Application of a non-homogeneous Markov chain with seasonal transition probabilities to ozone data

In this work, we assume that the sequence recording whether or not an ozone exceedance of an environmental threshold has occurred in a given day is ruled by a non-homogeneous Markov chain of order one. In order to account for the possible presence of cycles in the empirical transition probabilities, a parametric form incorporating seasonal components is considered. Results show that even though some covariates (namely, relative humidity and temperature) are not included explicitly in the model, their influence is captured in the behavior of the transition probabilities. Parameters are estimated using the Bayesian point of view via Markov chain Monte Carlo algorithms. The model is applied to ozone data obtained from the monitoring network of Mexico City, Mexico. An analysis of how the methodology could be used as an aid in the decision-making is also given.     

A test for abrupt change in hazard regression models with Weibull baselines

We develop a likelihood ratio test for an abrupt change point in Weibull hazard functions with covariates, including the two-piece constant hazard as a special case. We first define the log-likelihood ratio test statistic as the supremum of the profile log-likelihood ratio process over the interval which may contain an unknown change point. Using local asymptotic normality (LAN) and empirical measure, we show that the profile log-likelihood ratio process converges weakly to a quadratic form of Gaussian processes. We determine the critical values of the test and discuss how the test can be used for model selection. We also illustrate the method using the Chronic Granulomatous Disease (CGD) data.     

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.     

Change Point Detection in a General Class of Distributions

ABSTRACT

We consider the change point problem in a general class of distributions, and derive a test statistic T _n which reduces to the statistic obtained by Kander and Zacks (1966 Kander , Z. , Zacks , S. ( 1966 ). Test procedure for possible changes in parameter of statistical distributions occurring at unknown time points. Ann. Math. Statist. 37 : 1196 – 1210 . [CSA] [Crossref] , [Google Scholar]) for the exponential family. Properties of the test, including its asymptotic distribution, are discussed.     

Likelihood ratio test for testing order of continuous time finite markov chains

Anderson and Goodman ( 1957) have obtained the likelihood ratio tests and chi-square tests for testing the hypothesis about the order of discrete time finite Markov chains, On the similar lines we have obtained likeli¬hood ratio tests and chi-square tests (asymptotic) for testing hypotheses about the order of continuous time Markov chains (MC) with finite state space.     

On U-statistics and von Mises statistics for a special class of Markov chains

We derive the Berry-Esséen theorem with optimal convergence rate for U-statistics and von Mises statistics associated with a special class of Markov chains occuring in the theory of dependence with complete connections.     

Stochastic monotonicity and comparability of Markov chains with block-monotone transition matrices and their applications to queueing systems

ABSTRACT

Motivated by various applications in queueing theory, this article is devoted to the stochastic monotonicity and comparability of Markov chains with block-monotone transition matrices. First, we introduce the notion of block-increasing convex order for probability vectors, and characterize the block-monotone matrices in the sense of the block-increasing order and block-increasing convex order. Second, we characterize the Markov chain with general transition matrix by martingale and provide a stochastic comparison of two block-monotone Markov chains under the two block-monotone orders. Third, the stochastic comparison results for the Markov chains corresponding to the discrete-time GI/G/1 queue with different service distributions under the two block-monotone orders are given, and the lower bound and upper bound of the Markov chain corresponding to the discrete-time GI/G/1 queue in the sense of the block-increasing convex order are found.     

Number of occurrences in two-state Markov chains,with an application in linguistics

Let N_n be the number of occurrences in n trials of an event governed by a two-state Markov chain (of first order or second order). We obtain the distribution of N_n and apply it to a problem involving literary text.     

Detection of a changepoint,a mean-shift accompanied with a trend change,in short time-series with autocorrelation

In this study, a changepoint model, which can detect either a mean shift or a trend change when accounting for autocorrelation in short time-series, was investigated with simulations and a new method is proposed. The changepoint hypotheses were tested using a likelihood ratio test. The test statistic does not follow a known distribution and depends on the length of the time-series and the autocorrelation. The results imply that it is not possible to detect autocorrelation and that the estimate of the autocorrelation parameter is biased. It is therefore recommended to use critical values from the empirical distribution for a fixed autocorrelation.     

A stochastic model of cyber attacks with imperfect detection

Abstract

This paper introduces a cyber security model with imperfect detection, in which one attacker launches multiple attacks against the target with adjusted strength based on the previous attacking outcome. Several sufficient conditions leading to the usual stochastic order on the first time to observe a truly compromised target, to observe a successful attack and to compromise the target are developed, respectively. The probability for the target to be truly compromised before observing some number of successful attacks is proved to increase (decrease) in the attacking (defense) strength. Monte Carlo simulations are also conducted to empirically illustrate the theoretical results.     

