On the probability of ruin in a continuous risk model with two types of delayed claims

ABSTRACT

This article studies a risk model involving one type of main claims and two types of by-claims, which is an extension of the general risk model with delayed claims. We suppose that every main claim may not induce any by-claims or may induce one by-claim belonging to one of the two types of by-claims with a certain probability. In addition, assume that the by-claim and its associated main claim may occur at the same time and that the occurrence of the by-claim may be delayed. An integro-differential equation system for survival probabilities is derived by using two auxiliary risk models. The expression of the survival probability is obtained by applying Laplace transforms and Rouché theorem. Furthermore, we provide a method for solving the survival probability when the two by-claim amounts satisfy different exponential distributions. As a special case, an explicit expression of survival probability is given when all the claim amounts obey the same exponential distribution. Finally, numerical results are provided to examine the proposed method.     

On the Gerber–Shiu discounted penalty function in a risk model with delayed claims

In this paper, we consider an extension to the continuous time risk model for which the occurrence of the claim may be delayed and the time of delay for the claim is assumed to be random. Two types of dependent claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim. The time of occurrence of a by-claim is later than that of its associate main claim and the time of delay for the occurrence of a by-claim is random. An integro-differential equations system for the Gerber–Shiu discounted penalty function is established using the auxiliary risk models. Both the system of Laplace transforms of the Gerber–Shiu discounted penalty functions and the Gerber–Shiu discounted penalty functions with zero initial surplus are obtained. From Lagrange interpolating theorem, we prove that the Gerber–Shiu discounted penalty function satisfies a defective renewal equation. Exact representation for the solution of this equation is derived through an associated compound geometric distribution. Finally, examples are given with claim sizes that have exponential and a mixture of exponential distributions.     

Asymptotic ruin probability for a by-claim risk model with pTQAI claims and constant interest force

Abstract

In this paper, we consider a by-claim risk model with a constant rate of interest force, in which the main claims and the by-claims form a sequence of pTQAI nonnegative random variables and all their distributions belong to the dominatedly-varying heavy-tailed subclass. We obtain the asymptotically upper and lower bound formulas of the ultimate ruin probability for such a by-claim risk model. As its by-products, some interesting properties for pTQAI structure are also investigated. The results extend some existing ones in the literature.     

Dividend barrier and ruin problems for a risk model with delayed claims

In this paper, a compound Poisson risk model in the presence of a constant dividend barrier is considered. Two types of individual claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim and and the time of delay for the claim is assumed to be random. A system of integro-differential equations with certain boundary conditions for the expected discounted penalty function is derived. We show that its solution can be expressed as the solution to the expected discounted penalty function in the same risk model with the absence of a barrier plus a linear combination of two linearly independent solutions to the associated homogeneous integro-differential equation. Using systems of integro-differential equations for the moment-generating function as well as for the arbitrary moments of the sum of discounted dividend payments until ruin, a matrix version of the dividends–penalty type relationship is derived. We also prove that ruin is certain under constant dividend barrier strategy. The closed form expressions are given when the claim amounts from both classes are exponentially distributed. Finally, a numerical example is presented to illustrate the solution procedure.     

On tests detecting difference in means and variances simultaneously under normality

In this paper, we propose several tests for detecting difference in means and variances simultaneously between two populations under normality. First of all, we propose a likelihood ratio test. Then we obtain an expression of the likelihood ratio statistic by a product of two functions of random quantities, which can be used to test the two individual partial hypotheses for differences in means and variances. With those individual partial tests, we propose a union-intersection test. Also we consider two optimal tests by combining the p-values of the two individual partial tests. For obtaining null distributions, we apply the permutation principle with the Monte Carlo approach. Then we compare efficiency among the proposed tests with well-known ones through a simulation study. Finally, we discuss some interesting features related to the simultaneous tests and resampling methods as concluding remarks.     

