The 3/2th and 2nd order asymptotic efficiency of maximum probability estimators in non-regular cases

In this paper we consider the estimation problem on independent and identically distributed observations from a location parameter family generated by a density which is positive and symmetric on a finite interval, with a jump and a nonnegative right differential coefficient at the left endpoit. It is shown that the maximum probability estimator (MPE) is 3/2th order two-sided asymptotically efficient at a point in the sense that it has the most concentration probability around the true parameter at the point in the class of 3/2th order asymptotically median unbiased (AMU) estimators only when the right differential coefficient vanishes at the left endpoint. The second order upper bound for the concentration probability of second order AMU estimators is also given. Further, it is shown that the MPE is second order two-sided asymptotically efficient at a point in the above case only.Research supported by University of Tsukuba Project Research.     

On the lower bound in second order estimation for Poisson processes: Asymptotic efficiency

In the estimation problem of the mean function of an inhomogeneous Poisson process there is a class of kernel type estimators that are asymptotically efficient alongside with the empirical mean function. We start by describing such a class of estimators which we call first order efficient estimators. To choose the best one among them we prove a lower bound that compares the second order term of the mean integrated square error of all estimators. The proof is carried out under the assumption on the mean function Λ(·) that Λ(τ) = S, where S is a known positive number. In the end, we discuss the possibility of the construction of an estimator which attains this lower bound, thus, is asymptotically second order efficient.     

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

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.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 106–111, 1988.     

Asymptotic optimality of the generalized bayes estimator in multiparameter cases

The higher order asymptotic efficiency of the generalized Bayes estimator is discussed in multiparameter cases. For all symmetric loss functions, the generalized Bayes estimator is second order asymptotically efficient in the classA ₂ of the all second order asymptotically median unbiased (AMU) estimators and third order asymptotically efficient in the restricted classD of estimators.     

Second order asymptotics in nonlinear regression

It is a well known part of statistical knowledge that first order asymptotically efficient procedures can be misleading for moderate sample sizes. Usually this is demonstrated for some popular special cases including numerical comparisons. Typically the situation is worse if nuisance parameters are present. In this paper we give second order asymptotically efficient tests, confidence regions, and estimators for the nonlinear regression model which are based on the least-squares estimator and the residual sum of squares.     

On the asymptotic efficiency of estimators in an autoregressive process

Summary Let {X _t } be defined recursively byX _t =θX _t−1+U _t (t=1,2, ...), whereX ₀=0 and {U _t } is a sequence of independent identically distributed real random variables having a density functionf with mean 0 and varianceσ ². We assume that |θ|<1. In the present paper we obtain the bound of the asymptotic distributions of asymptotically median unbiased (AMU) estimators of θ and the sufficient condition that an AMU estimator be asymptotically efficient in the sense that its distribution attains the above bound. It is also shown that the least squares estimator of θ is asymptotically efficient if and only iff is a normal density function. University of Electro-Communications     

Second order asymptotic optimality of estimators for a density with finite cusps

We consider i.i.d. samples from a continuous density with finite cusps. Then we obtain the bound for the second order asymptotic distribution of all asymptotically median unbiased estimators. Further we get the second order asymptotic distribution of a bias-adjusted maximum likelihood estimator, and we see that it is not generally second order asymptotically efficient.     

A third order optimum property of the ML estimator in a linear functional relationship model and simultaneous equation system in econometrics

Summary The maximum likelihood (ML) estimator and its modification in the linear functional relationship model with incidental parameters are shown to be third-order asymptotically efficient among a class of almost median-unbiased and almost mean-unbiased estimators, respectively, in the large sample sense. This means that the limited information maximum likelihood (LIML) estimator in the simultaneous equation system is third-order asymptotically efficient when the number of excluded exogenous variables in a particular structural equation is growing along with the sample size. It implies that the LIML estimator has an optimum property when the system of structural equations is large. The research was partly supported by National Science Foundation Grant SES 79-13976 at the Institute for Mathematical Studies in the Social Sciences, Stanford University and Grant-in-Aid 60301081 of the Ministry of Education, Science and Culture at the Faculty of Economics, University of Tokyo. This paper was originally written as a part of the author's Ph.D. dissertation submitted to Stanford University in August, 1981. Some details of the paper were deleted at the suggestion of the associate editor of this journal.     

