Nonparametric Inference for a Partially Observed Compartmental Process

Methods of nonparametric inference are proposed for a process with two transient and three absorbing states. It is assumed that the time of transitions between the transient states are unobservable. One area of applications is in epidemiology where the transient states correspond to healthy and ill, while the absorbing states correspond to types of death. It is the onset of illness which is not observable. An estimate is given for a cumulative hazard rate between the transient states, the exit hazard rates are estimated at a specific point in time and a statistic for comparing exit rates from the transient states is given.     

Basis risk modelling: a cointegration-based approach

ABSTRACT

In this paper we propose a multivariate approach for forecasting pairwise mortality rates of related populations. The need for joint modelling of mortality rates is analysed using a causality test. We show that for the datasets considered, the inclusion of national mortality information enhances predictions on its subpopulations. The investigated approach links national population mortality to that of a subset population, using an econometric model that captures a long-term relationship between the two mortality dynamics. This model does not focus on the correlation between the mortality rates of the two populations, but rather their long-term behaviour, which suggests that the two times series cannot wander off in opposite directions for long before mean reverting, which is consistent with biological reasoning. The model can additionally capture short-term adjustments in the mortality dynamics of the two populations. An empirical comparison of the forecast of one-year death probabilities for policyholders is performed using both a classical factor-based model and the proposed approach. The robustness of the model is tested on mortality rate data for England and Wales, alongside the Continuous Mortality Investigation assured lives dataset, representing the subpopulation.     

Some state space models of hiv epidemic and its applications for the estimation of hiv infection and incubation

In this paper we have developed some state space models for the HIV epidemic for populations at risk for AIDS. By using these state space models, we have developed a general Bayesian procedure for estimating simultaneously the unknown parameters and the state variables. The unknown parameters include the immigration and recruitment rates, the death and retirement rates, the incidence of HIV infection ( and hence the HIV infection distribution ) and the incidence of HIV incubation ( and hence the HIV incubation distribution). The state variables are the numbers of susceptible people (S people), HIV-infected people (I people) and AIDS incidence over time. The basic approach is through multi-level Gibbs sampler combined with the weighted bootstrap method. We have applied the methods to the Swiss AIDS homosexual and IV drug data to estimate simultaneously the unknown parameters and the state variables. Our results show that in both populations, both the HIV infection and HIV incubation have multi-peaks indicating the mixture nature of these distributions. Our results have also shown that the estimates of the death and retirement rates for I people are greater than those of S people, suggesting that the infection by HIV may have increased the death and retirement rates of the individuals.     

A note on the passage time of finite-state Markov chains

Consider a Markov chain with finite state {0, 1, …, d}. We give the generation functions (or Laplace transforms) of absorbing (passage) time in the following two situations: (1) the absorbing time of state d when the chain starts from any state i and absorbing at state d; (2) the passage time of any state i when the chain starts from the stationary distribution supposed the chain is time reversible and ergodic. Example shows that it is more convenient compared with the existing methods, especially we can calculate the expectation of the absorbing time directly.     

Local polynomial estimation of Poisson intensities in the presence of reporting delays

Summary.  The system for monitoring suicides in Hong Kong has considerable delays in reporting as the cause of death needs to be determined by a coroner's investigation. However, timely estimates of suicide rates are desirable to assist in the formulation of public health policies. This motivated us to develop a non-parametric procedure to estimate the intensity function of a Poisson process in the presence of reporting delays. We give closed form estimators of the Poisson intensity and the delay distribution, conduct simulation studies to evaluate the method proposed and derive their asymptotic properties. The method proposed is applied to estimate the intensity of suicide in Hong Kong.     

On the two-parameter Bell–Touchard discrete distribution

Abstract

In this paper we introduce a new two-parameter discrete distribution which may be useful for modeling count data. Additionally, the probability mass function is very simple and it may have a zero vertex. We show that the new discrete distribution is a particular solution of a multiple Poisson process, and that it is infinitely divisible. Additionally, various structural properties of the new discrete distribution are derived. We also discuss two methods (moments and maximum likelihood) to estimate the model parameters. The usefulness of the proposed distribution is illustrated by means of real data sets to prove its versatility in practical applications.     

