EMPIRICAL BAYES ESTIMATION WITH NON-IDENTICAL COMPONENTS. CONTINUOUS CASE.

In this paper a variant of the standard empirical Bayes estimation problem is considered where the component problems in the sequence are not identical in that the conditional distribution of the observations may vary with the component problems. Let {(Θ_n, X_n)} be a sequence of independent random vectors where Θ_n? G and, given Θ_n=Θ_n, X_n -P_Θ,m(n) with {m(n)} a sequence of positive integers where m(n)≤m? < ∞ for all n. With P_Θ,m in a continuous exponential family of distributions, asymptotically optimal empirical Bayes procedures are exhibited which depend on uniform approximations of certain functions on compact sets by polynomials in e^Θ. Such approximations have been explicitly calculated in the case of normal and gamma families. In the case of normal families, asymptotically optimal linear empirical Bayes estimators in the class of all linear estimators are derived and shown to have rate O(n^-1/2).     

Estimation of the location of a 0-type or ∞-type singularity by Poisson observations

We consider an inhomogeneous Poisson process X on [0, T]. The intensity function of X is supposed to be strictly positive and smooth on [0, T] except at the point θ, in which it has either a 0-type singularity (tends to 0 like |x| ^p , p∈(0, 1)), or an ∞-type singularity (tends to ∞ like |x| ^p , p∈(?1, 0)). We suppose that we know the shape of the intensity function, but not the location of the singularity. We consider the problem of estimation of this location (shift) parameter θ based on n observations of the process X. We study the Bayesian estimators and, in the case p>0, the maximum-likelihood estimator. We show that these estimators are consistent, their rate of convergence is n ^1/(p+1), they have different limit distributions, and the Bayesian estimators are asymptotically efficient.     

Approximations for marginal densities of M-estimators

We develop a saddle-point approximation for the marginal density of a real-valued function p(), where is a general M-estimator of a p-dimensional parameter, that is, the solution of the system {n^-1_ljl (Y_l,) = 0}_j=1,…,p. The approximation is applied to several regression problems and yields very good accuracy for small samples. This enables us to compare different classes of estimators according to their finite-sample properties and to determine when asymptotic approximations are useful in practice.     

Fast accelerated failure time modeling for case-cohort data

Semiparametric accelerated failure time (AFT) models directly relate the expected failure times to covariates and are a useful alternative to models that work on the hazard function or the survival function. For case-cohort data, much less development has been done with AFT models. In addition to the missing covariates outside of the sub-cohort in controls, challenges from AFT model inferences with full cohort are retained. The regression parameter estimator is hard to compute because the most widely used rank-based estimating equations are not smooth. Further, its variance depends on the unspecified error distribution, and most methods rely on computationally intensive bootstrap to estimate it. We propose fast rank-based inference procedures for AFT models, applying recent methodological advances to the context of case-cohort data. Parameters are estimated with an induced smoothing approach that smooths the estimating functions and facilitates the numerical solution. Variance estimators are obtained through efficient resampling methods for nonsmooth estimating functions that avoids full blown bootstrap. Simulation studies suggest that the recommended procedure provides fast and valid inferences among several competing procedures. Application to a tumor study demonstrates the utility of the proposed method in routine data analysis.     

Penalized least-squares estimation for regression coefficients in high-dimensional partially linear models

We consider a partially linear model with diverging number of groups of parameters in the parametric component. The variable selection and estimation of regression coefficients are achieved simultaneously by using the suitable penalty function for covariates in the parametric component. An MM-type algorithm for estimating parameters without inverting a high-dimensional matrix is proposed. The consistency and sparsity of penalized least-squares estimators of regression coefficients are discussed under the setting of some nonzero regression coefficients with very small values. It is found that the root p_n/n-consistency and sparsity of the penalized least-squares estimators of regression coefficients cannot be given consideration simultaneously when the number of nonzero regression coefficients with very small values is unknown, where p_n and n, respectively, denote the number of regression coefficients and sample size. The finite sample behaviors of penalized least-squares estimators of regression coefficients and the performance of the proposed algorithm are studied by simulation studies and a real data example.     

Non-parametric recursive estimates of a probability density function and its derivatives

Recursive estimates of a probability density function (pdf) are known. This paper presents recursive estimates of a derivative of any desired order of a pdf. Let f be a pdf on the real line and p?0 be any desired integer. Based on a random sample of size n from f, estimators f^(p)_n of f^(p), the pth order derivatives of f, are exhibited. These estimators are of the form $\text{n}{}^{\mathrm{?1}}\text{∑n}\text{j=1}\text{δ}{}_{\mathrm{jp}}$, where δ_jp depends only on p and the jth observation in the sample, and hence can be computed recursively as the sample size increases. These estimators are shown to be asymptotically unbiased, mean square consistent and strongly consistent, both at a point and uniformly on the real line. For pointwise properties, the conditions on f^(p) have been weakened with a little stronger assumption on the kernel function.     

