On null and alternative distribution of statistic of two-side discordancy test for an extreme outlier

We find the joint distribution of Grubbs statistics for a normal sample. Those statistics are standardized maximum and standardized minimum. We note some properties of the joint distribution function. We apply the joint distribution function and find the exact distribution of the test statistic which uses in two-sided discordancy test for an extreme outlier. We obtain recursive relationships for the distribution function of the statistic, which uses in two-sided discordancy test. We obtain the region of critical values of the statistic, where the significance level of criteria equals the double significance level of the Grubbs criteria. We apply the joint distribution of Grubbs statistics and find the power function for the criteria in the case of a normal sample with a single outlier.     

A rank test based on the moments of order statistics of the modified Makeham distribution

Single moments of order statistics from the modified Makeham distribution (MMD) are derived, an identity about the single moments of order statistics is given, and the specific expected value and variance of the single moments of order statistics from the MMD are calculated. In this study, the order statistic from the MMD was applied to the rank sum test in a two-sample problem. The exact critical values of the designated statistics were evaluated. Simulations were used to investigate the power of these statistics for the two-sided alternative with several population distributions. The powers of the statistics were compared with the Wilcoxon rank sum statistic, the Lepage statistic, the modified Baumgartner statistic, the Savage test and the normal score test. The Edgeworth expansion was used to evaluate the upper tail probability for the preferred statistic, given finite sample sizes.     

Stability of Order Statistics under Dependence

We precisely evaluate the upper and lower deviations of the expectation of every order statistic from an i.i.d. sample under arbitrary violations of the independence assumption, measured in scale units generated by various central absolute moments of the parent distribution of a single observation. We also determine the distributions for which the bounds are attained. The proof is based on combining the Moriguti monotone approximation of functions with the Hölder inequality applied for proper integral representations of expected order statistics in the independent and dependent cases. The method allows us to derive analogous bounds for arbitrary linear combinations of order statistics.     

Reliability and expectation bounds for coherent systems with exchangeable components

Sharp upper and lower bounds are obtained for the reliability functions and the expectations of lifetimes of coherent systems based on dependent exchangeable absolutely continuous components with a given marginal distribution function, by use of the concept of Samaniego's signature. We first show that the distribution of any coherent system based on exchangeable components with absolutely continuous joint distribution is a convex combination of distributions of order statistics (equivalent to the k-out-of-n systems) with the weights identical with the values of the Samaniego signature of the system. This extends the Samaniego representation valid for the case of independent and identically distributed components. Combining the representation with optimal bounds on linear combinations of distribution functions of order statistics from dependent identically distributed samples, we derive the corresponding reliability and expectation bounds, dependent on the signature of the system and marginal distribution of dependent components. We also present the sequences of exchangeable absolutely continuous joint distributions of components which attain the bounds in limit. As an application, we obtain the reliability bounds for all the coherent systems with three and four exchangeable components, expressed in terms of the parent marginal reliability function and specify the respective expectation bounds for exchangeable exponential components, comparing them with the lifetime expectations of systems with independent and identically distributed exponential components.     

Bayesian approach to prediction and the spacings in the exponential distribution

Summary A series of independent samples are drawn from a general population with positive variationf(x,ϕ), x>0. Based on the Bayesian approach, a general predictive distribution is given, to predict a statistic in the future sample based on the statistics in the earlier samples (or stages). Few general classes of distributions of this type like Koopman-Pitman family, power function family and Burr's class of distributions are considered to show how this procedure works in predicting order statistics in the future sample. Also, the sum of the spacings in the future samples from an exponential population is predicted in terms of similar sum of spacings in the earlier samples. Discussion on the variance of this predictive distribution is dealt with. Finally, an illustrative example with simulated samples from an exponential population gives actual prediction of an order statistic as well as the sum of spacings in the future sample.     

Estimation with Sequential Order Statistics from Exponential Distributions

The lifetime of an ordinary k-out-of-n system is described by the (n–k+1)-st order statistic from an iid sample. This set-up is based on the assumption that the failure of any component does not affect the remaining ones. Since this is possibly not fulfilled in technical systems, sequential order statistics have been proposed to model a change of the residual lifetime distribution after the breakdown of some component. We investigate such sequential k-out-of-n systems where the corresponding sequential order statistics, which describe the lifetimes of these systems, are based on one- and two-parameter exponential distributions. Given differently structured systems, we focus on three estimation concepts for the distribution parameters. MLEs, UMVUEs and BLUEs of the location and scale parameters are presented. Several properties of these estimators, such as distributions and consistency, are established. Moreover, we illustrate how two sequential k-out-of-n systems based on exponential distributions can be compared by means of the probability P(X < Y). Since other models of ordered random variables, such as ordinary order statistics, record values and progressive type II censored order statistics can be viewed as sequential order statistics, all the results can be applied to these situations as well.     