Stein shrinkage and second-order efficiency for semiparametric estimation of the shift

The problem of estimating the shift (or, equivalently, the center of symmetry) of an unknown symmetric and periodic function f observed in Gaussian white noise is considered. Using the blockwise Stein method, a penalized profile likelihood with a data-driven penalization is introduced so that the estimator of the center of symmetry is defined as the maximizer of the penalized profile likelihood. This estimator has the advantage of being independent of the functional class to which the signal f is assumed to belong and, furthermore, is shown to be semiparametrically adaptive and efficient. Moreover, the second-order term of the risk expansion of the proposed estimator is proved to behave at least as well as the second-order term of the risk of the best possible estimator using monotone smoothing filter. Under mild assumptions, this estimator is shown to be second-order minimax sharp adaptive over the whole scale of Sobolev balls with smoothness β > 1. Thus, these results extend those of [10], where second-order asymptotic minimaxity is proved for an estimator depending on the functional class containing f and β ≥ 2 is required.      

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.     

Efficient simulation of tail probabilities of sums of correlated lognormals

We consider the problem of efficient estimation of tail probabilities of sums of correlated lognormals via simulation. This problem is motivated by the tail analysis of portfolios of assets driven by correlated Black-Scholes models. We propose two estimators that can be rigorously shown to be efficient as the tail probability of interest decreases to zero. The first estimator, based on importance sampling, involves a scaling of the whole covariance matrix and can be shown to be asymptotically optimal. A further study, based on the Cross-Entropy algorithm, is also performed in order to adaptively optimize the scaling parameter of the covariance. The second estimator decomposes the probability of interest in two contributions and takes advantage of the fact that large deviations for a sum of correlated lognormals are (asymptotically) caused by the largest increment. Importance sampling is then applied to each of these contributions to obtain a combined estimator with asymptotically vanishing relative error.     

On estimator efficiency in stochastic processes

It is shown, under mild regularity conditions on the random information matrix, that the maximum likelihood estimator is efficient in the sense of having asymptotically maximum probability of concentration about the true parameter value. In the case of a single parameter, the conditions are improvements of those used by Heyde (1978). The proof is based on the idea of maximum probability estimators introduced by Weiss and Wolfowitz (1967).     

A definition of estimator efficiency ink-parameter case

Summary The paper introduces a new definition of efficiency in the multiparameter case (θ₁,...,θ_k) when the variance-covariance matrix of the vector estimator (t ₁, ...t _k) exists. The definition is also applicable to the asymptotically unbiased estimators. The basic idea is that, as we want in general to estimate some function g(θ₁,...θ_k) of the parameters, efficiency of the vector estimator shall be defined as the smallest efficiency of the estimatorg(t ₁, ...t _k),g being regular. It is shown that this definition is asymptotically equivalent to the one obtained by any linear combination of the estimators, as it happens, naturally, for quantile estimation in the location-dispersion case. This efficiency is larger than Cramér efficiency which is, thus, not attained, apart from a very exceptional case. Finally, a lower bound for the asymptotic variance is obtained.     

On asymptotic efficiency and deficiency of some adaptive tests in the one-sample problem

In this paper, we consider adaptive tests for the one-sample problem and investigate the order of difference between the power of a given adaptive asymptotically efficient test and that of the most powerful test. Here adaptation means that the efficient score function of the test is estimated from the sample. A Fourier series estimator is used for the score function. Supported by the Russian Foundation for Fundamental Research (grant No. 93-01-01446). Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.     

