Extensions of Samuelson's Inequality

In this note we obtain upper and lower bounds for the kth largest number in a set of real numbers in terms of their mean and standard deviation. For each inequality necessary and sufficient conditions for equality are given.     

Comparison of Robust Estimators of Standard Deviation in Normal Distributions Within the Context of Quality Control

Pitman criterion is used in simulation to determine the “closer” estimator of the standard deviation among selected choices. The initial simulation utilizes a standard normal distribution from which samples are taken of specific sizes. Popular and commonly used estimators of standard deviation are compared with the known population standard deviation in this study. Closeness criterion is calculated for each comparison and sample size. A secondary simulation applies the findings to variables control charts, in order to verify the ability of each estimator to identify out-of-control conditions.     

Some New Results on Likelihood Ratio Orderings for Spacings of Heterogeneous Exponential Random Variables

Let X ₁, X ₂,…, X _n be independent exponential random variables with X _i having failure rate λ _i for i = 1,…, n. Denote by D _i:n  = X _i:n  ? X _i?1:n the ith spacing of the order statistics X _1:n  ≤ X _2:n  ≤ ··· ≤ X _n:n , i = 1,…, n, where X _0:n ≡ 0. It is shown that if λ _n+1 ≤ [≥] λ _k for k = 1,…, n then D _n:n  ≤ _lr D _n+1:n+1 and D _1:n  ≤ _lr D _2:n+1 [D _2:n+1 ≤ _lr D _2:n ], and that if λ _i  + λ _j  ≥ λ _k for all distinct i,j, and k then D _n?1:n  ≤ _lr D _n:n and D _n:n+1 ≤ _lr D _n:n , where ≤ _lr denotes the likelihood ratio order. We also prove that D _1:n  ≤ _lr D _2:n for n ≥ 2 and D _2:3 ≤ _lr D _3:3 for all λ _i 's.     

A General Recurrence Relation for the Moments of Bivariate and Trivariate Order Statistics

In this article, by using the dropping argument, a general recurrence relation satisfied by the joint cumulative distribution functions of order statistics from any arbitrary bivariate distribution function is established. This recurrence relation is the first bivariate version of the basic triangle rule for order statistics arisen from univariate distribution function. Finally, this relation is extended to the trivariate case. These lead to similar identities for product moments (of any order) of order statistics.     

A Note on the Moments and Computer Generation of the Shifted Gompertz Distribution

In this paper we give a class of row-column designs with the property that the i-th row and the j-th column have precisely r treatments in common. A conjecture that such designs are quasi-factorial is disproved by showing that the designs given in this paper are not quasi-factorial. It is also shown that the designs given here are nearly optimal.     

Moments of order statistics of the standard two-sided power distribution

The standard two-sided power distribution is a flexible distribution having uniform, power function and triangular as subdistributions, and it is a reasonable alternative to the Laplace distribution in some cases. In this work, computationally efficient expressions for moments of order statistics, expressions for L-moments, and asymptotic results for sample extrema are derived. Then a simulation study is performed for the location-scale estimation problem of a small data set by considering the maximum likelihood estimation method and the best linear unbiased estimation method based on the moments of order statistics.     

Asymptotic Distributions of the Generalized Range,Midrange, Extremal Quotient,and Extremal Product,with a Comparison Study

Necessary and sufficient conditions for the weak convergence of the generalized range, midrange, extremal quotient, and extremal product are obtained. The classes of possible non degenerate limit distribution functions of these simple statistics are characterized. Comparison study between these statistics with some examples for the most important distribution functions are given.     

Use of Spacings in the Estimation of Common Scale Parameter of Several Distributions

In this article, we consider the problem of best linear unbiased estimation and best linear invariant estimation of the common scale parameter of several distributions using spacing of the pooled sample of all observations of individual samples. We derived conditions for the non negativity of the scale estimator obtained by the above methods. Further, we obtained necessary and sufficient conditions for the derived estimators to be constant multiples of the pooled sample range.     