Higher-order approximations for pitman estimators and for optimal compromise estimators

Laplace approximations for the Pitman estimators of location or scale parameters, including terms O(n^?1), are obtained. The resulting expressions involve the maximum-likelihood estimate and the derivatives of the log-likelihood function up to order 3. The results can be used to refine the approximations for the optimal compromise estimators for location parameters considered by Easton (1991). Some applications and Monte Carlo simulations are discussed.     

Another look at linear programming for feature selection via methods of regularization

We consider statistical procedures for feature selection defined by a family of regularization problems with convex piecewise linear loss functions and penalties of l ₁ nature. Many known statistical procedures (e.g. quantile regression and support vector machines with l ₁-norm penalty) are subsumed under this category. Computationally, the regularization problems are linear programming (LP) problems indexed by a single parameter, which are known as ‘parametric cost LP’ or ‘parametric right-hand-side LP’ in the optimization theory. Exploiting the connection with the LP theory, we lay out general algorithms, namely, the simplex algorithm and its variant for generating regularized solution paths for the feature selection problems. The significance of such algorithms is that they allow a complete exploration of the model space along the paths and provide a broad view of persistent features in the data. The implications of the general path-finding algorithms are outlined for several statistical procedures, and they are illustrated with numerical examples.     

On the improved estimation of a function of the scale parameter of an exponential distribution based on doubly censored sample

In the present article, we have studied the estimation of entropy, that is, a function of scale parameter ln⁡σ of an exponential distribution based on doubly censored sample when the location parameter is restricted to positive real line. The estimation problem is studied under a general class of bowl-shaped non monotone location invariant loss functions. It is established that the best affine equivariant estimator (BAEE) is inadmissible by deriving an improved estimator. This estimator is non-smooth. Further, we have obtained a smooth improved estimator. A class of estimators is considered and sufficient conditions are derived under which these estimators improve upon the BAEE. In particular, using these results we have obtained the improved estimators for the squared error and the linex loss functions. Finally, we have compared the risk performance of the proposed estimators numerically. One data analysis has been performed for illustrative purposes.     

On minimum Lp-distance estimation for inhomogeneous Poisson processes

ABSTRACT

We consider the problem of parameter estimation by the observations of the inhomogeneous Poisson processes. We suppose that the intensity function of these processes is a smooth function of the unknown parameter and as a method of estimation we take the minimum distance approach. We are interested by the behavior of estimators in non Hilbertian situation and we define the minimum distance estimation (MDE) with the help of the L^p metrics. We show that (under regularity conditions) the MDE is consistent and we describe its limit distribution.     

Weak Convergence of the Regularization Path in Penalized M‐Estimation

Abstract. We consider a function defined as the pointwise minimization of a doubly index random process. We are interested in the weak convergence of the minimizer in the space of bounded functions. Such convergence results can be applied in the context of penalized M‐estimation, that is, when the random process to minimize is expressed as a goodness‐of‐fit term plus a penalty term multiplied by a penalty weight. This weight is called the regularization parameter and the minimizing function the regularization path. The regularization path can be seen as a collection of estimators indexed by the regularization parameter. We obtain a consistency result and a central limit theorem for the regularization path in a functional sense. Various examples are provided, including the ?¹‐regularization path for general linear models, the ?¹‐ or ?²‐regularization path of the least absolute deviation regression and the Akaike information criterion.     

Bayes minimax estimation of a bernoulli p in a restricted parameter space

In many estimation problems the parameter of interest is known,a priori, to belong to a proper subspace of the natural parameter space. Although useful in practice this type of additional information can lead to surprising theoretical difficulties. In this paper the problem of minimax estimation of a Bernoulli pwhen pis restricted to a symmetric subinterval of the natural parameter space is considered. For the sample sizes n = 1,2,3, and 4 least favorable priors with finite support are provided and the corresponding Bayes estimators are shown to be minimax. For n = 5 and 6 the usual constant risk minimax estimator is shown to be the Bayes minimax estimator corresponding to a least favorable prior with finite support, provided the restriction on the parameter space is not too tight.     

Asymptotic behaviour of M‐estimators in AR(p) models under nonstandard conditions

The authors derive the limiting distribution of M‐estimators in AR(p) models under nonstandard conditions, allowing for discontinuities in score and density functions. Unlike usual regularity assumptions, these conditions are satisfied in the context of L₁‐estimation and autoregression quantiles. The asymptotic distributions of the resulting estimators, however, are not generally Gaussian. Moreover, their bootstrap approximations are consistent along very specific sequences of bootstrap sample sizes only.     

