Graphical Demonstration of an Optimality Property of the Median

The property that the mean absolute deviation is a minimum when measured from a median is demonstrated with graphs. Deficiencies of the usual proof of the property are pointed out, and a new, general proof is suggested.     

MRL and MPL Functions of Parallel Systems with INID Components under Double Monitoring

This article investigates parallel systems with n independent but not identically distributed (INID) components. Under the condition that, at time t ₁ (t ₁ > 0) the total number of failures of the components is r (r < n), and at time t ₂ (t ₂ ≥ t ₁) the sys-tem is still working or it has failed, the mean residual life (MRL) function of the parallel system and the mean past lifetime (MPL) function of the components are conducted. Some representations and corresponding properties of the MRL function and the MPL function under several specific conditions are obtained.     

Nonparametrie recursive estimator for mean residual life and vitality function under dependence conditions

In this paper, we develop a nonparametrie recursive estimator for the vitality and mena residual life function, based on kernel density estimators under mixing dependence conditions. The consistency and asymptotic normality of the estimator are established, under suitable regularity conditions. It is also shown that the Integrated Mean Squared Error converges to zero. The paper is concluyed with some simulation results.     

Bayesian Prediction of the Median of the Burr Distribution with Fixed and Random Sample Sizes

sLingappaiah (1986) was the first to introduce the idea of Bayesian prediction in life testing when the size of the future sample is a random variable. In this paper we discuss the Bayesion prediction of the sample median when the parent distribution is a generalized Burr distribution (GBD), the old sample is censored type II and the size of the future sample is a random variable. A numerical illustration is provided.     

Nonparametric estimation of some important indicators in reliability for semi-Markov processes

In this article we present nonparametric estimators of the mean time to failure (MTTF), the mean up time (MUT), and the mean down time (MDT) of an arbitrary finite semi-Markov process. Strong consistency and weak convergence is proved for the estimators of MTTF, MUT, and MDT, as the time interval of observation becomes large.     

Bounds on Sample Variation Measures Based on Majorization

Using majorization theory, upper and lower bounds are derived for different measures of variation as progressively more items of information are available about the sample data. As a convenient starting point, bounds are first established for a one-parameter family of variation measures, which is a generalized mean difference measure of which Gini's mean difference, the standard deviation, and the range are particular cases. While, as pointed out, some of the derived bounds are well known, others do not appear to have been published and are tighter than established bounds. Some 40 different bounds are derived, besides any number of bounds given for the generalized family of variation measures. A number of interesting inequalities are also derived on the basis of some of the bounds. While the bounds have been developed in terms of real-valued sample data generally, the paper concludes with a brief discussion of the bounds for categorical data when the sample data consists of frequencies (counts).     

Online change detection of Markov chains with unknown post-change transition probabilities

ABSTRACT

In this paper, we investigate the performance of cumulative sum (CUSUM) stopping rules for the online detection of unknown change point in a time homogeneous Markov chain. Under the condition that the post-change transition probabilities are unknown, we proposed two CUSUM type schemes for the detection. The first scheme is based on the maximum likelihood estimates of the post-change transition probabilities. This scheme is limited by its computation burden, which is mitigated by another scheme based on the reference transition probabilities selected from a prior known region. We give the bounds of the mean delay time and the mean time between false alarms to illustrate the effectiveness of the proposed schemes. The results of the simulation also demonstrate the feasibility of the proposed schemes.     

On estimation of population mean based on two measurements

There are many estimators for population mean: arithmetic mean, median, root mean square, geometric mean, harmonic mean, and so forth. Which one is the best? Here, we answer this question given two measurements from a uniform distribution.     

碳排放的结构影响与效应分解

 本文从碳排放系数的角度推导了总量变化的SDA分解模型,从社会经济发展视角构建了LMDI分解公式,从而讨论不同时期碳排放量变化的影响因素。文章以云南省为例,运用云南省分行业能源消费数据以及 4个年份的投入产出表进行实证分析。SDA计算结果表明：消费与投资扩张效应是碳排放增长的贡献源泉,碳排放强度变动效应是节能减排的源动力;LMDI分解结果表明：人均GDP 增长是拉动云南省CO2排放增长的决定性因素,而能耗强度下降是抑制碳排放增长的主要原因。根据计算结果,文章提出了低碳经济条件下云南省产业结构调整的新方向。     

Influence and sensitivity measures in correspondence analysis

Since correspondence analysis appears to be sensitive to outliers, it is important to be able to evaluate the sensitivity of the data on the results. This article deals with measuring the influence of rows and columns on the results obtained with correspondence analysis. To establish the influence of individuals on the analysis, we use the notion of influence curve and we propose a general criterion based on the mean square error to measure the sensitivity of the correspondence analysis and its robustness. A numerical example is presented to illustrate the notions developed in this article.     

