On the accuracy of the bootstrap approximation of the joint distribution of sample quantiles

It is proved that the accuracy of the bootstrap approximation of the joint distribution of sample quantiles lies between O(n^?1/4) and O(n^?1/4 a_n), where (log(n))1/2=O(a_n). As an application, we investigated confidence intervals based on the bootstrap.     

Contrasting the Gini and Zenga indices of economic inequality

The current financial turbulence in Europe inspires and perhaps requires researchers to rethink how to measure incomes, wealth, and other parameters of interest to policy-makers and others. The noticeable increase in disparities between less and more fortunate individuals suggests that measures based upon comparing the incomes of less fortunate with the mean of the entire population may not be adequate. The classical Gini and related indices of economic inequality, however, are based exactly on such comparisons. It is because of this reason that in this paper we explore and contrast the classical Gini index with a new Zenga index, the latter being based on comparisons of the means of less and more fortunate sub-populations, irrespectively of the threshold that might be used to delineate the two sub-populations. The empirical part of the paper is based on the 2001 wave of the European Community Household Panel data set provided by EuroStat. Even though sample sizes appear to be large, we supplement the estimated Gini and Zenga indices with measures of variability in the form of normal, t-bootstrap, and bootstrap bias-corrected and accelerated confidence intervals.     

On the simple step-stress model for two-parameter exponential distribution

In this paper, we consider the simple step-stress model for a two-parameter exponential distribution, when both the parameters are unknown and the data are Type-II censored. It is assumed that under two different stress levels, the scale parameter only changes but the location parameter remains unchanged. It is observed that the maximum likelihood estimators do not always exist. We obtain the maximum likelihood estimates of the unknown parameters whenever they exist. We provide the exact conditional distributions of the maximum likelihood estimators of the scale parameters. Since the construction of the exact confidence intervals is very difficult from the conditional distributions, we propose to use the observed Fisher Information matrix for this purpose. We have suggested to use the bootstrap method for constructing confidence intervals. Bayes estimates and associated credible intervals are obtained using the importance sampling technique. Extensive simulations are performed to compare the performances of the different confidence and credible intervals in terms of their coverage percentages and average lengths. The performances of the bootstrap confidence intervals are quite satisfactory even for small sample sizes.     

Interval estimation in discriminant analysis for large dimension

This paper is concerned with the interval estimation for the log odds of the posterior probability that the observation vector belongs to one of two homoscedastic multivariate normal distributions (Π₁ and Π₂). We give the limiting distribution of the unbiased estimator for the log odds as the sample sizes and the dimension jointly tend to infinity, and approximate the confidence interval based on the asymptotic distribution. Small-scale simulations are performed to check the precision of the approximation.     

区间型符号数据的因子分析及其应用

符号数据分析是一种新兴的数据挖掘技术,区间数是最常用的一种符号数据。基于误差分析理论,研究针对区间数据的因子分析方法。将区间数看作一个由中点和半径构成的有序偶,并将半径视为区间数的极限误差。对中点样本阵进行因子分析,得到因子得分的中点值。然后将半径样本阵按照误差传递公式,得到因子得分的极限误差。由因子得分的中点值和极限误差最终得到因子得分的区间值。最后以股票的市场综合表现评价问题为案例,进行了应用研究。     

Optimal record-based statistical procedures for the two-parameter exponential distribution

There are a number of situations in which the experimental data observed are record statistics. In this paper, optimal confidence intervals as well as uniformly most powerful (MP) tests for one-sided alternatives are developed. Since a uniformly MP test for a two-sided alternative does not exist, generalized likelihood ratio and uniformly unbiased and invariant tests are derived for the two parameters of the exponential distribution based on record data. For illustrative purposes, a data set on the times between consecutive telephone calls to a company's switchboard is analysed using the proposed procedures. Finally, some open problems in this direction are pointed out.     

Unbiastld estimation of parametric functions in sampling from two two-truncation parameter families

We consider the problem of UMVU estimation of a U-estimable function of four unknown truncation parameters based on two independent random samples from two two-truncation parameter families. In particular, we obtain the UMVU estimator of functional, P (Y > X). Also the confidence intervals for some parametric functions are obtained.     

Inference for the scale parameter of lifetime distribution of k-unit parallel system based on progressively censored data

In this paper, inference for the scale parameter of lifetime distribution of a k-unit parallel system is provided. Lifetime distribution of each unit of the system is assumed to be a member of a scale family of distributions. Maximum likelihood estimator (MLE) and confidence intervals for the scale parameter based on progressively Type-II censored sample are obtained. A β-expectation tolerance interval for the lifetime of the system is obtained. As a member of the scale family, half-logistic distribution is considered and the performance of the MLE, confidence intervals and tolerance intervals are studied using simulation.     

基于区间数大小不能直接判定的灰矩阵博弈的策略优超及其最优解研究

对于区间灰数大小不能直接判定的灰矩阵博弈G(⊗)={S₁,S₂,A(⊗)}问题,其策略优超和纯策略求解问题的关键在于A(⊗)中区间灰数大小判定准则的设定与判定方法的设计。本文运用灰色系统思想和系统工程的理论,揭示了人们在灰信息条件下的博弈心理与博弈决策规则,根据区间灰数势关系的判定规则,提出了灰数势意义下的策略优超法则,定义了纯策略解。最后,以商业银行贷款动态损失准备金计提为案例,对其灰势意义下的策略优超和纯策略解问题进行了研究。     

Criticism of the predictive distribution

The predictive distribution is a mixture of the original distribution model and is used for predicting a future observation. Therein, the mixing distribution is the posterior distribution of the distribution parameters in the Bayesian inference. The mixture can also be computed for the frequentist inference because the Bayesian posterior distribution has the same meaning as a frequentist confidence interval. I present arguments against the concept of predictive distribution. Examples illustrate these. The most important argument is that the predictive distribution can depend on the parameterization. An improvement of the theory of the predictive distribution is recommended.