Ergodicity and truncation bounds for inhomogeneous birth and death processes with additional transitions from and to origin

ABSTRACT

In this paper, we present the extension of the analysis of time-dependent limiting characteristics the class of continuous-time birth and death processes defined on non-negative integers with special transitions from and to the origin. From the origin transitions can occur to any state. But being in any other state, besides ordinary transitions to neighboring states, a transition to the origin can occur. All possible transition intensities are assumed to be non-random functions of time and may depend on the state of the process. We improve previously known ergodicity and truncation bounds for this class of processes that were known only for the case when transitions from the origin decay exponentially (other intensities must have unique uniform upper bound). We show how the bounds can be obtained if the decay rate is slower than exponential. Numerical results are given in the queueing theory context.     

A hypothesis testing procedure for the evaluation on the lifetime performance index of products with Burr XII distribution under progressive type I interval censoring

ABSTRACT

It is a very important topic these days to assessing the lifetime performance of products in manufacturing or service industries. Lifetime performance indices C_L is used to measure the larger-the-better type quality characteristics to evaluate the process performance for the improvement of quality and productivity. The lifetimes of products are assumed to have Burr XII distribution. The maximum likelihood estimator is used to estimate the lifetime performance index based on the progressive type I interval censored sample. The asymptotic distribution of this estimator is also developed. We use this estimator to build the new hypothesis testing algorithmic procedure with respect to a lower specification limit. Finally, two practical examples are given to illustrate the use of this testing algorithmic procedure to determine whether the process is capable.     

Lindley first-order autoregressive model with applications

ABSTRACT

A new stationary first-order autoregressive process with Lindley marginal distribution, denoted as LAR(1) is introduced. We derive the probability function for the innovation process. We consider many properties of this process, involving spectral density, some multi-step ahead conditional measures, run probabilities, stationary solution, uniqueness and ergodicity. We estimate the unknown parameters of the process using three methods of estimation and investigate properties of the estimators with some numerical results to illustrate them. Some applications of the process are discussed to two real data sets and it is shown that the LAR(1) model fits better than other known non Gaussian AR(1) models.     

Asymptotic Methods of Enumeration and Applications to Markov Chain Models

Abstract

The purpose of this article is to present analytic methods for determining the asymptotic behaviour of the coefficents of power series that can be applied to homogeneous discrete quasi death and birth processes. It turns that there are in principle only three types for the asymptotic behaviour. The process either converges to the stationary distribution or it can be approximated in terms of a reflected Brownian motion or by a Brownian motion. In terms of Markov chains these cases correspond to positive recurrence, to null recurrence, and to non recurrence. The same results hold for the continuous case, too.     

Valuation of contingent claims with stochastic interest rate and mortality driven by Lévy processes

Abstract

In this paper, we consider a model with stochastic interest rate and stochastic mortality, which is driven by a Lévy process. Under the assumption that the stochastic mortality and interest rate are dependent, we discuss the valuation of life insurance contracts. Employing the method of change of measure together with the Bayes’ rule, we present the pricing formulas in closed form for the survival and death benefit models. Finally, numerical experiments illustrate the effects of some parameters.     

Residual life estimation under time-varying conditions based on a Wiener process

Residual life (RL) estimation plays an important role in prognostics and health management. In operating conditions, components usually experience stresses continuously varying over time, which have an impact on the degradation processes. This paper investigates a Wiener process model to track and predict the RL under time-varying conditions. The item-to-item variation is captured by the drift parameter and the degradation characteristic of the whole population is described by the diffusion parameter. The bootstrap method and Bayesian theorem are employed to estimate and update the distribution parameters of ‘a’ and ‘b’, which are the coefficients of the linear drifting process in the degradation model. Once new degradation information becomes available, the RL distributions considering the future operating condition are derived. The proposed method is tested on Lithium-ion battery devices under three levels of charging/discharging rates. The results are further validated by a simulation method.     

