A bimodal flexible distribution for lifetime data

ABSTRACT

A four-parameter extended bimodal lifetime model called the exponentiated log-sinh Cauchy distribution is proposed. It extends the log-sinh Cauchy and folded Cauchy distributions. We derive some of its mathematical properties including explicit expressions for the ordinary moments and generating and quantile functions. The method of maximum likelihood is used to estimate the model parameters. We implement the fit of the model in the GAMLSS package and provide the codes. The flexibility of the model is illustrated by means of three real data sets.     

Characterization of discrete scale invariant Markov sequences

ABSTRACT

Some special sampling of discrete scale invariant (DSI) processes are presented to provide a multi-dimensional self-similar process in correspondence. By imposing Markov property we show that the covariance functions of such Markov DSI sequences are characterized by variance, and covariance of adjacent samples in the first scale interval. We also provide a theoretical method for estimating spectral density matrix of corresponding multi-dimensional self-similar Markov process. Some examples such as simple Brownian motion (sBm) with drift and scale invariant autoregressive model are presented and these properties are investigated. We present two new method to estimate Hurst parameter of DSI processes and apply them to some sBm and also to the SP500 indices for some period which has DSI property. We compare our estimates with the maximum-likelihood and rescaled range (R/S) method which are applied to the corresponding multi-dimensional self-similar processes.     

Markov chain Monte Carlo exact tests for incomplete two-way contingency tables

We consider testing the quasi-independence hypothesis for two-way contingency tables which contain some structural zero cells. For sparse contingency tables where the large sample approximation is not adequate, the Markov chain Monte Carlo exact tests are powerful tools. To construct a connected chain over the two-way contingency tables with fixed sufficient statistics and an arbitrary configuration of structural zero cells, an algebraic algorithm proposed by Diaconis and Sturmfels [Diaconis, P. and Sturmfels, B. (1998). The Annals of statistics, 26, pp. 363–397.] can be used. However, their algorithm does not seem to be a satisfactory answer, because the Markov basis produced by the algorithm often contains many redundant elements and is hard to interpret. We derive an explicit characterization of a minimal Markov basis, prove its uniqueness, and present an algorithm for obtaining the unique minimal basis. A computational example and the discussion on further basis reduction for the case of positive sufficient statistics are also given.     

Saddlepoint approximations for subordinator processes

ABSTRACT

We develop the saddlepoint approximations in obtaining the transition functions for general subordinator processes. We derive explicit expressions of the first- and second-order approximations. Specifically, we consider some particular classes of subordinators including the Poisson processes, the Gamma processes, the α-stable subordinators, and the Poisson random integrals. We test this technique on the Poisson and Gamma processes, which have closed-form transition functions. Outcomes show that the approximate expressions are consistent with the true transition functions. We then use this method to predict transition density functions for the α-stable subordinator processes. Finally, we calculate approximated transition densities for some Poisson random integrations. Numerical analysis shows the perfect ability of the saddlepoint approximations to predict the transition densities of the α-stable processes and the Poisson random integrations.     

Concentration inequalities of the cross-validation estimator for empirical risk minimizer

ABSTRACT

We derive concentration inequalities for the cross-validation estimate of the generalization error for empirical risk minimizers. In the general setting, we show that the worst-case error of this estimate is not much worse that of training error estimate see Kearns M, Ron D. [Algorithmic stability and sanity-check bounds for leave-one-out cross-validation. Neural Comput. 1999;11:1427–1453]. General loss functions and class of predictors with finite VC-dimension are considered. Our focus is on proving the consistency of the various cross-validation procedures. We point out the interest of each cross-validation procedure in terms of rates of convergence. An interesting consequence is that the size of the test sample is not required to grow to infinity for the consistency of the cross-validation procedure.     

Stress-strength reliability estimation for exponentially distributed system with common minimum guarantee time

Abstract

In this article, we study the problem of estimating the stress-strength reliability, where the stress and strength variables follow independent exponential distributions with a common location parameter but different scale parameters. All parameters are assumed to be unknown. We derive the MLE, the UMVUE of the reliability parameter. We also derive the Bayes estimators considering conjugate prior distributions for the scale parameters and a dependent prior for the common location parameter. Monte Carlo simulations have been carried out to compare among the proposed estimators with respect to different loss functions.     

Spectral representation and autocovariance structure of Markov switching DSGE models

Abstract

We investigate the L²-structure of Markov switching Dynamic Stochastic General Equilibrium (MS DSGE) models and derive conditions for strict and second-order stationarity. Then we determine the autocovariance function of the process driven by a stationary MS DSGE model and give a stable VARMA representation of it. It turns out that the autocovariance structure of the process coincides with that of a standard VARMA. Finally, we propose a method to derive the spectral density in a matrix closed-form of MS DSGE models. Our results relate with the works of Francq and Zakoian, Krolzig, Zhang and Stine. Numerical and empirical illustrations complete the article.     

Estimation of weibull parameters for grouped data with competing risks

We consider the problem of estimating Weibull parameters for grouped data when competing risks are present. We propose two simple methods of estimation and derive their asymptotic properties. A Monte Carlo study was carried out to evaluate the performance of these two methods.     

Distribution of busy period in stochastic fluid models

We consider the busy period in a stochastic fluid flow model with infinite buffer where the input and output rates are controlled by a finite homogeneous Markov process. We derive an explicit expression for the distribution of the busy period and we obtain an algorithm to compute it which exhibits nice numerical properties.

