Asymptotic properties of conditional maximum likelihood estimator in a certain exponential model

The conditional maximum likelihood estimator is suggested as an alternative to the maximum likelihood estimator and is favorable for an estimator of a dispersion parameter in the normal distribution, the inverse-Gaussian distribution, and so on. However, it is not clear whether the conditional maximum likelihood estimator is asymptotically efficient in general. Consider the case where it is asymptotically efficient and its asymptotic covariance depends only on an objective parameter in an exponential model. This remand implies that the exponential model possesses a certain parallel foliation. In this situation, this paper investigates asymptotic properties of the conditional maximum likelihood estimator and compares the conditional maximum likelihood estimator with the maximum likelihood estimator. We see that the bias of the former is more robust than that of the latter and that two estimators are very close, especially in the sense of bias-corrected version. The mean Pythagorean relation is also discussed.     

Corrected version of AIC for selecting multivariate normal linear regression models in a general nonnormal case

This paper deals with the bias reduction of Akaike information criterion (AIC) for selecting variables in multivariate normal linear regression models when the true distribution of observation is an unknown nonnormal distribution. We propose a corrected version of AIC which is partially constructed by the jackknife method and is adjusted to the exact unbiased estimator of the risk when the candidate model includes the true model. It is pointed out that the influence of nonnormality in the bias of our criterion is smaller than the ones in AIC and TIC. We verify that our criterion is better than the AIC, TIC and EIC by conducting numerical experiments.     

On the limiting spectral distribution of the covariance matrices of time-lagged processes

We consider two continuous-time Gaussian processes, one being partially correlated to a time-lagged version of the other. We first give the limiting spectral distribution for the covariance matrices of the increments of the processes when the span between two observations tends to zero. Then, we derive the limiting distribution of the eigenvalues of the sample covariance matrices. This result is obtained when the number of paths of the processes is asymptotically proportional to the number of observations for each single path. As an application, we use the second moment of this distribution together with auxiliary volatility and correlation estimates to construct an adaptive estimator of the time lag between the two processes. Finally, we provide an asymptotic theory for our estimation procedure.     

Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero

The asymptotic distribution of the quasi-maximum likelihood (QML) estimator is established for generalized autoregressive conditional heteroskedastic (GARCH) processes, when the true parameter may have zero coefficients. This asymptotic distribution is the projection of a normal vector distribution onto a convex cone. The results are derived under mild conditions. For an important subclass of models, no moment condition is imposed on the GARCH process. The main practical implication of these results concerns the estimation of overidentified GARCH models.     

An adjusted maximum likelihood method for solving small area estimation problems

For the well-known Fay-Herriot small area model, standard variance component estimation methods frequently produce zero estimates of the strictly positive model variance. As a consequence, an empirical best linear unbiased predictor of a small area mean, commonly used in small area estimation, could reduce to a simple regression estimator, which typically has an overshrinking problem. We propose an adjusted maximum likelihood estimator of the model variance that maximizes an adjusted likelihood defined as a product of the model variance and a standard likelihood (e.g., a profile or residual likelihood) function. The adjustment factor was suggested earlier by Carl Morris in the context of approximating a hierarchical Bayes solution where the hyperparameters, including the model variance, are assumed to follow a prior distribution. Interestingly, the proposed adjustment does not affect the mean squared error property of the model variance estimator or the corresponding empirical best linear unbiased predictors of the small area means in a higher order asymptotic sense. However, as demonstrated in our simulation study, the proposed adjustment has a considerable advantage in small sample inference, especially in estimating the shrinkage parameters and in constructing the parametric bootstrap prediction intervals of the small area means, which require the use of a strictly positive consistent model variance estimate.     

Sieve maximum likelihood estimation for doubly semiparametric zero-inflated Poisson models

For nonnegative measurements such as income or sick days, zero counts often have special status. Furthermore, the incidence of zero counts is often greater than expected for the Poisson model. This article considers a doubly semiparametric zero-inflated Poisson model to fit data of this type, which assumes two partially linear link functions in both the mean of the Poisson component and the probability of zero. We study a sieve maximum likelihood estimator for both the regression parameters and the nonparametric functions. We show, under routine conditions, that the estimators are strongly consistent. Moreover, the parameter estimators are asymptotically normal and first order efficient, while the nonparametric components achieve the optimal convergence rates. Simulation studies suggest that the extra flexibility inherent from the doubly semiparametric model is gained with little loss in statistical efficiency. We also illustrate our approach with a dataset from a public health study.     