Property of bivariate poisson distribution and its application to stochastic processes

Assuming that the random vectors X ₁ and X ₂ have independent bivariate Poisson distributions, the conditional distribution of X ₁ given X ₁?+?X ₂?=?n is obtained. The conditional distribution turns out to be a finite mixture of distributions involving univariate binomial distributions and the mixing proportions are based on a bivariate Poisson (BVP) distribution. The result is used to establish two properties of a bivariate Poisson stochastic process which are the bivariate extensions of the properties for a Poisson process given by Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, Academic Press, New York.     

Pricing longevity-linked derivatives using a stochastic mortality model

Abstract

We propose a 2-factor MBMM model with exponential Lévy process to develop a stochastic mortality process. The two components are fitted by two independent NIG distributions. Compared to Lee–Carter model or 1-factor MBMM model, our mortality model explains more variation and improves the goodness of fit by including the second time component. Based on the improved model, we price three longevity-linked financial instruments, namely the longevity bond, q-forward and s-forward. The pricing is demonstrated on English and Welsh males aged 65 in 2013. Results indicate that the 2-factor MBMM model gives the highest price for mortality-related type of contract.     

Non-parametric Bayesian inference through MCMC method for Y-linked two-sex branching processes with blind choice

ABSTRACT

Dirichlet-process-based non-parametric Bayesian inference is developed for a Y-linked two-sex branching process with blind choice. This stochastic model is suitable for analysing the evolution of the number of carriers of two alleles of a Y-linked gene in a two-sex monogamous population where each female chooses her partner from among the male population without caring about his type (i.e. the allele he carries). The only data assumed to be available are the total number of females and males (regardless of their types) up to some generation and the numbers of each type of male in the last generation. A simulation method which is based on a Dirichlet process and a Gibbs sampler is developed to estimate the posterior distributions of the model's main parameters. Finally, the computational efficiency of the algorithm is illustrated with example simulations and an application to real data.     

Empirical Bayes procedures for stabilizing maps of U.S. cancer mortality rates

"The geographic mapping of age-standardized, cause-specific death rates is a powerful tool for identifying possible etiologic factors, because the spatial distribution of mortality risks can be examined for correlations with the spatial distribution of disease-specific risk factors. This article presents a two-stage empirical Bayes procedure for calculating age-standardized cancer death rates, for use in mapping, which are adjusted for the stochasticity of rates in small area populations. Using the adjusted rates helps isolate and identify spatial patterns in the rates. The model is applied to sex-specific data on U.S. county cancer mortality in the white population for 15 cancer sites for three decades: 1950-1959, 1960-1969, and 1970-1979. Selected results are presented as maps of county death rates for white males."     

Estimation of actuarial quantities at fractional ages with Kriging

ABSTRACT

Life tables used in life insurance are often calibrated to show the survival function of the age of death distribution at exact integer ages. Actuaries usually make fractional age assumptions (FAAs) when having to value payments that are not restricted to integer ages. Traditional FAAs have the advantage of simplicity but cannot guarantee to capture precisely the real trends of the survival functions and sometimes even result in a non intuitive overall shape of the force of mortality. In fact, an FAA is an interpolation between integer age values which are accepted as given. In this article, we introduce Kriging model, which is widely used as a metamodel for expensive simulations, to fit the survival function at integer ages, and furthermore use the precisely constructed survival function to build the force of mortality and the life expectancy. The experimental results obtained from a simulated life table (Makehamized life table) and two “real” life tables (the Chinese and US life tables) show that these actuarial quantities (survival function, force of mortality, and life expectancy) presented by Kriging model are much more accurate than those presented by commonly-used FAAs: the uniform distribution of death (UDD) assumption, the constant force assumption, and the Balducci assumption.     

A cold standby repairable system with the repairman having multiple vacations and operational,repair, and vacation times following phase-type distributions

ABSTRACT

This paper studies a cold standby repairable system with two identical components and one repairman having multiple vacations applying matrix-analytic methods. The lifetime of the component follows a phase-type distribution. The repair times and the vacation times of the repairman are governed by different phase-type distributions, respectively. For this system, the Markov process governing the system is constructed. The system is studied in a transient and stationary regime, the availability, the reliability, the rates of occurrence of the different types of failures, and the working probability of the repairman are calculated, respectively. A numerical application is performed to illustrate the calculations.