Rates of uniform convergence of extreme order statistics

Summary Bounds for the convergence uniformly over all Borel sets of the largest order statistic as well as of the joint distribution of extremes are established which reveal in which way these rates are determined by the distance of the underlying density from the density of the corresponding generalized Pareto distribution. The results are highlighted by several examples among which there is a bound for the rate at which the joint distribution of thek largest order statistics from a normal distribution converges uniformly to its limit.     

Asymptotic robustness of the normal theory likelihood ratio statistic for two-level covariance structure models

Data in social and behavioral sciences are often hierarchically organized. Special statistical procedures have been developed to analyze such data while taking into account the resulting dependence of observations. Most of these developments require a multivariate normality distribution assumption. It is important to know whether normal theory-based inference can still be valid when applied to nonnormal hierarchical data sets. Using an analytical approach for balanced data and numerical illustrations for unbalanced data, this paper shows that the likelihood ratio statistic based on the normality assumption is asymptotically robust for many nonnormal distributions. The result extends the scope of asymptotic robustness theory that has been established in different contexts.     

On Edgeworth Expansions in Generalized Urn Models

The random vector of frequencies in a generalized urn model can be viewed as conditionally independent random variables, given their sum. Such a representation is exploited here to derive Edgeworth expansions for a “sum of functions of such frequencies,” which are also called “decomposable statistics.” Applying these results to urn models such as with- and without-replacement sampling schemes as well as the multicolor Pólya–Egenberger model, new results are obtained for the chi-square statistic, for the sample sum in a without-replacement scheme, and for the so-called Dixon statistic that is useful in comparing two samples.     

Likelihood ratio tests of correlated multivariate samples

We develop methods to compare multiple multivariate normally distributed samples which may be correlated. The methods are new in the context that no assumption is made about the correlations among the samples. Three types of null hypotheses are considered: equality of mean vectors, homogeneity of covariance matrices, and equality of both mean vectors and covariance matrices. We demonstrate that the likelihood ratio test statistics have finite-sample distributions that are functions of two independent Wishart variables and dependent on the covariance matrix of the combined multiple populations. Asymptotic calculations show that the likelihood ratio test statistics converge in distribution to central Chi-squared distributions under the null hypotheses regardless of how the populations are correlated. Following these theoretical findings, we propose a resampling procedure for the implementation of the likelihood ratio tests in which no restrictive assumption is imposed on the structures of the covariance matrices. The empirical size and power of the test procedure are investigated for various sample sizes via simulations. Two examples are provided for illustration. The results show good performance of the methods in terms of test validity and power.     

Factors affecting economic output in developed countries: A copula approach to sample selection with panel data

This work aims at determining the factors affecting economic output in developed countries. However, the definition of development depends on the criteria by which different principles provide different criteria of level of development. Therefore, there exists uncertainty about choice of sample or real development country and if the selected samples are not representative of the underlying population of real developed countries then the ordinary least squares coefficients may be biased. This paper examines the determinants of economic output in the panel data of 22 developed countries from 1996 to 2008 utilizing econometric techniques that take into account the selective nature of the samples. In general, there are two approaches to estimate the sample selection model, namely the maximum likelihood method and the method proposed by Heckman (1979) [21]. Moreover, these two approaches require that the joint distribution to be known. In general the multivariate normal distribution is assumed. However, this assumption can often be seen as excessively restrictive and this lead to uncertainty about the structure or assumption of joint distribution. Smith (2003) [37] suggests applying the copula approach, especially the Archimedean copula to the sample selection model and the result also shows that the copula approach is well suited to apply to a model where the sample selection is biased, using cross-section data. In our work, we employ the copula approach to construct the sample selection model in the case of panel data, resulting in the identification of significant factors affecting economic output.     

Asymptotic linear prediction of extreme order statistics

Summary We consider the problem of predicting thesth order statistic using the lowestr order statistics from a large sample of sizen under the assumption that the sample minimum, appropriately normalized, has a non-degenerate limit distribution asn→∞. Assumingr, s fixed andn→∞ we obtain asymptotically best linear unbiased as well as asymptotically best linear invariant predictors of thesth order statistic.     

On the exact distribution of linear combinations of order statistics from dependent random variables

We study the exact distribution of linear combinations of order statistics of arbitrary (absolutely continuous) dependent random variables. In particular, we examine the case where the random variables have a joint elliptically contoured distribution and the case where the random variables are exchangeable. We investigate also the particular L-statistics that simply yield a set of order statistics, and study their joint distribution. We present the application of our results to genetic selection problems, design of cellular phone receivers, and visual acuity. We give illustrative examples based on the multivariate normal and multivariate Student t distributions.     