Estimation of a Discriminant Function from a Mixture of Two Burr Type III Distributions

In this article, we propose the finite mixture of two Burr Type-III distributions (MTBIIID). First, we formulate the proposed model with some properties and prove the identifiability property. Next, we obtain the maximum likelihood estimates (MLEs) of the unknown parameters of MTBIIID under classified and unclassified observations. Also, we estimate the nonlinear discriminant function of the underlying model. In addition, we calculate the total probabilities of misclassification as well as the percentage bias. Further, we investigate the performance of the all results through series of the simulation experiments by the means of the relative efficiencies.     

Stochastic Bounds with a Low Rank Decomposition

We investigate how we can bound a discrete time Markov chain (DTMC) by a stochastic matrix with a low rank decomposition. In the first part of the article, we show the links with previous results for matrices with a decomposition of size 1 or 2. Then we show how the complexity of the analysis for steady-state and transient distributions can be simplified when we take into account the decomposition. Finally, we show how we can obtain a monotone stochastic upper bound with a low rank decomposition.     

A Nonparametric Test Procedure Based on Quantiles for the Right Censored Data

In this study, we propose nonparametric tests using the several quantile statistics simultaneously for the right censored data. First of all, we consider statistics of the quadratic form with estimated covariance matrices. Then we derive the limiting distribution using the large sample approximation theory. Also we consider different forms of statistics such as the maximal and summing types with their limiting distributions. Then we illustrate our procedure with examples and compare performance among tests with empirical powers through a simulation study. Also we comment briefly on some interesting features including re-sampling methods as concluding remarks. Finally in Appendices, we provide proofs for the theoretic results needed for the derivation of the limiting distributions of the proposed test statistics.     

On a length-biased life distribution based on the sinh-normal model

In this article, we characterize a new life distribution based on the sinh-normal model. Specifically, we find the density, the distribution function, and the moments of the new model. In addition, we carry out a brief graphical analysis of its density. Furthermore, we derive some properties and transformations related to the new distribution. Moreover, we conduct a study of its hazard rate. Finally, we present an example that illustrates the obtained results and a computational implementation of these results.     

A Nonparametric Test Procedure Based on a Group of Quantile Tests

In this article, we consider nonparametric test procedures based on a group of quantile test statistics. We consider the quadratic form for the two-sided test and the maximal and summing types of statistics for the one-sided alternatives. Then we derive the null limiting distributions of the proposed test statistics using the large sample approximation theory. Also, we consider applying the permutation principle to obtain the null distribution. In this vein, we may consider the supremum type, which should use the permutation principle for obtaining the null distribution. Then we illustrate our procedure with an example and compare the proposed tests with other existing tests including the individual quantile tests by obtaining empirical powers through simulation study. Also, we comment on the related discussions to this testing procedure as concluding remarks. Finally we prove the lemmas and theorems in the appendices.     

A method for constructing asymmetric pair-copula and its application

Pair-copula has become a hot spot in the research of both theory and application of statistics. Therefore, proper construction of pair-copula is an area worthy of study. Considering the asymmetry of variate dependence in practical applications and the disadvantages of the widely used asymmetric copulas, we propose a method to construct asymmetric pair-copula. In our method, we treat the asymmetric bivariate copula constructed by Liebscher's method as a generator, using this generator to construct asymmetric pair-copula. Also, on the basis of our method, we propose and prove a reference for selecting copula family in the construction. To verify the method, we construct asymmetric copulas and asymmetric pair-copulas using the daily runoff data collected at Yichang hydrological station to describe the extreme drought events of Yangtze River. After comparing the models in some aspects, we accept a model we construct, and the result displays the feasibility and practicality of the method we propose.     

Median Control Charts Based on Bootstrap Method

In this study, we propose a median control chart. In order to determine the control limits, we consider using an estimate of the variance of sample median. Also, we consider applying the bootstrap methods. Then we illustrate the proposed median control chart with an example and compare the bootstrap methods by simulation study. Finally, we discuss some peculiar features for the median control chart as concluding remarks.     

