On an asymptotic relative efficiency concept based on expected volumes of confidence regions

ABSTRACT

The paper deals with an asymptotic relative efficiency concept for confidence regions of multidimensional parameters that is based on the expected volumes of the confidence regions. Under standard conditions the asymptotic relative efficiencies of confidence regions are seen to be certain powers of the ratio of the limits of the expected volumes. These limits are explicitly derived for confidence regions associated with certain plugin estimators, likelihood ratio tests and Wald tests. Under regularity conditions, the asymptotic relative efficiency of each of these procedures with respect to each one of its competitors is equal to 1. The results are applied to multivariate normal distributions and multinomial distributions in a fairly general setting.     

Testing dominance of biometric functions

The hazard rate (HR) and mean residual lifetime are two of the most practical and best-known functions in biometry, reliability, statistics and life testing. Recently, the reversed HR function is found to have interesting properties useful in additional areas such as censored data and forensic science. For these three biometric functions, we propose testing methods that they take on a known functional form against that they dominate or are dominated by this known form. This goodness-of-fit-type testing is wider in applications and more interesting than the long-standing testing procedures for exponentiality against the monotonicity of these functions or even the change point problems. This is so since we can test against any choice of the survival distribution and not just exponentiality. For this general testing, we present easy to implement tests and generalize them into classes of statistics that could lead to more powerful and efficient testing.     

Parameter identification for the discretely observed geometric fractional Brownian motion

This paper deals with the problem of estimating all the unknown parameters of geometric fractional Brownian processes from discrete observations. The estimation procedure is built upon the marriage of the quadratic variation and the maximum likelihood approach. The asymptotic properties of the estimators are provided. Moveover, we compare our derived method with the approach proposed by Misiran et al. [Fractional Black-Scholes models: complete MLE with application to fractional option pricing. In International conference on optimization and control; Guiyang, China; 2010. p. 573–586.], namely the complete maximum likelihood estimation. Simulation studies confirm theoretical findings and illustrate that our methodology is efficient and reliable. To show how to apply our approach in realistic contexts, an empirical study of Chinese financial market is also presented.     

Analysis of Type-II hybrid censored competing risks data

Kundu and Gupta [Analysis of hybrid life-tests in presence of competing risks. Metrica. 2007;65:159–170] provided the analysis of Type-I hybrid censored competing risks data, when the lifetime distributions of the competing cause of failures follows exponential distribution. In this paper, we consider the analysis of Type-II hybrid censored competing risks data. It is assumed that latent lifetime distributions of the competing causes of failures follow independent exponential distributions with different scale parameters. It is observed that the maximum likelihood estimators of the unknown parameters do not always exist. We propose the modified estimators of the scale parameters, which coincide with the corresponding maximum likelihood estimators when they exist, and asymptotically they are equivalent. We obtain the exact distribution of the proposed estimators. Using the exact distributions of the proposed estimators, associated confidence intervals are obtained. The asymptotic and bootstrap confidence intervals of the unknown parameters are also provided. Further, Bayesian inference of some unknown parametric functions under a very flexible Beta-Gamma prior is considered. Bayes estimators and associated credible intervals of the unknown parameters are obtained using the Monte Carlo method. Extensive Monte Carlo simulations are performed to see the effectiveness of the proposed estimators and one real data set has been analysed for the illustrative purposes. It is observed that the proposed model and the method work quite well for this data set.     

A note on the asymptotic distribution of the process capability index C pmk

Process capability indices have been widely used to evaluate the process performance to the continuous improvement of quality and productivity. The distribution of the estimator of the process capability index C _pmk is very complicated and the asymptotic distribution is proposed by Chen and Hsu [The asymptotic distribution of the processes capability index C _pmk , Comm. Statist. Theory Methods 24(5) (1995), pp. 1279–1291]. However, we found a critical error for the asymptotic distribution when the population mean is not equal to the midpoint of the specification limits. In this paper, a correct version of the asymptotic distribution is given. An asymptotic confidence interval of C _pmk by using the correct version of asymptotic distribution is proposed and the lower bound can be used to test if the process is capable. A simulation study of the coverage probability of the proposed confidence interval is shown to be satisfactory. The relation of six sigma technique and the index C _pmk is also discussed in this paper. An asymptotic testing procedure to determine if a process is capable based on the index of C _pmk is also given in this paper.     

A new confidence interval in errors-in-variables model with known error variance

This paper considers constructing a new confidence interval for the slope parameter in the structural errors-in-variables model with known error variance associated with the regressors. Existing confidence intervals are so severely affected by Gleser–Hwang effect that they are subject to have poor empirical coverage probabilities and unsatisfactory lengths. Moreover, these problems get worse with decreasing reliability ratio which also result in more frequent absence of some existing intervals. To ease these issues, this paper presents a fiducial generalized confidence interval which maintains the correct asymptotic coverage. Simulation results show that this fiducial interval is slightly conservative while often having average length comparable or shorter than the other methods. Finally, we illustrate these confidence intervals with two real data examples, and in the second example some existing intervals do not exist.     

Asymptotic properties of the maximum-likelihood estimator in zero-inflated binomial regression

The zero-inflated binomial (ZIB) regression model was proposed to account for excess zeros in binomial regression. Since then, the model has been applied in various fields, such as ecology and epidemiology. In these applications, maximum-likelihood estimation (MLE) is used to derive parameter estimates. However, theoretical properties of the MLE in ZIB regression have not yet been rigorously established. The current paper fills this gap and thus provides a rigorous basis for applying the model. Consistency and asymptotic normality of the MLE in ZIB regression are proved. A consistent estimator of the asymptotic variance–covariance matrix of the MLE is also provided. Finite-sample behavior of the estimator is assessed via simulations. Finally, an analysis of a data set in the field of health economics illustrates the paper.     

Affiliation networks with an increasing degree sequence

Affiliation network is one kind of two-mode social network with two different sets of nodes (namely, a set of actors and a set of social events) and edges representing the affiliation of the actors with the social events. Although a number of statistical models are proposed to analyze affiliation networks, the asymptotic behaviors of the estimator are still unknown or have not been properly explored. In this article, we study an affiliation model with the degree sequence as the exclusively natural sufficient statistic in the exponential family distributions. We establish the uniform consistency and asymptotic normality of the maximum likelihood estimator when the numbers of actors and events both go to infinity. Simulation studies and a real data example demonstrate our theoretical results.