Spectral analysis of Markov switching GARCH models with statistical inference

We derive matrix expressions in closed form for the autocovariance function and the spectral density of Markov switching GARCH models and their powers. For this, we apply the Riesz–Fischer theorem which defines the spectral representation as the Fourier transform of the autocovariance function. Under suitable assumptions, we prove that the sample estimator of the spectral density is consistent and asymptotically normally distributed. Further statistical implications in terms of order identification and parameter estimation are discussed. A simulation study confirms the validity of the asymptotic properties. These methods are also well suited for financial market applications, and in particular for the analysis of time series in the frequency domain, as shown in some proposed real-world examples.     

A Likelihood Integrated Method for Exploratory Graphical Analysis of Change Point Problem with Directional Data

In this article we introduce a new likelihood based method, called the likelihood integrated method, which is distinct from the well-known integrated likelihood method. We use the likelihood integrated to propose a simple exploratory graphical analysis for the change point problem in the context of directional data. The method is applied to analysis of two real life data sets. The results obtained by application of this simple method are seen to be quite similar to those obtained earlier by different formal methods in most cases. A small simulation study is conducted to assess the effectiveness of this procedure in indicating presence of change point.     

A class of small deviation theorems for the random variables associated with mth-order asymptotic circular Markov chains

ABSTRACT

In this article, we study a class of small deviation theorems for the random variables associated with mth-order asymptotic circular Markov chains. First, the definition of mth-order asymptotic circular Markov chain is introduced, then by applying the known results of the limit theorem for mth-order non homogeneous Markov chain, the small deviation theorem on the frequencies of occurrence of states for mth-order asymptotic circular Markov chains is established. Next, the strong law of large numbers and asymptotic equipartition property for this Markov chains are obtained. Finally, some results of mth-order nonhomogeneous Markov chains are given.     

The use of simple reparameterizations to improve the efficiency of Markov chain Monte Carlo estimation for multilevel models with applications to discrete time survival models

Summary.  We consider the application of Markov chain Monte Carlo (MCMC) estimation methods to random-effects models and in particular the family of discrete time survival models. Survival models can be used in many situations in the medical and social sciences and we illustrate their use through two examples that differ in terms of both substantive area and data structure. A multilevel discrete time survival analysis involves expanding the data set so that the model can be cast as a standard multilevel binary response model. For such models it has been shown that MCMC methods have advantages in terms of reducing estimate bias. However, the data expansion results in very large data sets for which MCMC estimation is often slow and can produce chains that exhibit poor mixing. Any way of improving the mixing will result in both speeding up the methods and more confidence in the estimates that are produced. The MCMC methodological literature is full of alternative algorithms designed to improve mixing of chains and we describe three reparameterization techniques that are easy to implement in available software. We consider two examples of multilevel survival analysis: incidence of mastitis in dairy cattle and contraceptive use dynamics in Indonesia. For each application we show where the reparameterization techniques can be used and assess their performance.     

Bayesian analysis of single-molecule experimental data

Summary.  Recent advances in experimental technologies allow scientists to follow biochemical processes on a single-molecule basis, which provides much richer information about chemical dynamics than traditional ensemble-averaged experiments but also raises many new statistical challenges. The paper provides the first likelihood-based statistical analysis of the single-molecule fluorescence lifetime experiment designed to probe the conformational dynamics of a single deoxyribonucleic acid (DNA) hairpin molecule. The conformational change is initially treated as a continuous time two-state Markov chain, which is not observable and must be inferred from changes in photon emissions. This model is further complicated by unobserved molecular Brownian diffusions. Beyond the simple two-state model, a competing model that models the energy barrier between the two states of the DNA hairpin as an Ornstein–Uhlenbeck process has been suggested in the literature. We first derive the likelihood function of the simple two-state model and then generalize the method to handle complications such as unobserved molecular diffusions and the fluctuating energy barrier. The data augmentation technique and Markov chain Monte Carlo methods are developed to sample from the posterior distribution desired. The Bayes factor calculation and posterior estimates of relevant parameters indicate that the fluctuating barrier model fits the data better than the simple two-state model.     

Hierarchical Bayesian modeling of marked non-homogeneous Poisson processes with finite mixtures and inclusion of covariate information

We investigate marked non-homogeneous Poisson processes using finite mixtures of bivariate normal components to model the spatial intensity function. We employ a Bayesian hierarchical framework for estimation of the parameters in the model, and propose an approach for including covariate information in this context. The methodology is exemplified through an application involving modeling of and inference for tornado occurrences.     

Efficient Semiparametric Bayesian Estimation of Multivariate Discrete Proportional Hazards Model with Random Effects

We incorporate a random effect into a multivariate discrete proportional hazards model and propose an efficient semiparametric Bayesian estimation method. By introducing a prior process for the parameters of baseline hazards, we consider a nonparametric estimation of baseline hazards function. Using a state space representation, we derive a dynamic modeling of baseline hazards function and propose an efficient block sampler for Markov chain Monte Carlo method. A numerical example using kidney patients data is given.