Assessing robustness of crossover designsto subjects dropping out

In some crossover experiments, particularly in medical applications, subjects may fail to complete their sequences of treatments for reasons unconnected with the treatments received. A method is described of assessing the robustness of a planned crossover design, with more than two periods, to subjects leaving the study prematurely. The method involves computing measures of efficiency for every possible design that can result, and is therefore very computationally intensive. Summaries of these measures are used to choose between competing designs. The computational problem is reduced to a manageable size by a software implementation of Polya theory. The method is applied to comparing designs for crossover studies involving four treatments and four periods. Designs are identified that are more robust to subjects dropping out in the final period than those currently favoured in medical and clinical trials.     

Weighted Approximations to Continuous Time Martingales with Applications

A weighted approximation to a sequence of continuous time martingales by a time transformed Wiener process is established. The basic tool of proof is the Skorohod imbedding for martingale difference sequences. As an application of the main result a useful weighted approximation to the randomly weighted uniform empirical process is derived. A number of other applications are also discussed.     

试论时间价值观

时间是一种重要的资源。时间价值观是个体以自身的需要为尺度对时间重要性的稳定态度和观念。时间价值观含有时间资源观、时间效率观、时机观、时间顺序观等方面的意义。时间价值观的结构可以有不同的划分。时间价值观受社会文化传统、个体的年龄因素、个体性格、社会环境等因素的影响。心理学对时间价值观的研究在理论上和应用中都有着重要的意义     

Modelling small and medium enterprise loan defaults as rare events: the generalized extreme value regression model

A pivotal characteristic of credit defaults that is ignored by most credit scoring models is the rarity of the event. The most widely used model to estimate the probability of default is the logistic regression model. Since the dependent variable represents a rare event, the logistic regression model shows relevant drawbacks, for example, underestimation of the default probability, which could be very risky for banks. In order to overcome these drawbacks, we propose the generalized extreme value regression model. In particular, in a generalized linear model (GLM) with the binary-dependent variable we suggest the quantile function of the GEV distribution as link function, so our attention is focused on the tail of the response curve for values close to one. The estimation procedure used is the maximum-likelihood method. This model accommodates skewness and it presents a generalisation of GLMs with complementary log–log link function. We analyse its performance by simulation studies. Finally, we apply the proposed model to empirical data on Italian small and medium enterprises.     

Extended generalised Pareto models for tail estimation

The most popular approach in extreme value statistics is the modelling of threshold exceedances using the asymptotically motivated generalised Pareto distribution. This approach involves the selection of a high threshold above which the model fits the data well. Sometimes, few observations of a measurement process might be recorded in applications and so selecting a high quantile of the sample as the threshold leads to almost no exceedances. In this paper we propose extensions of the generalised Pareto distribution that incorporate an additional shape parameter while keeping the tail behaviour unaffected. The inclusion of this parameter offers additional structure for the main body of the distribution, improves the stability of the modified scale, tail index and return level estimates to threshold choice and allows a lower threshold to be selected. We illustrate the benefits of the proposed models with a simulation study and two case studies.     

Changepoint Estimation in a Segmented Linear Regression via Empirical Likelihood

For a segmented regression system with an unknown changepoint over two domains of a predictor, a new empirical likelihood ratio statistic is proposed to test the null hypothesis of no change. Under the null hypothesis of no change, the proposed test statistic is shown empirically to be Gumbel distributed with robust location and scale estimators against various parameter settings and error distributions. A power analysis is conducted to illustrate the performance of the test. Under the alternative hypothesis with a changepoint, the test statistic is utilized to estimate the changepoint between the two domains. A comparison of the frequency distributions between the proposed estimator and two parametric methods indicates that the proposed method is effective in capturing the true changepoint.     

