Estimation of a quantile in some nonstandard cases

A necessary condition for the asymptotic normality of the sample quantile estimator isf(Q(p))=F(Q(p))>0, whereQ(p) is thep-th quantile of the distribution functionF(x). In this paper, we estimate a quantile by a kernel quantile estimator when this condition is violated. We have shown that the kernel quantile estimator is asymptotically normal in some nonstandard cases. The optimal convergence rate of the mean squared error for the kernel estimator is obtained with respect to the asymptotically optimal bandwidth. A law of the iterated logarithm is also established.This research was partially supported by the new faculty award from the University of Oregon.     

On an autoregressive model with time-dependent coefficients

Summary As one of the non-stationary time series model, we consider a firstorder autoregressive model in which the autoregressive coefficient is assumed to be a function,f _t (θ), of timet. We establish several assumptions onf _t (θ), not on the terms in the Taylor expansion of log-likelihood function, and show that the estimators of unknown parameters involved inf _t (θ) have strong consistency and asymptotic normality under these assumptions when sample size tends to infinity.     

Weighted L quantile distance estimators for randomly censored data

The asymptotic properties of a family of minimum quantile distance estimators for randomly censored data sets are considered. These procedures produce an estimator of the parameter vector that minimizes a weighted L² distance measure between the Kaplan-Meier quantile function and an assumed parametric family of quantile functions. Regularity conditions are provided which insure that these estimators are consistent and asymptotically normal. An optimal weight function is derived for single parameter families, which, for location/scale families, results in censored sample analogs of estimators such as those suggested by Parzen.     

Statistical inference based on aligned ranks for two-way MANOVA with interaction

Multiresponse experiments in two-way layouts with interactions, having equal number of observations per cell, are considered. Robust procedures based on aligned ranks for statistical inference of interactions, main effects and an overall mean response in the models are proposed. Large sample properties of the proposed tests, estimators and confidence regions as the cell size tends to infinity are investigated. For the univariate case, it is found that the asymptotic relative efficiencies (ARE's) of the proposed procedures relative to classical procedures agree with the ARE-results of the two-sample rank test relative to the t-test. In addition, robustness due to Huber (1981, Robust Statistics, Wiley, New York) can be drawn.     

Exact convergence rate of bootstrap quantile variance estimator

Summary It is shown that the relative error of the bootstrap quantile variance estimator is of precise order n ^-1/4, when n denotes sample size. Likewise, the error of the bootstrap sparsity function estimator is of precise order n ^-1/4. Therefore as point estimators these estimators converge more slowly than the Bloch-Gastwirth estimator and kernel estimators, which typically have smaller error of order at most n ^-2/5.     

Estimation of probability characteristics by generalized Bernstein polynomials

Approximations based on Bernstein polynomials are used for smoothing a sample quantile function and estimating the underlying distribution and its characteristics. Generalized Bernstein-type polynomials are introduced to reduce the bias of estimation under various types of distributions including finite distributions. The asymptotic behavior of the expectations of these estimators is studied. Bibliography: 10 titles.Published in Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 127–138.     

A Berry-Esseen theorem for the kernel quantile estimator with application to studying the deficiency of quantile estimators

A Berry-Esseen bound is established for the kernel quantile estimator under various conditions. The results improve an earlier result of Falk (1985,Ann. Statist.,13, 428–433) and rely on the local smoothness of the quantile function. This new Berry-Esseen bound is applied to studying the deficiency of the sample quantile estimator with respect to the kernel quantile estimator. A new result is obtained which is an extension of that in Falk (1985).     

Minimax estimation of means of multivariate normal mixtures

Assume X = (X₁, …, X_p)′ is a normal mixture distribution with density w.r.t. Lebesgue measure, , where Σ is a known positive definite matrix and F is any known c.d.f. on (0, ∞). Estimation of the mean vector under an arbitrary known quadratic loss function _Q(θ, a) = (a − θ)′ Q(a − θ), Q a positive definite matrix, is considered. An unbiased estimator of risk is obatined for an arbitrary estimator, and a sufficient condition for estimators to be minimax is then achieved. The result is applied to modifying all the Stein estimators for the means of independent normal random variables to be minimax estimators for the problem considered here. In particular the results apply to the Stein class of limited translation estimators.     