A goodness-of-fit statistic for Pareto-type behaviour

The fit of a statistical model can be visually assessed by inspection of a quantile–quantile or QQ plot. For the strict Pareto distribution, since log-transformed Pareto random variables are exponentially distributed, it is natural to consider an exponential quantile plot based on the log-transformed data. In case the data originate from a Pareto-type distribution, the Pareto quantile plot will be linear but only in some of the largest observations. In this paper we modify the Jackson statistic, originally proposed as a goodness-of-fit statistic for testing exponentiality, in such a way that it measures the linearity of the k largest observations on the Pareto quantile plot. Further, by taking the second-order tail behaviour of a Pareto-type model into account we construct a bias-corrected Jackson statistic. For both statistics the limiting distribution is derived. Next to these asymptotic results we also evaluate the small sample behaviour on the basis of a simulation study. The method is illustrated on two practical case studies.     

股票价格序列中随机跳跃存在性的一种检验方法

提出股票价格序列跳跃的一种检验方法.假设价格具有连续样本路径,建立一个关于股票价格样本观察的统计量,利用中心极限定理求得该统计量的极限分布为正态分布,这样,当该统计量超出基于极限分布算出的临界水平时,可以拒绝原假设,认为样本中存在跳跃.用此方法来应用于中国股市沪深股票指数,得到了中国股市存在随机跳跃的直接证据.提出的跳跃检验方法无需对连续部分的波动率形式作过多的假设,克服了波动率模型对检验准确性的影响.结果对金融资产的定价、投资和风险管理都具有积极的意义.     

Testing equality between several populations covariance operators

In many situations, when dealing with several populations, equality of the covariance operators is assumed. An important issue is to study whether this assumption holds before making other inferences. In this paper, we develop a test for comparing covariance operators of several functional data samples. The proposed test is based on the Hilbert–Schmidt norm of the difference between estimated covariance operators. In particular, when dealing with two populations, the test statistic is just the squared norm of the difference between the two covariance operators estimators. The asymptotic behaviour of the test statistic under both the null hypothesis and local alternatives is obtained. The computation of the quantiles of the null asymptotic distribution is not feasible in practice. To overcome this problem, a bootstrap procedure is considered. The performance of the test statistic for small sample sizes is illustrated through a Monte Carlo study and on a real data set.     

Distribution of Bids for Buildings and Roads Contracts

This paper describes the analysis of data relating to bids for 535 contracts to determine the distribution of these bids. The findings are that bids for building contracts can be treated as samples from a normal distribution and that for roads, although the assumption is statistically less valid, it is adequate for practical purposes. The tests for normality used were the studentized range and the Anderson-Darling statistic. Comments on the uses of these distributions in testing for unrealistic bids and for predicting the lowest bid are given.     

On high-dimensional change point problem

New statistics are proposed to estimate and test the structural change when the data dimension is comparable to or larger than the sample size. Consistency of the new statistic in estimating the change point position is established under the alternative hypothesis. The asymptotic distribution of the new statistic in testing the existence of a change point is obtained under the null hypothesis. Some simulation results are presented which show that the numerical performance of our method is satisfactory. The method is illustrated via the analysis of the house price index of US.     

有限混合模型有限制Log极大似然比统计量的极限分布

有限混合模型的Log极大似然比统计量极限分布不是平常x2分布,1985年已为Hartigan指出.在这篇文章我们限制了混合比大于一正数下,讨论了两个含单个未知参数混合模型的Log极大似然比统计量的极限分布,它是零与x2分布的混合分布.     

An evaluation of the multivariate dispersion charts with estimated parameters under non‐normality

Various charts such as |S|, W, and G are used for monitoring process dispersion. Most of these charts are based on the normality assumption, while exact distribution of the control statistic is unknown, and thus limiting distribution of control statistic is employed which is applicable for large sample sizes. In practice, the normality assumption of distribution might be violated, while it is not always possible to collect large sample size. Furthermore, to use control charts in practice, the in‐control state usually has to be estimated. Such estimation has a negative effect on the performance of control chart. Non‐parametric bootstrap control charts can be considered as an alternative when the distribution is unknown or a collection of large sample size is not possible or the process parameters are estimated from a Phase I data set. In this paper, non‐parametric bootstrap multivariate control charts |S|, W, and G are introduced, and their performances are compared against Shewhart‐type control charts. The proposed method is based on bootstrapping the data used for estimating the in‐control state. Simulation results show satisfactory performance for the bootstrap control charts. Ultimately, the proposed control charts are applied to a real case study.