Bias reduction of the maximum-likelihood estimator for a conditional Gaussian MA(1) model

In this paper, we consider an estimation for the unknown parameters of a conditional Gaussian MA(1) model. In the majority of cases, a maximum-likelihood estimator is chosen because the estimator is consistent. However, for small sample sizes the error is large, because the estimator has a bias of O(n^{? 1}). Therefore, we provide a bias of O(n^{? 1}) for the maximum-likelihood estimator for the conditional Gaussian MA(1) model. Moreover, we propose new estimators for the unknown parameters of the conditional Gaussian MA(1) model based on the bias of O(n^{? 1}). We investigate the properties of the bias, as well as the asymptotical variance of the maximum-likelihood estimators for the unknown parameters, by performing some simulations. Finally, we demonstrate the validity of the new estimators through this simulation study.     

Bias-corrected estimators for dispersion models with dispersion covariates

In this paper we discuss bias-corrected estimators for the regression and the dispersion parameters in an extended class of dispersion models (Jørgensen, 1997b). This class extends the regular dispersion models by letting the dispersion parameter vary throughout the observations, and contains the dispersion models as particular case. General formulae for the O(n⁻¹) bias are obtained explicitly in dispersion models with dispersion covariates, which generalize previous results obtained by Botter and Cordeiro (1998), Cordeiro and McCullagh (1991), Cordeiro and Vasconcellos (1999), and Paula (1992). The practical use of the formulae is that we can derive closed-form expressions for the O(n⁻¹) biases of the maximum likelihood estimators of the regression and dispersion parameters when the information matrix has a closed-form. Various expressions for the O(n⁻¹) biases are given for special models. The formulae have advantages for numerical purposes because they require only a supplementary weighted linear regression. We also compare these bias-corrected estimators with two different estimators which are also bias-free to order O(n⁻¹) that are based on bootstrap methods. These estimators are compared by simulation.     

Finite sample properties of ml and reml estimators in time series regression models with long memory noise

Finite sample properties of ML and REML estimators in time series regression models with fractional ARIMA noise are examined. In particular, theoretical approximations for bias of ML and REML estimators of the noise parameters are developed and their accuracy is assessed through simulations. The impact of noise parameter estimation on performance of t -statistics and likelihood ratio statistics for testing regression parameters is also investigated.     

Super efficient frequency estimation

In this paper we propose a modified Newton-Raphson method to obtain super efficient estimators of the frequencies of a sinusoidal signal in presence of stationary noise. It is observed that if we start from an initial estimator with convergence rate O_p(n⁻¹) and use Newton-Raphson algorithm with proper step factor modification, then it produces super efficient frequency estimator in the sense that its asymptotic variance is lower than the asymptotic variance of the corresponding least squares estimator. The proposed frequency estimator is consistent and it has the same rate of convergence, namely O_p(n^−3/2), as the least squares estimator. Monte Carlo simulations are performed to observe the performance of the proposed estimator for different sample sizes and for different models. The results are quite satisfactory. One real data set has been analyzed for illustrative purpose.     

A note on estimating parameters of partial prior information

In an empirical Bayes model, examples for estimators of parameters of partial prior information are given. A typical application is the estimation of a probability ?(p≤a),p the “fraction defective”, which is used as prior information in Quality Control.     

Asymptotically optimum estimation of a probability in inverse binomial sampling under general loss functions

The optimum quality that can be asymptotically achieved in the estimation of a probability p using inverse binomial sampling is addressed. A general definition of quality is used in terms of the risk associated with a loss function that satisfies certain assumptions. It is shown that the limit superior of the risk for p asymptotically small has a minimum over all (possibly randomized) estimators. This minimum is achieved by certain non-randomized estimators. The model includes commonly used quality criteria as particular cases. Applications to the non-asymptotic regime are discussed considering specific loss functions, for which minimax estimators are derived.     

A comparison of estimators for the proportion in the tail of a normal distribution

Three estimators of the proportion in a tail of the normal distribution are compared using the criteria of mean squared error and mean absolute error. The estimators that we compare are the maximum likelihood estimator, the minimum variance unbiased estimator, and an intuitive estimator that is frequently used in practice. The intuitive estimator is similar to the MLE but uses the usual unbiased estimator of σ² rather than the MLE of σ². We show that the intuitive estimator has low efficiency, and for this reason it is not recommended. For very smallp and for largep the MVUE has the highest efficiency. The MLE is best for moderate values ofp.