior of the moments of both statistics.