Asymptotic properties for the moment estimators in Rayleigh distribution with two parameters

ABSTRACT

We consider the asymptotic properties for the moment estimators in Rayleigh distribution with two parameters. The law of the iterated logarithm for the estimators can be obtained. Moreover, we can give a simple proof of the asymptotic normality which has been obtained by Li and Li (2012) Li, Y.W., Li, M.H. (2012). Moment estimation of the parameters in Rayleigh distribution with two parameters. Commun. Stat.-Theor. Methods 41:2643–2660.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar].     

Negative moments of a modified power series distribution and bias of the maximum likelihood estimator

The class of Modified Power Series distributions (MPSD) containing Lagrangian Poisson (LPD) (Consul and Jain, 1973) and Lagrangian binomial distributions (LBD) (Jain and Consul, 1971) was studied by Gupta (1974). We investigate the problem of finding the negative momentsE[X^-r ], of displaced and decapitated Modified Power Series Distributions. We derive the relationship between rand (r-1) negative moments. The negative moments of the decapitated and displaced LPD are obtained. These results are, then, used to find the exact amount of bias in the ML estimators of the parameters in the LPD and the LBD. We have also given the variances of the ML estimator and the minimum variance unbiased estimator of the parameter in the LPD.     

On estimating a function of the mean of a multivariate normal distribution

The unique minimum variance of unbiased estimator is obtained for analysis functions of the mean of a multivariate normal distribution with either unknown covariance matrix or with covariance matrix of the form σ²v where σ² is unknown.     

Inference in flexible families of distributions with normal kernel

This paper addresses the inference problem for a flexible class of distributions with normal kernel known as skew-bimodal-normal family of distributions. We obtain posterior and predictive distributions assuming different prior specifications. We provide conditions for the existence of the maximum-likelihood estimators (MLE). An EM-type algorithm is built to compute them. As a by product, we obtain important results related to classical and Bayesian inferences for two special subclasses called bimodal-normal and skew-normal (SN) distribution families. We perform a Monte Carlo simulation study to analyse behaviour of the MLE and some Bayesian ones. Considering the frontier data previously studied in the literature, we use the skew-bimodal-normal (SBN) distribution for density estimation. For that data set, we conclude that the SBN model provides as good a fit as the one obtained using the location-scale SN model. Since the former is a more parsimonious model, such a result is shown to be more attractive.     

The Effect of Verification Bias in the Naïve Estimators of Accuracy of a Binary Diagnostic Test

The assessment of a binary diagnostic test requires a knowledge of the disease status of all the patients in the sample through the application of a gold standard. In practice, the gold standard is not always applied to all of the patients, which leads to the problem of partial verification of the disease. When the accuracy of the diagnostic test is assessed using only those patients whose disease status has been verified using the gold standard, the estimators obtained in this way, known as Naïve estimators, may be biased. In this study, we obtain the explicit expressions of the bias of the Naïve estimators of sensitivity and specificity of a binary diagnostic test. We also carry out simulation experiments in order to study the effect of the verification probabilities on the Naïve estimators of sensitivity and specificity.     

Parametric and nonparametric confidence intervals for estimating the difference of means of two skewed populations

In this paper, we have reviewed and proposed several interval estimators for estimating the difference of means of two skewed populations. Estimators include the ordinary-t, two versions proposed by Welch [17] and Satterthwaite [15], three versions proposed by Zhou and Dinh [18], Johnson [9], Hall [8], empirical likelihood (EL), bootstrap version of EL, median t proposed by Baklizi and Kibria [2] and bootstrap version of median t. A Monte Carlo simulation study has been conducted to compare the performance of the proposed interval estimators. Some real life health related data have been considered to illustrate the application of the paper. Based on our findings, some possible good interval estimators for estimating the mean difference of two populations have been recommended for the researchers.     

Mobile safety cameras: estimating casualty reductions and the demand for secondary healthcare

We consider a fully Bayesian analysis of road casualty data at 56 designated mobile safety camera sites in the Northumbria Police Force area in the UK. It is well documented that regression to the mean (RTM) can exaggerate the effectiveness of road safety measures and, since the 1980s, an empirical Bayes (EB) estimation framework has become the gold standard for separating real treatment effects from those of RTM. In this paper we suggest some diagnostics to check the assumptions underpinning the standard estimation framework. We also show that, relative to a fully Bayesian treatment, the EB method is over-optimistic when quantifying the variability of estimates of casualty frequency. Implementing a fully Bayesian analysis via Markov chain Monte Carlo also provides a more flexible and complete inferential procedure. We assess the sensitivity of estimates of treatment effectiveness, as well as the expected monetary value of prevention owing to the implementation of the safety cameras, to different model specifications, which include the estimation of trend and the construction of informative priors for some parameters.     

Optimal estimators for the importance sampling method

The Monte Carlo method gives some estimators to evaluate the expectation [ILM0001] based on samples from either the true density f or from some instrumental density. In this paper, we show that the Riemann estimators introduced by Philippe (1997) can be improved by using the importance sampling method. This approach produces a class of Monte Carlo estimators such that the variance is of order O(n ^?2). The choice of an optimal estimator among this class is discussed. Some simulations illustrate the improvement brought by this method. Moreover, we give a criterion to assess the convergence of our optimal estimator to the integral of interest.     

A Local Linear Least-Absolute-Deviations Estimator of Volatility

An important empirical characteristic of financial time series is that the unconditional distribution of the returns tends to possess heavy tails. This is the motivation for the particular local kernel volatility estimator proposed in this work. Whereas least-square-deviations (LSD) estimators are strongly affected by heavy-tailed distributions, the performance of least-absolute-deviations (LAD) estimators is not. This robustness to heavy tails is evidenced by the more flexible assumptions made on the distributional moments of the observable variable. The simulation examples also highlight the superior performances of the LAD estimator when compared to the LSD estimator under heavy tails conditions. The full nonparametric model is described and the asymptotic properties of the LAD estimator are derived. Extensive Monte Carlo studies strongly suggest that the LAD estimator is asymptotically adaptive to the unknown conditional first moment. The LAD estimator is also used to estimate the volatility of the S&P500 and the BOVESPA returns.