A Goodness-of-Fit Test for Location-Scale Max-Stable Distributions

In this article, a technique based on the sample correlation coefficient to construct goodness-of-fit tests for max-stable distributions with unknown location and scale parameters and finite second moment is proposed. Specific details to test for the Gumbel distribution are given, including critical values for small sample sizes as well as approximate critical values for larger sample sizes by using normal quantiles. A comparison by Monte Carlo simulation shows that the proposed test for the Gumbel hypothesis is substantially more powerful than some other known tests against some alternative distributions with positive skewness coefficient.     

Correlation-Type Goodness of Fit Test for Extreme Value Distribution Based on Simultaneous Closeness

In reliability studies, one typically would assume a lifetime distribution for the units under study and then carry out the required analysis. One popular choice for the lifetime distribution is the family of two-parameter Weibull distributions (with scale and shape parameters) which, through a logarithmic transformation, can be transformed to the family of two-parameter extreme value distributions (with location and scale parameters). In carrying out a parametric analysis of this type, it is highly desirable to be able to test the validity of such a model assumption. A basic tool that is useful for this purpose is a quantile–quantile (QQ) plot, but in its use, the issue of the choice of plotting position arises. Here, by adopting the optimal plotting points based on Pitman closeness criterion proposed recently by Balakrishnan et al. (2010b Balakrishnan , N. , Davies , K. F. , Keating , J. P. , Mason , R. L. ( 2010b ). Computation of optimal plotting points based on Pitman Closeness with an application to goodness of fit for location-scale families. Submitted to Computational Statistics & Data Analysis.  [Google Scholar]), and referred to as simultaneous closeness probability (SCP) plotting points, we propose a correlation-type goodness of fit test for the extreme value distribution. We compute the SCP plotting points for various sample sizes and use them to determine the mean, standard deviation and critical values for the proposed correlation-type test statistic. Using these critical values, we carry out a power study, similar to the one carried out by Kinnison (1989 Kinnison , R. ( 1989 ). Correlation coefficient goodness of fit test for extreme value distribution . The American Statistician 43 : 98 – 100 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), through which we demonstrate that the use of SCP plotting points results in better power than with the use of mean ranks as plotting points and nearly the same power as with the use of median ranks. We then demonstrate the use of the SCP plotting points and the associated correlation-type test for Weibull analysis with an illustrative example. Finally, for the sake of comparison, we also adapt two statistics proposed by Gan and Koehler (1990 Gan , F. F. , Koehler , K. J. ( 1990 ). Goodness of fit based on P-P probability plots . Technometrics 32 : 289 – 303 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), in the context of probability–probability (PP) plots, based on SCP plotting points and compare their performance to those based on mean ranks. The empirical study also reveals that the tests from the QQ plot have better power than those from the PP plot.     

The rank distribution of the maximin in a random matrix

Let S be a set of tm distinct real numbers and R a random t × m matrix of these tm numbers with rows {r_i} and columns (c_i}. Define b = Max Min x. l≤i≤t x?r_i. Let c be the event Max Min x = Min Max x. l≤i≤t x?r_i l≤i≤m x?c_i. This paper derives the probability distribution of the rank of b in S, as well as the same distribution conditional on c.     

The best subset of parameters in min-max linear regression

These Fortran-77 subroutines provide building blocks for Generalized Cross-Validation (GCV) (Craven and Wahba, 1979) calculations in data analysis and data smoothing including ridge regression (Golub, Heath, and Wahba, 1979), thin plate smoothing splines (Wahba and Wendelberger, 1980), deconvolution (Wahba, 1982d), smoothing of generalized linear models (O'sullivan, Yandell and Raynor 1986, Green 1984 and Green and Yandell 1985), and ill-posed problems (Nychka et al., 1984, O'sullivan and Wahba, 1985). We present some of the types of problems for which GCV is a useful method of choosing a smoothing or regularization parameter and we describe the structure of the subroutines.Ridge Regression: A familiar example of a smoothing parameter is the ridge parameter X in the ridge regression problem which we write.