Robust nonparametric estimators of monotone boundaries

This paper revisits some asymptotic properties of the robust nonparametric estimators of order-m and order-α quantile frontiers and proposes isotonized version of these estimators. Previous convergence properties of the order-m frontier are extended (from weak uniform convergence to complete uniform convergence). Complete uniform convergence of the order-m (and of the quantile order-α) nonparametric estimators to the boundary is also established, for an appropriate choice of m (and of α, respectively) as a function of the sample size. The new isotonized estimators share the asymptotic properties of the original ones and a simulated example shows, as expected, that these new versions are even more robust than the original estimators. The procedure is also illustrated through a real data set.     

Consistent least squares fitting of ellipsoids

Summary. A parameter estimation problem for ellipsoid fitting in the presence of measurement errors is considered. The ordinary least squares estimator is inconsistent, and due to the nonlinearity of the model, the orthogonal regression estimator is inconsistent as well, i.e., these estimators do not converge to the true value of the parameters, as the sample size tends to infinity. A consistent estimator is proposed, based on a proper correction of the ordinary least squares estimator. The correction is explicitly given in terms of the true value of the noise variance.Mathematics Subject Classification (2000): 65D15, 65D10, 15A63Revised version received August 15, 2003     

Functional estimation for Lévy measures of semimartingales with Poissonian jumps

We consider semimartingales with jumps that have finite Lévy measures. The purpose of this article is to estimate integral-type functionals of the Lévy measures from discrete observations. We propose two types of estimators: kernel-type and empirical-type estimators, both of which are obtained by direct discretization from asymptotically efficient estimators of the target based on continuous observations. We show the asymptotic efficiency in the asymptotic minimax sense of our estimators as the sample size tends to infinity and the sampling interval tends to zero.     

Weighted L2 quantile distance estimators for randomly censored data

The asymptotic properties of a family of minimum quantile distance estimators for randomly censored data sets are considered. These procedures produce an estimator of the parameter vector that minimizes a weighted L² distance measure between the Kaplan-Meier quantile function and an assumed parametric family of quantile functions. Regularity conditions are provided which insure that these estimators are consistent and asymptotically normal. An optimal weight function is derived for single parameter families, which, for location/scale families, results in censored sample analogs of estimators such as those suggested by Parzen.     

Confidence intervals for quantile estimation using Jackknife techniques

We consider the inference on quantiles, Q _y (β), with jackknife techniques, in finite populations of a variable, Y, using the quantile information on an auxiliary variable, X. Jackknife techniques are applied to estimate quantiles and the behaviour of these estimators is analyzed. Their properties are studied for simple random sampling. We also examine the confidence intervals obtained with jackknife variances.     

Asymptotic Normality of the Minimum Non-Hilbertian Distance Estimators for a Diffusion Process with Small Noise

A one-dimensional diffusion type process with small noise is observed up to the time T. It depends on an unknown real parameter. Some minimum distance estimators of this parameter are considered. These estimators are defined using the L ^p-metric or the uniform metric. The limiting distribution of the normalizing minimum distance estimators (as the noise vanishing) is known to be the distribution of a random variable. The distribution of this random variable is studied as the time T goes to the infinity. We will prove under some conditions that it has a limiting Gaussian law. This revised version was published online in June 2006 with corrections to the Cover Date.     

Zeta functions related to the group SL2(ℤ p )

An explicit formula is given for the number of subgroups of indexp ⁿ in the principle congruence subgroups of SL₂(ℤ _p ) (for odd primesp), and for the zeta function associated with the group. Asymptotically this number iscnp ⁿ , wherec is a constant depending on the congruence subgroup. Also, the zeta function of thei-th congruence subgroup coincides with the partial zeta function of the 3-generated subgroups of thei+1-th congruence subgroup, and for each indexp ⁿ the ratio between 2-generated subgroups and 3-generated subgroups tends top - 1:1, asn tends to infinity. This work is part of the author’s Ph.D. thesis carried out at the Hebrew University of Jerusalem under the supervision of Prof. A. Lubotzky. I wish to thank Prof. Lubotzky for his continual interest and encouragement without which this paper would not have been published.     

