One and two sample confidence intervals for estimating the mean of skewed populations: an empirical comparative study

In this paper we consider and propose some confidence intervals for estimating the mean or difference of means of skewed populations. We extend the median t interval to the two sample problem. Further, we suggest using the bootstrap to find the critical points for use in the calculation of median t intervals. A simulation study has been made to compare the performance of the intervals and a real life example has been considered to illustrate the application of the methods.     

Testing for mean and variance differences with samples from distributions that may be non-normal with unequal variances

The problem of comparing several samples to decide whether the means and/or variances are significantly different is considered. It is shown that with very non-normal distributions even a very robust test to compare the means has poor properties when the distributions have different variances, and therefore a new testing scheme is proposed. This starts by using an exact randomization test for any significant difference (in means or variances) between the samples. If a non-significant result is obtained then testing stops. Otherwise, an approximate randomization test for mean differences (but allowing for variance differences) is carried out, together with a bootstrap procedure to assess whether this test is reliable. A randomization version of Levene's test is also carried out for differences in variation between samples. The five possible conclusions are then that (i) there is no evidence of any differences, (ii) evidence for mean differences only, (iii) evidence for variance differences only, (iv) evidence for mean and variance differences, or (v) evidence for some indeterminate differences. A simulation experiment to assess the properties of the proposed scheme is described. From this it is concluded that the scheme is useful as a robust, conservative method for comparing samples in cases where they may be from very non-normal distributions.     

On the distribution of the correlation coefficient when sampling from a mixture of two bivariate normal densities: Robustness and the effect of outliers

The distribution of the sample correlation coefficient is derived when the population is a mixture of two bivariate normal distributions with zero mean but different covariances and mixing proportions 1 - λ and λ respectively; λ will be called the proportion of contamination. The test of ρ = 0 based on Student's t, Fisher's z, arcsine, or Ruben's transformation is shown numerically to be nonrobust when λ, the proportion of contamination, lies between 0.05 and 0.50 and the contaminated population has 9 times the variance of the standard (bivariate normal) population. These tests are also sensitive to the presence of outliers.     

Estimation of a Conditional Copula and Association Measures

Abstarct. This paper is concerned with studying the dependence structure between two random variables Y ₁ and Y ₂ conditionally upon a covariate X. The dependence structure is modelled via a copula function, which depends on the given value of the covariate in a general way. Gijbels et al. (Comput. Statist. Data Anal., 55, 2011, 1919) suggested two non‐parametric estimators of the ‘conditional’ copula and investigated their numerical performances. In this paper we establish the asymptotic properties of the proposed estimators as well as conditional association measures derived from them. Practical recommendations for their use are then discussed.     

Estimation and testing procedures for the reliability functions of a general class of distributions

We consider here the general class of distributions proposed by Sankaran and Gupta (2005) by zeroing in on two measures of reliability, R(t) = P(X > t) and P = P(X > Y). Thereafter, we develop point estimation for R(t) and ‘P’ and develop uniformly minimum variance unbiased estimators (UMVUES). Then we derive testing procedures for the hypotheses related to different parametric functions. Finally, we compare the results using the Monte Carlo simulation method. Using real data set, we illustrate the procedure clearly.     

Bayes Estimation Based on Random Censored Data for Some Life Time Models Under Symmetric and Asymmetric Loss Functions

Censored data arise naturally in a number of fields, particularly in problems of reliability and survival analysis. There are several types of censoring; in this article, we shall confine ourselves to the right randomly censoring type. Under the Bayesian framework, we study the estimation of parameters in a general framework based on the random censored observations under Linear-Exponential (LINEX) and squared error loss (SEL) functions. As a special case, Weibull model is discussed and the admissibility of estimators of parameters verified. Finally, a simulation study is conducted based on Monte Carlo (MC) method for comparing estimated risks of the estimators obtained.