A Study of Some Different Concepts of Symmetry on the Real Line

Recently, different concepts of symmetry on R ⁺ such as R-symmetry, log-symmetry, and double symmetry are studied. Analogous concepts and their properties of these symmetries on R will be studied in this work. Based on skewing representation and previous studies, characterizations of double symmetry on R will be given. Among others, some interesting examples of the so-called I-symmetry, that is the analogue of log-symmetry on R, will also be presented.     

Some Small-Sample Properties of the Student t Confidence Intervals

For the two-sided Student t confidence interval for the mean of a normal distribution there is, for any sample size, a sufficiently large confidence level that ensures that the interval covers all the observations; there are also sufficiently small confidence levels guaranteeing, respectively, that (a) the interval does not cover all the observations and (b) the interval lies within the extreme observations. Necessary and sufficient conditions are also obtained for the width of the confidence interval to always exceed the sample range, as well as for the reverse inequality. Some implications of the results are discussed.     

Program of the Section on Economic and Business Statistics

Suppose that in the situation of a paired t test natural pairing, such as the use of twins, is not possible. Reduction in variability is then often achieved artificially, for example by pairing animals of similar birth weight. This article points out that, unless such pairing is ineffective, the usual assumptions underlying the paired t test are violated. Nevertheless, simulation indicates that, with randomization in the allocation of treatments, the standard procedure gives good results. Our bivariate normal model provides the factor by which the length of the confidence interval for the mean treatment difference is reduced as a result of the pairing. Another form of pairing sometimes used is shown to be incorrect. Nonparametric analogs are also briefly considered.     

A Note on the Use of Some Functions of Spacings in the Estimation of Common Scale Parameter of Several Symmetric Distributions

In this article, we consider the problem of best linear unbiased estimation and best linear invariant estimation of the common scale parameter of several symmetric distributions using some functions of spacings of all observations taken from individual samples. We also proved a sufficient condition for the non negativity of the common scale estimator obtained by the above method. Furthermore, we obtained necessary and sufficient conditions for the derived estimators to be constant multiple of the sum of first and last spacings of the pooled sample.     

Estimation of location and scale parameters of a distribution by U-statistics based on best linear functions of order statistics

In this work we propose a technique of estimating the location parameter μ$\mu$ and scale parameter σ$\sigma$ of a distribution by U-statistics constructed by taking best linear functions of order statistics as kernels. The method has been illustrated for estimating the location and scale parameters of type-I extreme value distribution. We have computed the asymptotic relative efficiencies of the proposed U-statistics with the appropriate maximum likelihood estimators based on samples drawn from each of type-I extreme value, logistic and normal distributions. In all cases very high asymptotic relative efficiencies are obtained.     

Estimation for Type II domain of attraction based on the W statistic

The paper presents an estimating equation approach to the estimation of high quantiles of a distribution in the Type II domain of attraction based on the k largest order statistics. The estimators are shown to be consistent. The method fits neatly into a general scheme for estimating high quantiles irrespective of the domain of attraction, which includes Wang's approach to optimally choosing k .     

A method for estimating parameters and quantiles of the three-parameter inverse Gaussian distribution based on statistics invariant to unknown location

The inverse Gaussian (IG) distribution, also known as the Wald distribution, is a long-tailed positively skewed distribution and a well-known lifetime distribution. In this paper, we propose an efficient method of estimation for the parameters and quantiles of the three-parameter IG distribution, which is based on statistics invariant to unknown location. Through a Monte Carlo simulation study, we then show that the proposed method performs well compared with other prominent methods in terms of bias and variance. Finally, we present two illustrative examples.     

一类新的数据变换及其对提高灰预测精度的有效性研究

提出了经过非负函数cg(x)+d变换来提高GM(1,1)模型的预测精度,并从理论上证明,这类变换为级比偏差压缩变换,保证了数据变换后的序列有非负上凹的特性,还原后的误差不会增大等性质,并通过实例对比说明了这种变换的有效性。     

Effects of nonnormality on the spart for the correlation coefficient: bivariate edgeworth series distribution

The effects of sampling from the bivariate Edgeworth series distribution (BVESD) on the sequential probability ratio test (SPRT) for the correlation coefficient are assessed. The values of the average sample number (ASN) and the operating characteristic (OC) are determined by simulation from such populations. The robustness of the SPRT to this type of nonnormality is demonstrated.