Composite hierachical linear quantile regression

Multilevel （hierarchical） modeling is a generalization of linear and generalized linear modeling in which regression coefficients are modeled through a model, whose parameters are also estimated from data. Multilevel model fails to fit well typically by the use of the EM algorithm once one of level error variance （like Cauchy distribution） tends to infinity. This paper proposes a composite multilevel to combine the nested structure of multilevel data and the robustness of the composite quantile regression, which greatly improves the efficiency and precision of the estimation. The new approach, which is based on the Gauss-Seidel iteration and takes a full advantage of the composite quantile regression and multilevel models, still works well when the error variance tends to infinity, We show that even the error distribution is normal, the MSE of the estimation of composite multilevel quantile regression models nearly equals to mean regression. When the error distribution is not normal, our method still enjoys great advantages in terms of estimation efficiency.     

Edgeworth expansion for the kernel quantile estimator

Using the kernel estimator of the pth quantile of a distribution brings about an improvement in comparison to the sample quantile estimator. The size and order of this improvement is revealed when studying the Edgeworth expansion of the kernel estimator. Using one more term beyond the normal approximation significantly improves the accuracy for small to moderate samples. The investigation is non- standard since the influence function of the resulting L-statistic explicitly depends on the sample size. We obtain the expansion, justify its validity and demonstrate the numerical gains in using it.     

Variable selection via quantile regression with the process of Ornstein-Uhlenbeck type

Based on the data-cutoff method,we study quantile regression in linear models,where the noise process is of Ornstein-Uhlenbeck type with possible jumps.In single-level quantile regression,we allow the noise process to be heteroscedastic,while in composite quantile regression,we require that the noise process be homoscedastic so that the slopes are invariant across quantiles.Similar to the independent noise case,the proposed quantile estimators are root-n consistent and asymptotic normal.Furthermore,the adaptive least absolute shrinkage and selection operator(LASSO)is applied for the purpose of variable selection.As a result,the quantile estimators are consistent in variable selection,and the nonzero coefficient estimators enjoy the same asymptotic distribution as their counterparts under the true model.Extensive numerical simulations are conducted to evaluate the performance of the proposed approaches and foreign exchange rate data are analyzed for the illustration purpose.     

Bias reduction and efficiency of reconstructed ratio estimators for a finite universe

Summary The bias of ratio estimators based on a simple random sample ofn units drawn from a finite universe ofN units, and reconstructed according to, or in ways similar to Quenouille's method, is of order 1/n only ifN≧n ², and of order 1/n ^Q , where 1<Q<2, ifN<n ². Further it is shown that the device of splitting samples, either for bias reduction and/or convenience of variance estimation, except for the special case noted in the paper, yields estimators that are inefficient and can be improved.     

On the errors committed by sequences of estimator functionals

Consider a sequence of estimators [^(q)] _n\hat \theta _n which converges almost surely to θ ₀ as the sample size n tends to infinity. Under weak smoothness conditions, we identify the asymptotic limit of the last time [^(q)] _n\hat \theta _n is further than ɛ away from θ ₀ when ɛ → 0⁺. These limits lead to the construction of sequentially fixed width confidence regions for which we find analytic approximations. The smoothness conditions we impose is that [^(q)] _n\hat \theta _n is to be close to a Hadamard-differentiable functional of the empirical distribution, an assumption valid for a large class of widely used statistical estimators. Similar results were derived in Hjort and Fenstad (1992) for the case of Euclidean parameter spaces; part of the present contribution is to lift these results to situations involving parameter functionals. The apparatus we develop is also used to derive appropriate limit distributions of other quantities related to the far tail of an almost surely convergent sequence of estimators, like the number of times the estimator is more than ɛ away from its target. We illustrate our results by giving a new sequential simultaneous confidence set for the cumulative hazard function based on the Nelson-Aalen estimator and investigate a problem in stochastic programming related to computational complexity.