Asymptotic consistency of fixed-width sequential confidence intervals for a multiple regression function

Summary Letm _n (x) be the recursive kernel estimator of the multiple regression functionm(x)=E[Y|X=x]. For given α (0<α<1) andd>0 we define a certain class of stopping timesN=N(α,d, x) and takeI _N,d (x)=[m _N (x)−d, m _N (x)+d] as a 2d-width confidence interval form(x) at a given pointx. In this paper it is shown that the probability P{m(x)∈I _N,d (x)} converges to α asd tends to zero.     

On sequential fixed-width confidence intervals for the mean and second-order expansions of the associated coverage probabilities

In order to construct fixed-width (2d) confidence intervals for the mean of an unknown distribution function F, a new purely sequential sampling strategy is proposed first. The approach is quite different from the more traditional methodology of Chow and Robbins (1965, Ann. Math. Statist., 36, 457–462). However, for this new procedure, the coverage probability is shown (Theorem 2.1) to be at least (1-)+Ad ²+o(d²) as d0 where (1-) is the preassigned level of confidence and A is an appropriate functional of F, under some regularity conditions on F. The rates of convergence of the coverage probability to (1-) obtained by Csenki (1980, Scand. Actuar. J., 107–111) and Mukhopadhyay (1981, Comm. Statist. Theory Methods, 10, 2231–2244) were merely O(d^1/2-q), with 0<q<1/2, under the Chow-Robbins stopping time ^*. It is to be noted that such considerable sharpening of the rate of convergence of the coverage probability is achieved even though the new stopping variable is O_p(^*). An accelerated version of the stopping rule is also provided together with the analogous second-order characteristics. In the end, an example is given for the mean estimation problem of an exponential distribution.     

Asymptotic properties of a maximum likelihood estimator with data from a Gaussian process

We consider an estimation problem with observations from a Gaussian process. The problem arises from a stochastic process modeling of computer experiments proposed recently by Sacks, Schiller, and Welch. By establishing various representations and approximations to the corresponding log-likelihood function, we show that the maximum likelihood estimator of the identifiable parameter θσ² is strongly consistent and converges weakly (when normalized by √n) to a normal random variable, whose variance does not depend on the selection of sample points. Some extensions to regression models are also obtained.     

On normal approximation for maximum likelihood estimation from single server queues

This paper is concerned with the rate of convergence of the distribution of the maximum likelihood estimators of the arrival and the service rates in a GI/G/1 queueing system. This revised version was published online in June 2006 with corrections to the Cover Date.     

On large deviation expansion of distribution of maximum likelihood estimator and its application in large sample estimation

For estimating an unknown parameter , the likelihood principle yields the maximum likelihood estimator. It is often favoured especially by the applied statistician, for its good properties in the large sample case. In this paper, a large deviation expansion for the distribution of the maximum likelihood estimator is obtained. The asymptotic expansion provides a useful tool to approximate the tail probability of the maximum likelihood estimator and to make statistical inference. Theoretical and numerical examples are given. Numerical results show that the large deviation approximation performs much better than the classical normal approximation.This work is supported in part by the Natural Science and Engineering Research Council of Canada under grant NSERC A-9216.This author is also partially supported by the National Science Foundation of China.     

Convergence rates for two-stage confidence intervals based on U-statistics

We consider a modified two-stage procedure for constructing a fixed-width confidence interval for the mean of a U-statistic. First, we discuss a few asymptotic results with the associated rates of convergence. The main result gives the rate of convergence for the coverage probability of our proposed confidence interval which is seen to be slower than that for the purely sequential procedure.     

On progressively truncated maximum likelihood estimators

Summary The basic regularity conditions pertaining to the asymptotic theory of progressively truncated likelihood functions and maximum likelihood estimators are considered, and the uniform strong consistency and weak convergence of progressively truncated maximum likelihood estimators are studied systematically. Work done during the first author's visit (as a visiting scholar) to the University of North Carolina at Chapel Hill, supported by the Ministry of Education of the Japanese Government. Work supported by the (U.S.) National Heart, Lung and Blood Institute, Contact NIH-NHLBI-F1-2243-L.     