UMVU estimation of the ratio of powers of normal generalized variances under correlation

We consider estimation of the ratio of arbitrary powers of two normal generalized variances based on two correlated random samples. First, the result of Iliopoulos [Decision theoretic estimation of the ratio of variances in a bivariate normal distribution, Ann. Inst. Statist. Math. 53 (2001) 436-446] on UMVU estimation of the ratio of variances in a bivariate normal distribution is extended to the case of the ratio of any powers of the two variances. Motivated by these estimators’ forms we derive the UMVU estimator in the multivariate case. We show that it is proportional to the ratio of the corresponding powers of the two sample generalized variances multiplied by a function of the sample canonical correlations. The mean squared errors of the derived UMVU estimator and the maximum likelihood estimator are compared via simulation for some special cases.     

A robust and efficient adaptive reweighted estimator of multivariate location and scatter

This article proposes a reweighted estimator of multivariate location and scatter, with weights adaptively computed from the data. Its breakdown point and asymptotic behavior under elliptical distributions are established. This adaptive estimator is able to attain simultaneously the maximum possible breakdown point for affine equivariant estimators and full asymptotic efficiency at the multivariate normal distribution. For the special case of hard-rejection weights and the MCD as initial estimator, it is shown to be more efficient than its non-adaptive counterpart for a broad range of heavy-tailed elliptical distributions. A Monte Carlo study shows that the adaptive estimator is as robust as its non-adaptive relative for several types of bias-inducing contaminations, while it is remarkably more efficient under normality for sample sizes as small as 200.     

A family of estimators for multivariate kurtosis in a nonnormal linear regression model

In this paper, we propose a new estimator for a kurtosis in a multivariate nonnormal linear regression model. Usually, an estimator is constructed from an arithmetic mean of the second power of the squared sample Mahalanobis distances between observations and their estimated values. The estimator gives an underestimation and has a large bias, even if the sample size is not small. We replace this squared distance with a transformed squared norm of the Studentized residual using a monotonic increasing function. Our proposed estimator is defined by an arithmetic mean of the second power of these squared transformed squared norms with a correction term and a tuning parameter. The correction term adjusts our estimator to an unbiased estimator under normality, and the tuning parameter controls the sizes of the squared norms of the residuals. The family of our estimators includes estimators based on ordinary least squares and predicted residuals. We verify that the bias of our new estimator is smaller than usual by constructing numerical experiments.     

Improving on the mle of a bounded location parameter for spherical distributions

For the problem of estimating under squared error loss the location parameter of a p-variate spherically symmetric distribution where the location parameter lies in a ball of radius m, a general sufficient condition for an estimator to dominate the maximum likelihood estimator is obtained. Dominance results are then made explicit for the case of a multivariate student distribution with d degrees of freedom and, in particular, we show that the Bayes estimator with respect to a uniform prior on the boundary of the parameter space dominates the maximum likelihood estimator whenever and d?p. The sufficient condition matches the one obtained by Marchand and Perron (Ann. Statist. 29 (2001) 1078) in the normal case with identity covariance matrix. Furthermore, we derive an explicit class of estimators which, for , dominate the maximum likelihood estimator simultaneously for the normal distribution with identity covariance matrix and for all multivariate student distributions with d degrees of freedom, d?p. Finally, we obtain estimators which dominate the maximum likelihood estimator simultaneously for all distributions in the subclass of scale mixtures of normals for which the scaling random variable is bounded below by some positive constant with probability one.     

A new projection estimate for multivariate location with minimax bias

The maximum asymptotic bias of an estimator is a global robustness measure of its performance. The projection median estimator for multivariate location shows a remarkable behavior regarding asymptotic bias. In this paper we consider a modification of the projection median estimator which renders an estimate with better bias performance for point mass contaminations (the worst situation for the projection median estimator). Moreover, it achieves the lowest bound for an equivariant estimate for point mass contaminations.     

Stein's phenomenon in estimation of means restricted to a polyhedral convex cone

This paper treats the problem of estimating the restricted means of normal distributions with a known variance, where the means are restricted to a polyhedral convex cone which includes various restrictions such as positive orthant, simple order, tree order and umbrella order restrictions. In the context of the simultaneous estimation of the restricted means, it is of great interest to investigate decision-theoretic properties of the generalized Bayes estimator against the uniform prior distribution over the polyhedral convex cone. In this paper, the generalized Bayes estimator is shown to be minimax. It is also proved that it is admissible in the one- or two-dimensional case, but is improved on by a shrinkage estimator in the three- or more-dimensional case. This means that the so-called Stein phenomenon on the minimax generalized Bayes estimator can be extended to the case where the means are restricted to the polyhedral convex cone. The risk behaviors of the estimators are investigated through Monte Carlo simulation, and it is revealed that the shrinkage estimator has a substantial risk reduction.     

