A backward construction and simulation of correlated Poisson processes

In this paper, we consider a generalisation of the backward simulation method of Duch et al. [New approaches to operational risk modeling. IBM J Res Develop. 2014;58:1–9] to build bivariate Poisson processes with flexible time correlation structures, and to simulate the arrival times of the processes. The proposed backward construction approach uses the Marshall–Olkin bivariate binomial distribution for the conditional law and some well-known families of bivariate copulas for the joint success probability in lieu of the typical conditional independence assumption. The resulting bivariate Poisson process can exhibit various time correlation structures which are commonly observed in real data.     

On confidence intervals for the mean past lifetime function under random censorship

The mean past lifetime (MPL) function (also known as the expected inactivity time function) is of interest in many fields such as reliability theory and survival analysis, actuarial studies and forensic science. For estimation of the MPL function some procedures have been proposed in the literature. In this paper, we give a central limit theorem result for the estimator of MPL function based on a right-censored random sample from an unknown distribution. The limiting distribution is used to construct normal approximation-based confidence interval for MPL. Furthermore, we use the empirical likelihood ratio procedure to obtain confidence interval for the MPL function. These two intervals are compared with each other through simulation study in terms of coverage probability. Finally, a couple of numerical example illustrating the theory is also given.     

New goodness of fit tests for the Cauchy distribution

Some goodness-of-fit procedures for the Cauchy distribution are presented. The power comparisons indicate that the new tests possess good performances among the competitors, especially against symmetric alternatives. A financial data set is analyzed for illustration.     

Estimation of P{X < Y} for geometric—exponential model based on complete and censored samples

This article deals with the estimation of R = P{X < Y}, where X and Y are independent random variables from geometric and exponential distribution, respectively. For complete samples, the MLE of R, its asymptotic distribution, and confidence interval based on it are obtained. The procedure for deriving bootstrap-p confidence interval is presented. The UMVUE of R and UMVUE of its variance are derived. The Bayes estimator of R is investigated and its Lindley's approximation is obtained. A simulation study is performed in order to compare these estimators. Finally, all point estimators for right censored sample from the exponential distribution, are obtained.     

On robustness of the Shiryaev–Roberts change-point detection procedure under parameter misspecification in the post-change distribution

The gist of the quickest change-point detection problem is to detect the presence of a change in the statistical behavior of a series of sequentially made observations, and do so in an optimal detection-speed-versus-“false-positive”-risk manner. When optimality is understood either in the generalized Bayesian sense or as defined in Shiryaev's multi-cyclic setup, the so-called Shiryaev–Roberts (SR) detection procedure is known to be the “best one can do”, provided, however, that the observations’ pre- and post-change distributions are both fully specified. We consider a more realistic setup, viz. one where the post-change distribution is assumed known only up to a parameter, so that the latter may be misspecified. The question of interest is the sensitivity (or robustness) of the otherwise “best” SR procedure with respect to a possible misspecification of the post-change distribution parameter. To answer this question, we provide a case study where, in a specific Gaussian scenario, we allow the SR procedure to be “out of tune” in the way of the post-change distribution parameter, and numerically assess the effect of the “mistuning” on Shiryaev's (multi-cyclic) Stationary Average Detection Delay delivered by the SR procedure. The comprehensive quantitative robustness characterization of the SR procedure obtained in the study can be used to develop the respective theory as well as to provide a rational for practical design of the SR procedure. The overall qualitative conclusion of the study is an expected one: the SR procedure is less (more) robust for less (more) contrast changes and for lower (higher) levels of the false alarm risk.     

A comparison study of computational methods of Kolmogorov–Smirnov statistic in credit scoring

Kolmogorov–Smirnov statistic (KS) is a standard measure in credit scoring. Currently, there are three computational methods of KS: method with equal-width binning, method with equal-size binning and method without binning. This paper compares the three methods in three aspects: Values, Rank Ordering of Scores and Geometrical Way. The computational results on the German Credit Data show that only the method without binning can produce a unique value of KS. It is further proved analytically that the method without binning yields the maximum value of KS among the three methods. The computational results also show that only the method with equal-size binning can be used to evaluate rank ordering of scores. Moreover, it is proved that all the three methods can be used to calculate KS in a geometric way.     

Riemann manifold Langevin methods on stochastic volatility estimation

In this article, we perform Bayesian estimation of stochastic volatility models with heavy tail distributions using Metropolis adjusted Langevin (MALA) and Riemman manifold Langevin (MMALA) methods. We provide analytical expressions for the application of these methods, assess the performance of these methodologies in simulated data, and illustrate their use on two financial time series datasets.     

The discrepancy of P-values and posterior probability: Two-sided hypothesis of variance of normal distribution with unknown mean

This article is concerned with the comparison of P-value and Bayesian measure in point null hypothesis for the variance of Normal distribution with unknown mean. First, using fixed prior for test parameter, the posterior probability is obtained and compared with the P-value when an appropriate prior is used for the mean parameter. In the second, lower bounds of the posterior probability of H₀ under a reasonable class of prior are compared with the P-value. It has been shown that even in the presence of nuisance parameters, these two approaches can lead to different results in the statistical inference.