Ching-Hui Chang

A Revisit to the Behrens–Fisher Problem: Comparison of Five Test Methods

We revisit the well-known Behrens–Fisher problem and apply a newly developed ‘Computational Approach Test’ (CAT) to test the equality of two population means where the populations are assumed to be normal with unknown and possibly unequal variances. An advantage of the CAT is that it does not require the explicit knowledge of the sampling distribution of the test statistic. The CAT is then compared with three widely accepted tests—Welch–Satterthwaite test (WST), Cochran–Cox test (CCT), ‘Generalized p-value’ test (GPT)—and a recently suggested test based on the jackknife procedure, called Singh–Saxena–Srivastava test (SSST). Further, model robustness of these five tests are studied when the data actually came from t-distributions, but wrongly perceived as normal ones. Our detailed study based on a comprehensive simulation indicate some interesting results including the facts that the GPT is quite conservative, and the SSST is not as good as it has been claimed in the literature. To the best of our knowledge, the trends observed in our study have not been reported earlier in the existing literature.

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.

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.

A comparison of usual indices and extended TOPSIS methods in mutual funds’ performance evaluation

Abstract The purpose of our study is to evaluate the performance of mutual funds. ‘Treynor Ratio’, ‘Sharpe Ratio’, ‘Information Ratio’ and ‘Jensen’s Alpha’ are four commonly used indices for evaluating the competing mutual funds. However, it is not clear which measure is the most robust. This paper not only focuses on investigating the four indices (criteria) separately but also combining all the indices at the same time in making a final ranking of the mutual funds. In this work, the extended TOPSIS with four objective weighing choices under Minkowski’s Lλ metric with λ=2 (Euclidean) distance measure are adopted for the multi-criteria decision making. The aforementioned four usual indices and the extended TOPSIS with four choices of weight methods are then applied to evaluate the performance of 82 Taiwanese mutual funds for 34 consecutive months. Roughly speaking, all the indices perform pretty well in terms of Spearman’s rank correlation coefficient. Although multicriteria methods are not noticeably more accurate than uni-criteria methods, but it offers an alternative to whom wants to take all four criteria into consideration simultaneously.

Estimation Of A Multivariate Normal Mean Vector And Local Improvements

Let X be a (p × 1) random vector following a multivariate normal distribution with mean vector 0 and a known dispersion matrix. Also suppose we have an extra observation U independent of X, whose distribution is completely known. In this article we first develop estimators of θ combining both X and U, which are similar to the James-Stein estimator but can give substantial risk improvement over it when θ is assumed to be in a neighborhood of a known value. Estimators based only on Xare then developed which can give further improvements.

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).

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.