Second order asymptotic comparison of estimators of a common parameter in the double exponential case

Summary The problem to estimate a common parameter for the pooled sample from the double exponential distributions is discussed in the presence of nuisance parameters. The maximum likelihood estimator, a weighted median, a weighted mean and others are asymptotically compared up to the second order, i.e. the ordern ^−1/2 with the asymptotic expansions of their distributions. University of Electro-communications     

Minimum disparity estimation for continuous models: Efficiency,distributions and robustness

A general class of minimum distance estimators for continuous models called minimum disparity estimators are introduced. The conventional technique is to minimize a distance between a kernel density estimator and the model density. A new approach is introduced here in which the model and the data are smoothed with the same kernel. This makes the methods consistent and asymptotically normal independently of the value of the smoothing parameter; convergence properties of the kernel density estimate are no longer necessary. All the minimum distance estimators considered are shown to be first order efficient provided the kernel is chosen appropriately. Different minimum disparity estimators are compared based on their characterizing residual adjustment function (RAF); this function shows that the robustness features of the estimators can be explained by the shrinkage of certain residuals towards zero. The value of the second derivative of theRAF at zero,A ₂, provides the trade-off between efficiency and robustness. The above properties are demonstrated both by theorems and by simulations.     

Efficient Density Estimation for Ergodic Diffusion Processes

The problem of asymptotically efficient estimation of the density of invariant measure of a diffusion process is considered. The efficient estimator is defined with the help of the minimax lower bound on the risk of all estimators. We show that the local–time and kernel–type estimators are asymptotically efficient for the loss functions with polynomial majorants. The asymptotic behavior of a wide class of unbiased estimators with the same limit variances is also discussed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Discretized likelihood methods—asymptotic properties of discretized likelihood estimators (DLE's)

Suppose thatX ₁,X ₂, ...,X _n , ... is a sequence of i.i.d. random variables with a densityf(x, θ). Letc _n be a maximum order of consistency. We consider a solution \(\hat \theta _n \) of the discretized likelihood equation $$\sum\limits_{i = 1}^n {\log f(X_i ,\hat \theta _n + rc_n^{ - 1} ) - } \sum\limits_{i = 1}^n {\log f(X_i ,\hat \theta _n ) = a_n (\hat \theta _n ,r)} $$ wherea _n (θ,r) is chosen so that \(\hat \theta _n \) is asymptotically median unbiased (AMU). Then the solution \(\hat \theta _n \) is called a discretized likelihood estimator (DLE). In this paper it is shown in comparison with DLE that a maximum likelihood estimator (MLE) is second order asymptotically efficient but not third order asymptotically efficient in the regular case. Further it is seen that the asymptotic efficiency (including higher order cases) may be systematically discussed by the discretized likelihood methods.     

纵向数据非参数混合效应模型的一个局部不变估计

非参数核回归方法近年来已被用于纵向数据的分析(Lin和Carroll,2000)．一个颇具争议性的问题是在非参数核回归中是否需要考虑纵向数据间的相关性．Lin和Carroll (2000)证明了基于独立性(即忽略相关性)的核估计在一类核GEE估计量中是(渐近)最有效的．基于混合效应模型方法作者提出了一个不同的核估计类,它自然而有效地结合了纵向数据的相关结构．估计量达到了与Lin和Carroll的估计量相同的渐近有效性,且在有限样本情形下表现更好．由此方法可以很容易地获得对于总体和个体的非参数曲线估计．所提出的估计量具有较好的统计性质,且实施方便,从而对实际工作者具有较大的吸引力．     

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This paper concerns with the estimation of a fixed effects panel data partially linear regression model with the idiosyncratic errors being an autoregressive process. For fixed effects short time series panel data, the commonly used autoregressive error structure fitting method will not result in a consistent estimator of the autoregressive coefficients. Here we propose an alternative estimation and show that the resulting estimator of the autoregressive coefficients is consistent and this method is workable for any order autoregressive error structure. Moreover, combining the B-spline approximation, profile least squares dummy variable (PLSDV) technique and consistently estimated the autoregressive error structure, we develop a weighted PLSDV estimator for the parametric component and a weighted B-spline series (BS) estimator for the nonparametric component. The weighted PLSDV estimator is shown to be asymptotically normal and more asymptotically efficient than the one which ignores the error autoregressive structure. In addition, this paper derives the asymptotic bias of the weighted BS estimator and establish its asymptotic normality as well. Simulation studies and an example of application are conducted to illustrate the finite sample performance of the proposed procedures.