Bivariate Birnbaum-Saunders accelerated lifetime model: estimation and diagnostic analysis

In this paper, we discuss the bivariate Birnbaum-Saunders accelerated lifetime model, in which we have modeled the dependence structure of bivariate survival data through the use of frailty models. Specifically, we propose the bivariate model Birnbaum-Saunders with the following frailty distributions: gamma, positive stable and logarithmic series. We present a study of inference and diagnostic analysis for the proposed model, more concisely, are proposed a diagnostic analysis based in local influence and residual analysis to assess the fit model, as well as, to detect influential observations. In this regard, we derived the normal curvatures of local influence under different perturbation schemes and we performed some simulation studies for assessing the potential of residuals to detect misspecification in the systematic component, the presence in the stochastic component of the model and to detect outliers. Finally, we apply the methodology studied to real data set from recurrence in times of infections of 38 kidney patients using a portable dialysis machine, we analyzed these data considering independence within the pairs and using the bivariate Birnbaum-Saunders accelerated lifetime model, so that we could make a comparison and verify the importance of modeling dependence within the times of infection associated with the same patient.     

Influence diagnostics on the coefficient of variation of elliptically contoured distributions

In this article, we study the behavior of the coefficient of variation (CV) of a random variable that follows a symmetric distribution in the real line. Specifically, we estimate this coefficient using the maximum-likelihood (ML) method. In addition, we provide asymptotic inference for this parameter, which allows us to contrast hypothesis and construct confidence intervals. Furthermore, we produce influence diagnostics to evaluate the sensitivity of the ML estimate of this coefficient when atypical data are present. Moreover, we illustrate the obtained results by using financial real data. Finally, we carry out a simulation study to detect the potential influence of atypical observations on the ML estimator of the CV of a symmetric distribution. The illustration and simulation demonstrate the robustness of the ML estimation of this coefficient.     

A permutation test for multivariate data with grouped components

In this paper, we consider a nonparametric test procedure for multivariate data with grouped components under the two sample problem setting. For the construction of the test statistic, we use linear rank statistics which were derived by applying the likelihood ratio principle for each component. For the null distribution of the test statistic, we apply the permutation principle for small or moderate sample sizes and derive the limiting distribution for the large sample case. Also we illustrate our test procedure with an example and compare with other procedures through simulation study. Finally, we discuss some additional interesting features as concluding remarks.     

Shrinkage and LASSO strategies in high-dimensional heteroscedastic models

ABSTRACT

In this paper, we consider the estimation problem of the parameter vector in the linear regression model with heteroscedastic errors. First, under heteroscedastic errors, we study the performance of shrinkage-type estimators and their performance as compared to theunrestricted and restricted least squares estimators. In order to accommodate the heteroscedastic structure, we generalize an identity which is useful in deriving the risk function. Thanks to the established identity, we prove that shrinkage estimators dominate the unrestricted estimator. Finally, we explore the performance of high-dimensional heteroscedastic regression estimator as compared to classical LASSO and shrinkage estimators.     

典型相关分析的延拓研究

在典型相关分析中,求得典型相关变量的表达式并没有全部完成任务,例如需要确定典型相关变量的个数和变量选择。针对典型相关变量的个数问题,发现了常用的卡方检验和冗余分析方法的不足,进而提出了一种新的算法。针对原始变量的选择问题,提出了三种可能的路径。最后利用人体尺寸数据对相关结论进行了验证。     

Likelihood ratio tests for fuzzy hypotheses testing

In hypotheses testing, such as other statistical problems, we may confront imprecise concepts. One case is a situation in which hypotheses are imprecise. In this paper, we recall and redefine some concepts about fuzzy hypotheses testing, and then we introduce the likelihood ratio test for fuzzy hypotheses testing. Finally, we give some applied examples.     

Homogeneity test for functional data

In the context of functional data analysis, we propose new sample tests for homogeneity. Based on some well-known depth measures, we construct four different statistics in order to measure distance between the two samples. A simulation study is performed to check the efficiency of the tests when confronted with shape and magnitude perturbation. Finally, we apply these tools to measure the homogeneity in some samples of real data, and we obtain good results using this new method.