Jyh-Jiuan Lin

A Note on Improved Approximation of the Binomial Distribution by the Skew-Normal Distribution

It is a common practice to approximate a binomial distribution by a suitable normal distribution when n, the number of trials, is moderately large. But when p, the probability of success, is not close to 0.5, the binomial distribution can be heavily skewed, and hence the usual normal approximation may not be a good idea. In this note we show that the skew-normal distribution can provide a far better approximation due to its flexibility, and it can be used to approximate distributions other than the binomial one.

Estimation of a normal variance—A critical review

Consider the problem of estimating the variance based on a random sample from a normal distribution with unknown mean. In this article we review the rich literature developed over the last three decades on the problem of variance estimation in a decision theoretic setup. While examining the developments we point out a few errors that exist in the literature.

A Revisit to Contingency Table and Tests of Independence: Bootstrap is Preferred to Chi-Square Approximations as Well as Fisher’s Exact Test

To test the mutual independence of two qualitative variables (or attributes), it is a common practice to follow the Chi-square tests (Pearson’s as well as likelihood ratio test) based on data in the form of a contingency table. However, it should be noted that these popular Chi-square tests are asymptotic in nature and are useful when the cell frequencies are “not too small.” In this article, we explore the accuracy of the Chi-square tests through an extensive simulation study and then propose their bootstrap versions that appear to work better than the asymptotic Chi-square tests. The bootstrap tests are useful even for small-cell frequencies as they maintain the nominal level quite accurately. Also, the proposed bootstrap tests are more convenient than the Fisher’s exact test which is often criticized for being too conservative. Finally, all test methods are applied to a few real-life datasets for demonstration purposes.

Short communication: A revisit to the common mean problem: Comparing the maximum likelihood estimator with the Graybill-Deal estimator

For estimating the common mean of two normal populations with unknown and possibly unequal variances the well-known Graybill-Deal estimator (GDE) has been a motivating factor for research over the last five decades. Surprisingly the literature does not have much to show when it comes to the maximum likelihood estimator (MLE) and its properties compared to those of the GDE. The purpose of this note is to shed some light on the structure of the MLE, and compare it with the GDE. While studying the asymptotic variance of the GDE, we provide an upgraded set of bounds for its variance. A massive simulation study has been carried out with very high level of accuracy to compare the variances of the above two estimators results of which are quite interesting.

A Note on Comparing Several Poisson Means

There has been a renewed interest lately in comparing the means (rates) of several Poisson distributions (processes) due to its wide applicability in various fields, especially in life sciences. In this article we first review the recent developments in the literature, and then propose a parametric bootstrap method which performs as good as, if not better than, the existing methods. Results of our comprehensive simulation study have been provided to compare the relevant methods. Finally these methods have been used to four real-life datasets including a most recent one obtained from a clinical trial on the Intrinsa hormone patch developed by Proctor & Gamble.

Improvements over the james-stein estimator: A risk analysis

Consider the problem of estimating a p-variate (p≥3) normal mean vector under the squared error loss when the dispersion matrix is assumed to be the identity matrix. Here we study the risk functions of several estimators which are uniformly better than the James-Stein estimator.

A note on selling distressed loans with bank bailouts: modelling of bank interest margins with default probabilities

This article extends the framework of Merton (1974) with Vassalou and Xing (2004) to value a troubled but solvent banks equity by explicitly incorporating distressed assets purchased by the government in an imperfectly competitive loan market. We show that the bank may be willing to take this bailout when the purchased amount is relatively small and the margin is relatively low. However, the bank may be harder to entice even when the unit price of the bailed-out assets subsidized by the government is relatively high. As a consequence, most of the first half of the Troubled Asset Relief Programs money is not used to buy troubled assets (Wilson, 2010).

Alternative estimation procedure in SPC when the process data are correlated

If the process observations are autocorrelated, the performance of control chart is influenced significantly. This autocorrelation leads to a large false-alarm rate. This article considers the problem of monitoring the mean of AR(1) process with random error. We provide a simple algorithm to improve the estimation results of process parameters. Simulation results show that the proposed method can produce stable and adequate estimates for the AR(1) process with random error, even though the sample size is small.

A Note on Robustness of Normal Variance Estimators Under t-Distributions

ABSTRACT In many real life problems one assumes a normal model because the sample histogram looks unimodal, symmetric, and/or the standard tests like the Shapiro-Wilk test favor such a model. However, in reality, the assumption of normality may be misplaced since the normality tests often fail to detect departure from normality (especially for small sample sizes) when the data actually comes from slightly heavier tail symmetric unimodal distributions. For this reason it is important to see how the existing normal variance estimators perform when the actual distribution is a t-distribution with k degrees of freedom (d.f.) (t k -distribution). This note deals with the performance of standard normal variance estimators under the t k -distributions. It is shown that the relative ordering of the estimators is preserved for both the quadratic loss as well as the entropy loss irrespective of the d.f. and the sample size (provided the risks exist).

Applications of Improved Variance Estimators in a Multivariate Normal Mean Vector Estimation

Consider the problem of estimating a normal mean vector when i.i.d observations are available from a p-dimensional normal distribution with an unknown mean vector and an unknown diagonal dispersion matrix proportional to the identity matrix. By using the improved variance estimation techniques we propose wide classes of shrinkage mean estimators which are uniformly better than the James-Stein estimator. Some of our improved mean estimators are completely new and are not covered by Kubokawas (1994; A Unified Approach to Improving Equivariant Estimators. Annals of Statistics) result. Numerical results are provided to study the risk performance of some of these improved mean estimators.