Robust parameter estimation with a small bias against heavy contamination

In this paper we consider robust parameter estimation based on a certain cross entropy and divergence. The robust estimate is defined as the minimizer of the empirically estimated cross entropy. It is shown that the robust estimate can be regarded as a kind of projection from the viewpoint of a Pythagorean relation based on the divergence. This property implies that the bias caused by outliers can become sufficiently small even in the case of heavy contamination. It is seen that the asymptotic variance of the robust estimator is naturally overweighted in proportion to the ratio of contamination. One may surmise that another form of cross entropy can present the same behavior as that discussed above. It can be proved under some conditions that no cross entropy can present the same behavior except for the cross entropy considered here and its monotone transformation.     

Constructing priors based on model size for nondecomposable Gaussian graphical models: A simulation based approach

A method for constructing priors is proposed that allows the off-diagonal elements of the concentration matrix of Gaussian data to be zero. The priors have the property that the marginal prior distribution of the number of nonzero off-diagonal elements of the concentration matrix (referred to below as model size) can be specified flexibly. The priors have normalizing constants for each model size, rather than for each model, giving a tractable number of normalizing constants that need to be estimated. The article shows how to estimate the normalizing constants using Markov chain Monte Carlo simulation and supersedes the method of Wong et al. (2003) [24] because it is more accurate and more general. The method is applied to two examples. The first is a mixture of constrained Wisharts. The second is from Wong et al. (2003) [24] and decomposes the concentration matrix into a function of partial correlations and conditional variances using a mixture distribution on the matrix of partial correlations. The approach detects structural zeros in the concentration matrix and estimates the covariance matrix parsimoniously if the concentration matrix is sparse.     

Boundary kernels for adaptive density estimators on regions with irregular boundaries

In some applications of kernel density estimation the data may have a highly non-uniform distribution and be confined to a compact region. Standard fixed bandwidth density estimates can struggle to cope with the spatially variable smoothing requirements, and will be subject to excessive bias at the boundary of the region. While adaptive kernel estimators can address the first of these issues, the study of boundary kernel methods has been restricted to the fixed bandwidth context. We propose a new linear boundary kernel which reduces the asymptotic order of the bias of an adaptive density estimator at the boundary, and is simple to implement even on an irregular boundary. The properties of this adaptive boundary kernel are examined theoretically. In particular, we demonstrate that the asymptotic performance of the density estimator is maintained when the adaptive bandwidth is defined in terms of a pilot estimate rather than the true underlying density. We examine the performance for finite sample sizes numerically through analysis of simulated and real data sets.     

Parametric tail copula estimation and model testing

Parametric models for tail copulas are being used for modeling tail dependence and maximum likelihood estimation is employed to estimate unknown parameters. However, two important questions seem unanswered in the literature: (1) What is the asymptotic distribution of the MLE and (2) how does one test the parametric model? In this paper, we answer these two questions in the case of a single parameter for ease of illustration. A simulation study is provided to investigate the finite sample performance of the proposed estimator and test.     

A robust estimator of multivariate location based on projection

This paper presents an estimator of location vector based on one-dimensional projection of high dimensional data. The properties of the new estimator including consistency ,asymptotic normality and robustness are discussed. It is proved that the estimator is not only stronglyconsistent and asymptotically normal but also with a breakdown point 1/2 and a bounded influence function.     

Bias of estimator of change point detected by a CUSUM procedure

For independent observations from a standard one-parameter exponential family, the estimator of change point after being detected by a CUSUM procedure is defined as the last zero point of the CUSUM process before the alarm time. By assuming that the change occurs far away from beginning and the control limit is large, an explicit form for the bias of estimator is derived conditioning on the change being detected. By further assuming that the change magnitude and its reference value approach zero at the same order, the local second order expansion of the bias is obtained for numerical evaluation. It is found that, surprisingly, even in the normal distribution case, the bias is non-zero when the change magnitude equals to its reference value, in contrast to the continuous time analog and the fixed sample size case. Numerical results show that the approximations are quite satisfactory.     

The multivariate least-trimmed squares estimator

In this paper we introduce the least-trimmed squares estimator for multivariate regression. We give three equivalent formulations of the estimator and obtain its breakdown point. A fast algorithm for its computation is proposed. We prove Fisher-consistency at the multivariate regression model with elliptically symmetric error distribution and derive the influence function. Simulations investigate the finite-sample efficiency and robustness of the estimator. To increase the efficiency of the estimator, we also consider a one-step reweighted estimator.     

Sensitivity of minimaxity and admissibility in the estimation of a positive normal mean

On the problem of estimating a positive normal mean with known variance, it is well known that one minimax admissible estimator is the generalized Bayes one with respect to the non-informative prior measure, the Lebesgue measure, restricted on the positive half-line. When the true variance is misspecified, however, it is shown that this estimator does not always retain minimaxity and admisssibility. In particular, it is almost surely inadmissible in the misspecification case.