Geometrical expansions for the distributions of the score vector and the maximum likelihood estimator

In the present note, asymptotic expansions for conditional and unconditional distributions of the score vector are derived. Our aim is to consider these expansions in the light of differential geometry, particularly the theory of derivative strings. Expansions for the distributions of the maximum likelihood estimator are obtained from those for the score vector via transformation, with a view to interpreting from the standpoint of differential geometry the various terms entering the expansions.The present work was carried out at the Department of Theoretical Statistics, University of Aarhus, Denmark, with support from the Danish-French Cultural Exchange Programme.     

面板数据公共变点的非参数最大似然估计

Gao Pengli;Xia Zhiming(School of Mathematics,Northwest University,Xi'an 710127,China)     

Information amount and higher-order efficiency in estimation

By means of second-order asymptotic approximation, the paper clarifies the relationship between the Fisher information of first-order asymptotically efficient estimators and their decision-theoretic performance. It shows that if the estimators are modified so that they have the same asymptotic bias, the information amount can be connected with the risk based on convex loss functions in such a way that the greater information loss of an estimator implies its greater risk. The information loss of the maximum likelihood estimator is shown to be minimal in a general set-up. A multinomial model is used for illustration.     

Empirical likelihood confidence intervals for the differences of quantiles with missing data

Suppose that there are two nonparametric populations x and y with missing data on both of them. We are interested in constructing confidence intervals on the quantile differences of x and y. Random imputation is used. Empirical likelihood confidence intervals on the differences are constructed. Supported by the National Natural Science Foundation of China (No. 10661003) and Natural Science Foundation of Guangxi (No. 0728092).     

Semi-empirical likelihood confidence intervals for the differences of quantiles with missing data

Detecting population （group） differences is useful in many applications, such as medical research. In this paper, we explore the probabilistic theory for identifying the quantile differences .between two populations. Suppose that there are two populations x and y with missing data on both of them, where x is nonparametric and y is parametric. We are interested in constructing confidence intervals on the quantile differences of x and y. Random hot deck imputation is used to fill in missing data. Semi-empirical likelihood confidence intervals on the differences are constructed.     

An approximate maximum likelihood estimation for non-Gaussian non-minimum phase moving average processes

An approximate maximum likelihood procedure is proposed for the estimation of parameters in possibly nonminimum phase (noninvertible) moving average processes driven by independent and identically distributed non-Gaussian noise. Under appropriate conditions, parameter estimates that are solutions of likelihood-like equations are consistent and are asymptotically normal. A simulation study for MA(2) processes illustrates the estimation procedure.     

Quasi-maximum likelihood estimators in generalized linear models with autoregressive processes

The paper studies a generalized linear model(GLM)y_t = h(x_t~T β) + ε_t,t = l,2,...,n,where ε_1 = η_1,ε_1 =ρε_t +η_t,t = 2,3,...;n,h is a continuous differentiable function,η_t's are independent and identically distributed random errors with zero mean and finite variance σ~2.Firstly,the quasi-maximum likelihood(QML) estimators of β,p and σ~2 are given.Secondly,under mild conditions,the asymptotic properties(including the existence,weak consistency and asymptotic distribution) of the QML estimators are investigated.Lastly,the validity of method is illuminated by a simulation example.     

