Folded overlapping variance estimators for simulation

We propose and analyze a new class of estimators for the variance parameter of a steady-state simulation output process. The new estimators are computed by averaging individual estimators from “folded” standardized time series based on overlapping batches composed of consecutive observations. The folding transformation on each batch can be applied more than once to produce an entire set of estimators. We establish the limiting distributions of the proposed estimators as the sample size tends to infinity while the ratio of the sample size to the batch size remains constant. We give analytical and Monte Carlo results showing that, compared to their counterparts computed from nonoverlapping batches, the new estimators have roughly the same bias but smaller variance. In addition, these estimators can be computed with order-of-sample-size work.     

On exponential rates of likelihood ratio estimators for location parameters

In this paper the exponential rates, bounds, and local exponential rates for likelihood ratio estimators are studied. Under certain regularity conditions, a family of likelihood ratio estimators is shown to be admissible in exponential rate. It is also shown that the maximum likelihood estimator is the limit of this family of estimators.     

指数分布抽样基本定理及在三参数一般指数分布参数估计中的应用

讨论三参数一般指数分布的参数估计,首先讨论了三参数一般指数分布参数的最大似然估计的求解问题,当其中参数α=1时,应用指数分布抽样基本定理,得到了三参数一般指数分布其它参数的一致最小方差无偏估计;并且由此给出求解三参数一般指数分布参数最大似然估计的迭代方法,得到了三参数一般指数分布参数最大似然估计的近似值,给出了模拟结果以说明迭代方法的收敛性;并以相关文献的观察数据作为样本,得到了三参数一般指数分布的参数估计,从而说明了迭代方法的有效性.     

Limit theorems for maximum likelihood estimators in the Curie-Weiss-Potts model

The Curie-Weiss-Potts model, a model in statistical mechanics, is parametrized by the inverse temperature β and the external magnetic field h. This paper studies the asymptotic behavior of the maximum likelihood estimator of the parameter β when h = 0 and the asymptotic behavior of the maximum likelihood estimator of the parameter h when β is known and the true value of h is 0. The limits of these maximum likelihood estimators reflect the phase transition in the model; i.e., different limits depending on whether β < β_c, β = β_c or β > β_c, where β_c ε (0, ∞) is the critical inverse temperature of the model.     

Asymptotic risk behavior of mean vector and variance estimators and the problem of positive normal mean

Asymptotic risk behavior of estimators of the unknow variance and of the unknown mean vector in a multivariate normal distribution is considered for a general loss. It is shown that in both problems this characteristic is related to the risk in an estimation problem of a positive normal mean under quadratic loss function. A curious property of the Brewster-Zidek variance estimator of the normal variance is also noticed.Research supported by NSF Grant DMS 9000999 and by Alexander von Humboldt Foundation Senior Distinguished Scientist Award.University of Münster     

Empirical likelihood estimators for the error distribution in nonparametric regression models

The aim of this paper is to show that existing estimators for the error distribution in non-parametric regression models can be improved when additional information about the distribution is included by the empirical likelihood method. The weak convergence of the resulting new estimator to a Gaussian process is shown and the performance is investigated by comparison of asymptotic mean squared errors and by means of a simulation study.      

A projection method of estimation for a subfamily of exponential families

Summary This paper is concerned with estimation for a subfamily of exponential-type, which is a parametric model with sufficient statistics. The family is associated with a surface in the domain of a sufficient statistic. A new estimator, termed a projection estimator, is introduced. The key idea of its derivation is to look for a one-to-one transformation of the sufficient statistic so that the subfamily can be associated with a flat subset in the transformed domain. The estimator is defined by the orthogonal projection of the transformed statistic onto the flat surface. Here the orthogonality is introduced by the inverse of the estimated variance matrix of the statistic on the analogy of Mahalanobis's notion (1936,Proc. Nat. Inst. Sci. Ind.,2, 49–55). Thus the projection estimator has an explicit representation with no iterations. On the other hand, the MLE and classical estimators have to be sought as numerical solutions by some algorithm with a choice of an initial value and a stopping rule. It is shown that the projection estimator is first-order efficient. The second-order property is also discussed. Some examples are presented to show the utility of the estimator.     

Gaining efficiency via weighted estimators for multivariate failure time data

Multivariate failure time data arise frequently in survival analysis. A commonly used technique is the working independence estimator for marginal hazard models. Two natural questions are how to improve the efficiency of the working independence estimator and how to identify the situations under which such an estimator has high statistical efficiency. In this paper, three weighted estimators are proposed based on three different optimal criteria in terms of the asymptotic covariance of weighted estimators. Simplified close-form solutions are found, which always outperform the working independence estimator. We also prove that the working independence estimator has high statistical efficiency, when asymptotic covariance of derivatives of partial log-likelihood functions is nearly exchangeable or diagonal. Simulations are conducted to compare the performance of the weighted estimator and working independence estimator. A data set from Busselton population health surveys is analyzed using the proposed estimators. This work was supported by National Natural Science Foundation of China (Grant No. 10628104), Fan was also supported by National Institutes of Health (Grant No. R01-GM072611) and Natural Science Foundation (Grant No. DMS-0714554), Zhou was supported by National Natural Science Funds for Distinguisheel Young Scholar (Grant No. 70825004), National Natural Science Foundation of China (Grant Nos. 10731010, 10628104), the National Basic Research Program (Grant No. 2007CB814902), Creative Research Groups of China (Grant No. 10721101) and Leading Academic Disipline Program, the 10 ^th five year plan of 211 Project for Shanghai University of Finance and Economics (the 3 ^rd phase), Cai was supported by National Institutes of Health (Grant No. R01-HL57444)     

Characterization of the simple cubic multivariate exponential families

The Letac-Mora class of real cubic natural exponential families has been characterized by a property of 2-orthogonality of an associated sequence of polynomials (see [G. Letac, M. Mora, Natural real exponential families with cubic variance functions, Ann. Statist. 18 (1990) 1-37; A. Hassairi, M. Zarai, Characterization of the cubic exponential families by orthogonality of polynomials, Ann. Probab. 32 (2004) 2463-2476]). The present paper introduces a notion of transorthogonality for a sequence of polynomial on Rd to extend the characterization to the multivariate version of the Letac-Mora class of real natural exponential families.     

