Testing Equality of Mean Vectors in Two Sample Problem with Missing Data

Finding the influence of traffic accident on the road is helpful to analyze the characteristics of traffic flow, and take reasonable and effective control measures. Here, the detrended fluctuation analysis method is applied to investigate the complexity of time series in mixed traffic flow with a blockage induced by an accident. As a parameter to depict the long-term evolutionary behavior of the time series in traffic flow, the scaling exponent is analyzed. According to the scaling exponent, it is shown that the traffic flow time series can display long-range correlation characteristics, short-range correlation characteristics, and non-power-law relation in the long-range correlation characteristics, which is strongly dependent on the entering probability of vehicle, the ratio of slow vehicle and the blockage duration time.     

Parameter measures of skewness

Several different measures of skewness are commonly used in place of γ₁, the third central moment divided by the cube of the standard deviation. The numerical values of these measures are compared in this paper for members of the gamma, lognormal or Weibull family of distributions and shown to vary considerably in most cases even when skewness and kurtosis are moderate.     

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.     

Additive Outlier Detection and Estimation for the Logarithmic Autoregressive Conditional Duration Model

This study investigates the influences of additive outliers on financial durations. An outlier test statistic and an outlier detection procedure are proposed to detect and estimate outlier effects for the logarithmic Autoregressive Conditional Duration (Log-ACD) model. The proposed test statistic has an exact sampling distribution and performs very well, in terms of size and power, in a series of Monte Carlo simulations. Furthermore, the test statistic is robust to several alternative distribution assumptions. An empirical application shows that parameter estimates without considering outliers tend to be biased.     

An Approximate Bayesian Marginal Likelihood Approach for Estimating Finite Mixtures

Estimation of finite mixture models when the mixing distribution support is unknown is an important problem. This article gives a new approach based on a marginal likelihood for the unknown support. Motivated by a Bayesian Dirichlet prior model, a computationally efficient stochastic approximation version of the marginal likelihood is proposed and large-sample theory is presented. By restricting the support to a finite grid, a simulated annealing method is employed to maximize the marginal likelihood and estimate the support. Real and simulated data examples show that this novel stochastic approximation and simulated annealing procedure compares favorably with existing methods.     

Sample Size Calculation and Blinded Sample Size Recalculation in Clinical Trials Where the Treatment Effect is Measured by the Relative Risk

In clinical trials with binary endpoints, the required sample size does not depend only on the specified type I error rate, the desired power and the treatment effect but also on the overall event rate which, however, is usually uncertain. The internal pilot study design has been proposed to overcome this difficulty. Here, nuisance parameters required for sample size calculation are re-estimated during the ongoing trial and the sample size is recalculated accordingly. We performed extensive simulation studies to investigate the characteristics of the internal pilot study design for two-group superiority trials where the treatment effect is captured by the relative risk. As the performance of the sample size recalculation procedure crucially depends on the accuracy of the applied sample size formula, we firstly explored the precision of three approximate sample size formulae proposed in the literature for this situation. It turned out that the unequal variance asymptotic normal formula outperforms the other two, especially in case of unbalanced sample size allocation. Using this formula for sample size recalculation in the internal pilot study design assures that the desired power is achieved even if the overall rate is mis-specified in the planning phase. The maximum inflation of the type I error rate observed for the internal pilot study design is small and lies below the maximum excess that occurred for the fixed sample size design.     

Graphical and Numerical Methods for Detecting Influential Observations in Complex Bingham Data

We describe methods to detect influential observations in a sample of pre-shapes when the underlying distribution is assumed to be complex Bingham. One of these methods is based on Cook's distance, which is derived from the likelihood of the complex Bingham distribution. Other method is related to the tangent space, which is based on the local influence for the multivariate normal distribution. A method to detect outliers is also explained. The application of the methods is illustrated in both a real dataset and a simulated sample.     

Mixture Models in View of Evidential Analysis

In some practical inferential situations, it is needed to mix some finite sort of distributions to fit an adequate model for multi-modal observations. In this article, using evidential analysis, we determine the sample size for supporting hypotheses about the mixture proportion and homogeneity. An Expectation-Maximization algorithm is used to evaluate the probability of strong misleading evidence based on modified likelihood ratio as a measure of support.     

Bayes Estimation Based on Joint Progressive Type II Censored Data Under LINEX Loss Function

In the lifetime experiments, the joint censoring scheme is useful for planning comparative purposes of two identical products manufactured coming from different lines. In this article, we will confine ourselves to the data obtained by conducting a joint progressive Type II censoring scheme on the basis of the two combined samples selected from the two lines. Moreover, it is supposed that the distributions of lifetimes of the two products satisfy in a proportional hazard model. A general form for the distributions is considered, and we tackle the problem of obtaining Bayes estimates under the squared error and linear-exponential (LINEX) loss functions. As a special case, the Weibull distribution is discussed in more detail. Finally, the estimated risks of the various estimators obtained are compared using the Monte Carlo method.