Asymptotics for non-parametric likelihood estimation with doubly censored multivariate failure times

This paper considers non-parametric estimation of a multivariate failure time distribution function when only doubly censored data are available, which occurs in many situations such as epidemiological studies. In these situations, each of multivariate failure times of interest is defined as the elapsed time between an initial event and a subsequent event and the observations on both events can suffer censoring. As a consequence, the estimation of multivariate distribution is much more complicated than that for multivariate right- or interval-censored failure time data both theoretically and practically. For the problem, although several procedures have been proposed, they are only ad-hoc approaches as the asymptotic properties of the resulting estimates are basically unknown. We investigate both the consistency and the convergence rate of a commonly used non-parametric estimate and show that as the dimension of multivariate failure time increases or the number of censoring intervals of multivariate failure time decreases, the convergence rate for non-parametric estimate decreases, and is slower than that with multivariate singly right-censored or interval-censored data.     

On a robust and efficient maximum depth estimator

The best breakdown point robustness is one of the most outstanding features of the univariate median. For this robustness property, the median, however, has to pay the price of a low efficiency at normal and other light-tailed models. Affine equivariant multivariate analogues of the univariate median with high breakdown points were constructed in the past two decades. For the high breakdown robustness, most of them also have to sacrifice their efficiency at normal and other models, nevertheless. The affine equivariant maximum depth estimator proposed and studied in this paper turns out to be an exception. Like the univariate median, it also possesses a highest breakdown point among all its multivariate competitors. Unlike the univariate median, it is also highly efficient relative to the sample mean at normal and various other distributions, overcoming the vital low-efficiency shortcoming of the univariate and other multivariate generalized medians. The paper also studies the asymptotics of the estimator and establishes its limit distribution without symmetry and other strong assumptions that are typically imposed on the underlying distribution. This work was supported by Natural Science Foundation of USA (Grant Nos. DMS-0071976, DMS-0234078) and by the Southwestern University of Finance and Economics Third Period Construction Item Funds of the 211 Project (Grant No. 211D3T06)     

On the rate of convergence of the maximum likelihood estimator of a <Emphasis Type="Italic">k</Emphasis>-monotone density

Bounds for the bracketing entropy of the classes of bounded k-monotone functions on [0, A] are obtained under both the Hellinger distance and the L ^p (Q) distance, where 1 ⩽ p < ∞ and Q is a probability measure on [0,A]. The result is then applied to obtain the rate of convergence of the maximum likelihood estimator of a k-monotone density. This work was supported by National Science Foundation of USA (Grant No. DMS-0405855, DMS-0804587)     

Exact maximum likelihood estimator for drift fractional Brownian motion at discrete observation

This paper deals with the problems of consistency and strong consistency of the maximum likelihood estimators of the mean and variance of the drift fractional Brownian motions observed at discrete time instants. Both the central limit theorem and the Berry-Ess′een bounds for these estimators are obtained by using the Stein’s method via Malliavin calculus.     

On bounded maximum width sequential confidence ellipsoids based on generalized U-statistics

For a vector of (estimable) functionals of several independent distributions, sequential confidence ellipsoids (of bounded maximum width) based on a class of generalized U-statistics are studied. A stopping rule along with a procedure for choosing the component sample sizes at each stage is developed, so that the proposed confidence ellipsoid has a confidence coefficient asymptotically (as the prescribed maximum width shrinks to zero) equal to a preassigned 1 - α (0 < α < 1), and the expected total sample size is minimized for the procedure. Asymptotic efficiency of the procedure is also studied. The case of von Mises' functionals is treated briefly at the end.     

排序集抽样下指数分布的产品可靠度研究

为了提高指数分布产品可靠度的估计效率,研究了基于排序集抽样方法的极大似然估计量(Maximum likelihood estimator,MLE),证明了新MLE具有存在性、唯一性和渐近正态性,并通过排序集样本的Fisher信息得到MLE的渐近方差。针对似然方程没有显式解的问题,利用部分期望法对MLE进行修正,并给出其具体表达式。渐近相对效率和模拟相对效率的研究结果表明:排序集抽样下MLE和修正MLE的估计效率都一致高于简单随机抽样下MLE。最后,将推荐方法应用到转移性肾癌的临床研究中。