Methods of constructing characterizations by constancy of regression on the sample mean and related problems for NEF’s

Characterizations of a distribution by zero (or constant) regression properties of arbitrary degree polynomial statistics on the sample mean are discussed. Various practical steps collected from the relevant literature are put together in this framework into a comprehensive guideline for constructing such characterizations. Applications are provided for natural exponential families (NEF’s). In particular, two reciprocal NEF’s associated with the continuous time symmetric Bernoulli random walk are characterized using this guideline. Moreover, a class of infinitely divisible NEF’s having some polynomial variance function structure is discussed in this framework.      

A class of infinitely divisible variance functions with an application to the polynomial case

Let be a natural exponential family on ??? with variance function (V, Ω). Here, Ω is the mean domain of and V is its variance expressed in terms of the mean μ ε Ω. In this note we prove the following result. Consider an open interval Ω = (0, b), 0 < b ∞, and a positive real analytic function V on Ω. If V² is absolutely monotone on [0, b) and V has the form μt(μ), where 1 and t is real analytic in a neighborhood of zero, then there exits an infinitely divisible natural exponential family with variance function (V, Ω). We illustrate this result with several examples of general nature.     

Asymptotic equivalence of the jackknife and infinitesimal jackknife variance estimators for some smooth statistics

The jackknife variance estimator and the infinitesimal jackknife variance estimator are shown to be asymptotically equivalent if the functional of interest is a smooth function of the mean or a trimmed L-statistic with Hölder continuous weight function.     

A Markov Chain Monte Carlo comparison of variance estimators for the sampling of particulate mixtures

During the sampling of particulate mixtures, samples taken are analyzed for their mass concentration, which generally has non‐zero sample‐to‐sample variance. Bias, variance, and mean squared error (MSE) of a number of variance estimators, derived by Geelhoed, were studied in this article. The Monte Carlo simulation was applied using an observable first‐order Markov Chain with transition probabilities that served as a model for the sample drawing process. Because the bias and variance of a variance estimator could depend on the specific circumstances under which it is applied, Monte Carlo simulation was performed for a wide range of practically relevant scenarios. Using the ‘smallest mean squared error’ as a criterion, an adaptation of an estimator based on a first‐order Taylor linearization of the sample concentration is the best. An estimator based on the Horvitz–Thompson estimator is not practically applicable because of the potentially high MSE for the cases studied. The results indicate that the Poisson estimator leads to a biased estimator for the variance of fundamental sampling error (up to 428% absolute value of relative bias) in case of low levels of grouping and segregation. The uncertainty of the results obtained by the simulations was also addressed and it was found that the results were not significantly affected. The potentials of a recently described other approach are discussed for extending the first‐order Markov Chain described here to account also for higher levels of grouping and segregation. Copyright © 2013 John Wiley & Sons, Ltd.     

Precise asymptotics of error variance estimator in partially linear models

In this paper, we focus our attention on the precise asymptotics of error variance estimator in partially linear regression models, y _i = x _i ^τ β + g(t _i ) + ε _i , 1 ≤ i ≤ n, {ε _i , i = 1, ⋯ n} are i.i.d random errors with mean 0 and positive finite variance σ ². Following the ideas of Allan Gut and Aurel Spătaru^[7,8] and Zhang^[21], on precise asymptotics in the Baum-Katz and Davis laws of large numbers and precise rate in laws of the iterated logarithm, respectively, and subject to some regular conditions, we obtain the corresponding results in partially linear regression models.      

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.     

Generalized Bayes minimax estimators of the mean of multivariate normal distribution with unknown variance

We construct a broad class of generalized Bayes minimax estimators of the mean of a multivariate normal distribution with covariance equal to σ²I_p, with σ² unknown, and under the invariant loss δ(X)−θ²/σ². Examples that illustrate the theory are given. Most notably it is shown that a hierarchical version of the multivariate Student-t prior yields a Bayes minimax estimate.     

Minimum divergence estimators,maximum likelihood and exponential families

The dual representation formula of the divergence between two distributions in a parametric model is presented. Resulting estimators do not make use of any grouping or smoothing. For smooth divergences they all coincide with the MLE on any regular exponential family.     

A new proof of admissibility of tests in the multivariate analysis of variance

A new proof of admissibility of tests in MANOVA is given using Stein's theorem [7]. The convexity condition of Stein's theorem is proved directly by means of majorization rather than by the supporting hyperplane approach. This makes the geometrical meaning of the admissibility result clearer.     

Asymptotic variance estimation in multivariate distributions

A version of an asymptotic estimation problem of the unknown variance in a multivariate location-scale parameter family is studied under a general loss function. The asymptotic inadmissibility of the traditional estimator is established. In a particular case we derive an admissible improvement on this estimator.     

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.