Inverse Circular–Linear/Linear–Circular Regression

We propose two distance-based methods and two likelihood-based methods of inversely regressing a linear predictor on a circular variable, and of inversely regressing a circular predictor on a linear variable. An asymptotic result on least circular distance estimators is provided in the Appendix. We present likelihood-based methods for symmetrical and asymmetrical errors in each situation. The utility of our methodology in each situation is illustrated by applying it to real data sets in engineering and environmental science. We then compare their performances using a cross validation method.     

Selection of the Ridge Parameter Using Mathematical Programming

Ridge regression solves multicollinearity problems by introducing a biasing parameter that is called ridge parameter; it shrinks the estimates and their standard errors in order to reach acceptable results. Selection of the ridge parameter was done using several subjective and objective techniques that are concerned with certain criteria. In this study, selection of the ridge parameter depends on other important statistical measures to reach a better value of the ridge parameter. The proposed ridge parameter selection technique depends on a mathematical programming model and the results are evaluated using a simulation study. The performance of the proposed method is good when the error variance is greater than or equal to one; the sample consists of 20 observations, the number of explanatory variables in the model is 2, and there is a very strong correlation between the two explanatory variables.     

Empirical bayes estimation of reliability characteristics for an exponential family

This paper considers empirical Bayes (EB) squared-error-loss estimations of mean lifetime, variance and reliability function for failure-time distributions belonging to an exponential family, which includes gamma and Weibull distributions as special cases. EB estimators are proposed when the prior distribution of the lifetime parameter is completely unknown but has a compact (known or unknown) support. Asymptotic optimality and rates of convergence of these estimators are investigated. The rates established here under the compact support restriction are better than the polynomial rates of convergence obtained previously.     

Restricted ridge estimator in the logistic regression model

It is known that when the multicollinearity exists in the logistic regression model, variance of maximum likelihood estimator is unstable. As a remedy, Schaefer et al. presented a ridge estimator in the logistic regression model. Making use of the ridge estimator, when some linear restrictions are also present, we introduce a restricted ridge estimator in the logistic regression model. Statistical properties of this newly defined estimator will be studied and comparisons are done in the simulation study in the sense of mean squared error criterion. A real-data example and a simulation study are introduced to discuss the performance of this estimator.     

从统计发展史谈统计的精确度

随着统计教育的普及,统计方法在现实生活中的使用越来越广泛。同时,由于现实生活中各种现象的复杂性,人们难免对统汁方法的精确度表示怀疑。本文从统计史的角度并结合现实生活对统计方法的精确度给予了充分的说明与论证。     

Estimation of a parametric function associated with the lognormal distribution

Estimation of the mean of the lognormal distribution has received much attention in the literature beginning with Finney (1941 Finney, D.J. (1941). On the distribution of a variate whose logarithm is normally distributed. Supp. J. Royal Stat. Soc. 7(2):155–161.[Crossref] , [Google Scholar]). The problem is of significant practical importance because of the ubiquitous use of log-transformation. In this article, we consider the estimation of a parametric function associated with the lognormal distribution of which the mean, median, and moments are special cases. We generalize various estimators from the literature for the mean to this parametric function and propose a new simple estimator. We present the estimators in a unified framework and use this framework to derive asymptotic expressions for their biases and mean square errors (MSEs). Next, we make asymptotic and small-sample comparisons via simulations between them in terms of their MSEs. Our proposed estimator outperforms many of the previously proposed estimators. A numerical example is given to illustrate the various estimators.     

Estimation of Mode Using Auxiliary Information

In this article, we propose naïve estimator, ratio estimator, and difference estimator of the mode of a study variable by using the known mode of an auxiliary variable. The asymptotic properties of the proposed estimators are studied analytically as well as empirically for different situations. The proper use of auxiliary information is found to result in efficient ratio and difference estimators than the naïve estimator.     

Target estimation for the logistic regression model

The method of target estimation developed by Cabrera and Fernholz [(1999). Target estimation for bias and mean square error reduction. The Annals of Statistics, 27(3), 1080–1104.] to reduce bias and variance is applied to logistic regression models of several parameters. The expectation functions of the maximum likelihood estimators for the coefficients in the logistic regression models of one and two parameters are analyzed and simulations are given to show a reduction in both bias and variability after targeting the maximum likelihood estimators. In addition to bias and variance reduction, it is found that targeting can also correct the skewness of the original statistic. An example based on real data is given to show the advantage of using target estimators for obtaining better confidence intervals of the corresponding parameters. The notion of the target median is also presented with some applications to the logistic models.     

Improved estimation of population mean using known median of auxiliary variable

In this article, we consider the problem of estimation of population mean using the known median of auxiliary variable. We proposed an estimator and its efficiency is studied analytically as well as empirically for different conditions. The proposed estimator is found to be more efficient than traditional estimators such as sample mean and linear regression estimator.