Efficient estimators for the good family

We consider the problem of estimating the two parameters of the discrete Good distribution. We first show that the sufficient statistics for the parameters are the arithmetic and the geometric means. The maximum likelihood estimators (MLE's) of the parameters are obtained by solving numerically a system of equations involving the Lerch zeta function and the sufficient statistics. We find an expression for the asymptotic variance-covariance matrix of the MLE's, which can be evaluated numerically. We show that the probability mass function satisfies a simple recurrence equation linear in the two parameters, and propose the quadratic distance estimator (QDE) which can be computed with an ineratively reweighted least-squares algorithm. the QDE is easy to calculate and admits a simple expression for its asymptotic variance-covariance matrix. We compute this matrix for the MLE's and the QDE for various values of the parameters and see that the QDE has very high asymptotic efficiency. Finally, we present a numerical example.     

AN IMPROVED COMPOUND ESTIMATOR FOR ROBUST REGRESSION

ABSTRACT

Advances in statistical computing software have led to a substantial increase in the use of ordinary least squares (OLS) regression models in the engineering and applied statistics communities. Empirical evidence suggests that data sets can routinely have 10% or more outliers in many processes. Unfortunately, these outliers typically will render the OLS parameter estimates useless. The OLS diagnostic quantities and graphical plots can reliably identify a few outliers; however, they significantly lose power with increasing dimension and number of outliers. Although there have been recent advances in the methods that detect multiple outliers, improvements are needed in regression estimators that can fit well in the presence of outliers. We introduce a robust regression estimator that performs well regardless of outlier quantity and configuration. Our studies show that the best available estimators are vulnerable when the outliers are extreme in the regressor space (high leverage). Our proposed compound estimator modifies recently published methods with an improved initial estimate and measure of leverage. Extensive performance evaluations indicate that the proposed estimator performs the best and consistently fits the bulk of the data when outliers are present. The estimator, implemented in standard software, provides researchers and practitioners a tool for the model-building process to protect against the severe impact from multiple outliers.     

A note on estimation in bernoulli trials with dependence

Finite sample properties of estimators for the parameters of a dependent Bernoulli process are investigated using Monte Carlo techniques. A ratio estimator is proposed for the dependence parameter of the model and is compared to the approximate maximum likelihood estimator given by Klotz. It is shown that both estimators have a downward bias that is extreme in certain cases and that samples well in excess of 200 may be necessary before the asymptotic theory can be applied.     

Marshall–Olkin Extended Lomax Distribution and Its Application to Censored Data

The problem of making statistical inference about θ =P(X > Y) has been under great investigation in the literature using simple random sampling (SRS) data. This problem arises naturally in the area of reliability for a system with strength X and stress Y. In this study, we will consider making statistical inference about θ using ranked set sampling (RSS) data. Several estimators are proposed to estimate θ using RSS. The properties of these estimators are investigated and compared with known estimators based on simple random sample (SRS) data. The proposed estimators based on RSS dominate those based on SRS. A motivated example using real data set is given to illustrate the computation of the newly suggested estimators.     

Accent on Teaching Materials

Estimation of covariance components in the multivariate random-effect model with nested covariance structure is discussed. There are two covariance matrices to be estimated, namely, the between-group and the within-group covariance matrices. These two covariance matrices are most often estimated by forming a multivariate analysis of variance and equating mean square matrices to their expectations. Such a procedure involves taking the difference between the between-group mean square and the within-group mean square matrices, and often produces an estimated between-group covariance matrix that is not nonnegative definite. We present estimators of the two covariance matrices that are always proper covariance matrices. The estimators are the restricted maximum likelihood estimators if the random effects are normally distributed. The estimation procedure is extended to more complicated models, including the twofold nested and the mixed-effect models. A numerical example is presented to illustrate the use of the estimation procedure.     

Asymptotically minimax non–parametric function estimation with positivity constraints I

The spatially inhomogeneous smoothness of the non-parametric density or regression-function to be estimated by non-parametric methods is often modelled by Besov- and Triebel-type smoothness constraints. For such problems, Donoho and Johnstone [D.L. Donoho and I.M. Johnstone, Minimax estimation via wavelet shrinkage. Ann. Stat. 26 (1998), pp. 879–921.], Delyon and Juditsky [B. Delyon and A. Juditsky, On minimax wavelet estimators, Appl. Comput. Harmon. Anal. 3 (1996), pp. 215–228.] studied minimax rates of convergence for wavelet estimators with thresholding, while Lepski et al. [O.V. Lepski, E. Mammen, and V.G. Spokoiny, Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimators with variable bandwidth selectors, Ann. Stat. 25 (1997), pp. 929–947.] proposed a variable bandwidth selection for kernel estimators that achieved optimal rates over Besov classes. However, a second challenge in many real applications of non-parametric curve estimation is that the function must be positive. Here, we show how to construct estimators under positivity constraints that satisfy these constraints and also achieve minimax rates over the appropriate smoothness class.     

On minimaxity of the normal precision matrix estimator of Krishnamoorthy and Gupta

We consider the orthogonally invariant estimation problem of the inverse of the scale matrix of Wishart distribution using Stein's loss (entropy loss). In this problem Krishnamoorthy and Gupta [2] Krishnamoorthy, K. and Gupta, A. K. (1989). Improved minimax estimation of a normal precision matrix. Canad. J. Statist., 17: 91–102. [Crossref], [Web of Science ®] , [Google Scholar] proposed an estimator and showed its good performance in a Monte Carlo simulation. They conjectured their estimator is minimax. Perron [3] Perron, F. (1997). On a conjecture of Krishnamoorthy and Gupta. J. Multivariate Anal., 62: 110–120.  [Google Scholar] proved its minimaxity for p?=?2. In this paper we prove it for p?=?3 by using a new method.     

Second-order asymptotic comparison of estimators under φ-divergence loss

Our main concern is on the second-order asymptotic optimality problem of estimators. The φ-divergence loss is used as a criterion for evaluating the performance of estimators. In the comparison problem of any two estimators, the condition that one estimator dominates another estimator under the φ-divergence risk is given by evaluating the second-order term in the difference between the risks. As a result, it is proved that the condition is characterized by a peculiar value of the φ-divergence loss, which is called the divergence-loss coefficient. Furthermore, it is shown that the comparison based on the φ-divergence loss does not correspond with that based on any standard loss functions including the mean squared error, the absolute loss and the 0-1 loss. In addition, a necessary and sufficient condition for an estimator to be second-order admissible is derived.