BIAS-CORRECTED MAXIMUM LIKELIHOOD ESTIMATION OF THE PARAMETERS OF THE WEIGHTED LINDLEY DISTRIBUTION

This report discusses the calculation of analytic second-order bias techniques for the maximum likelihood estimates (for short, MLEs) of the unknown parameters of the distribution in quality and reliability analysis. It is well-known that the MLEs are widely used to estimate the unknown parameters of the probability distributions due to their various desirable properties; for example, the MLEs are asymptotically unbiased, consistent, and asymptotically normal. However, many of these properties depend on an extremely large sample sizes. Those properties, such as unbiasedness, may not be valid for small or even moderate sample sizes, which are more practical in real data applications. Therefore, some bias-corrected techniques for the MLEs are desired in practice, especially when the sample size is small. Two commonly used popular techniques to reduce the bias of the MLEs, are ‘preventive’ and ‘corrective’ approaches. They both can reduce the bias of the MLEs to order O(n−2), whereas the ‘preventive’ approach does not have an explicit closed form expression. Consequently, we mainly focus on the ‘corrective’ approach in this report. To illustrate the importance of the bias-correction in practice, we apply the bias-corrected method to two popular lifetime distributions: the inverse Lindley distribution and the weighted Lindley distribution. Numerical studies based on the two distributions show that the considered bias-corrected technique is highly recommended over other commonly used estimators without bias-correction. Therefore, special attention should be paid when we estimate the unknown parameters of the probability distributions under the scenario in which the sample size is small or moderate.

RATE OF CONVERGENCE OF MAXIMUM LIKELIHOOD ESTIMATORS UNDER RELAXED SMOOTHNESS CONDITIONS ON THE LIKELIHOOD FUNCTION

This report reviews literature on the rate of convergence of maximum likelihood estimators and establishes a Central Limit Theorem, which yields an O(1/sqrt(n)) rate of convergence of the maximum likelihood estimator under somewhat relaxed smoothness conditions. These conditions include the existence of a one-sided derivative in θ of the pdf, compared to up to three that are classically required. A verification through simulation is included in the end of the report.

Estimation of the degree of polarization through computational sensing

The degree of polarization of a refected field from active laser illumination can be used for object identifcation and classifcation. The goal of this study is to investigate methods for estimating the degree of polarization for refected fields with active laser illumination, which involves the measurement and processing of two orthogonal field components (complex amplitudes), two orthogonal intensity components, and the total field intensity. We propose to replace interferometric optical apparatuses with a computational approach for estimating the degree of polarization from two orthogonal intensity data and total intensity data. Cramer-Rao bounds for each of the three sensing modalities with various noise models are computed. Algebraic estimators and maximum-likelihood (ML) estimators are proposed. Active-set algorithm and expectation-maximization (EM) algorithm are used to compute ML estimates. The performances of the estimators are compared with each other and with their corresponding Cramer-Rao bounds. Estimators for four-channel polarimeter (intensity interferometer) sensing have a better performance than orthogonal intensities estimators and total intensity estimators. Processing the four intensities data from polarimeter, however, requires complicated optical devices, alignment, and four CCD detectors. It only requires one or two detectors and a computer to process orthogonal intensities data and total intensity data, and the bounds and estimator performances demonstrate that reasonable estimates may still be obtained from orthogonal intensities or total intensity data. Computational sensing is a promising way to estimate the degree of polarization.