     

The multivariate Markov and multiple Chebyshev inequalities

Abstract

A sharp probability inequality named the multivariate Markov inequality is derived for the intersection of the survival functions for non-negative random variables as an extension of the Markov inequality for a single variable. The corresponding result in Chebyshev’s inequality is also obtained as a special case of the multivariate Markov inequality, which is called the multiple Chebyshev inequality to distinguish from the multivariate Chebyshev inequality for a quadratic form of standardized uncorrelated variables. Further, the results are extended to the inequalities for the union of the survival functions and those with lower bounds.     

Strong law of large numbers of the delayed sums for Markov Chains indexed by a Cayley tree

Abstract

In this paper, we will study the strong law of large numbers of the delayed sums for Markov chains indexed by a Cayley tree with countable state spaces. Firstly, we prove a strong limit theorem for the delayed sums of the bivariate functions for Markov chains indexed by a Cayley tree. Secondly, the strong law of large numbers for the frequencies of occurrence of states of the delayed sums is obtained. As a corollary, we obtain the strong law of large numbers for the frequencies of occurrence of states for countable Markov chains indexed by a Cayley tree.     

A Bayesian approach for misclassified ordinal response data

ABSTRACT

Motivated by a longitudinal oral health study, the Signal-Tandmobiel^® study, a Bayesian approach has been developed to model misclassified ordinal response data. Two regression models have been considered to incorporate misclassification in the categorical response. Specifically, probit and logit models have been developed. The computational difficulties have been avoided by using data augmentation. This idea is exploited to derive efficient Markov chain Monte Carlo methods. Although the method is proposed for ordered categories, it can also be implemented for unordered ones in a simple way. The model performance is shown through a simulation-based example and the analysis of the motivating study.     

Parallel tempering for dynamic generalized linear models

ABSTRACT

Markov chain Monte Carlo (MCMC) methods can be used for statistical inference. The methods are time-consuming due to time-vary. To resolve these problems, parallel tempering (PT), as a parallel MCMC method, is tried, for dynamic generalized linear models (DGLMs), as well as the several optimal properties of our proposed method. In PT, two or more samples are drawn at the same time, and samples can exchange information with each other. We also present some simulations of the DGLMs in the case and provide two applications of Poisson-type DGLMs in financial research.     

On the functional central limit theorem for first passage time of nonlinear semi-Markov reward processes

Abstract

In this article we examine the functional central limit theorem for the first passage time of reward processes defined over a finite state space semi-Markov process. In order to apply this process for a wider range of real-world applications, the reward functions, considered in this work, are assumed to have general forms instead of the constant rates reported in the other studies. We benefit from the martingale theory and Poisson equations to prove and establish the convergence of the first passage time of reward processes to a zero mean Brownian motion. Necessary conditions to derive the results presented in this article are the existence of variances for sojourn times in each state and second order integrability of reward functions with respect to the distribution of sojourn times. We finally verify the presented methodology through a numerical illustration.     

R2 Bounds for Predictive Models: What Univariate Properties Tell us About Multivariate Predictability

ABSTRACT

A long-standing puzzle in macroeconomic forecasting has been that a wide variety of multivariate models have struggled to out-predict univariate models consistently. We seek an explanation for this puzzle in terms of population properties. We derive bounds for the predictive R² of the true, but unknown, multivariate model from univariate ARMA parameters alone. These bounds can be quite tight, implying little forecasting gain even if we knew the true multivariate model. We illustrate using CPI inflation data. Supplementary materials for this article are available online.     

Statistical Inference for Partially Hidden Markov Models

ABSTRACT

In this article we introduce a new missing data model, based on a standard parametric Hidden Markov Model (HMM), for which information on the latent Markov chain is given since this one reaches a fixed state (and until it leaves this state). We study, under mild conditions, the consistency and asymptotic normality of the maximum likelihood estimator. We point out also that the underlying Markov chain does not need to be ergodic, and that identifiability of the model is not tractable in a simple way (unlike standard HMMs), but can be studied using various technical arguments.     

The discretely observed immigration-death process: Likelihood inference and spatiotemporal applications

ABSTRACT

We consider a stochastic process, the homogeneous spatial immigration-death (HSID) process, which is a spatial birth-death process with as building blocks (i) an immigration-death (ID) process (a continuous-time Markov chain) and (ii) a probability distribution assigning iid spatial locations to all events. For the ID process, we derive the likelihood function, reduce the likelihood estimation problem to one dimension, and prove consistency and asymptotic normality for the maximum likelihood estimators (MLEs) under a discrete sampling scheme. We additionally prove consistency for the MLEs of HSID processes. In connection to the growth-interaction process, which has a HSID process as basis, we also fit HSID processes to Scots pine data.     

A new look at Markov processes of G/M/1-type

We present a new method for deriving the stationary distribution of an ergodic Markov process of G/M/1-type in continuous-time, by deriving and making use of a new representation for each element of the rate matrices contained in these distributions. This method can also be modified to derive the Laplace transform of each transition function associated with Markov processes of G/M/1-type.     

Mixing and moments properties of a non-stationary copula-based Markov process

Abstract

We provide conditions under which a non-stationary copula-based Markov process is geometric β-mixing and geometric ρ-mixing. Our results generalize some results of Beare who considers the stationary case. As a particular case we introduce a stochastic process, that we call convolution-based Markov process, whose construction is obtained by using the C-convolution operator which allows the increments to be dependent. Within this subclass of processes we characterize a modified version of the standard random walk where copulas and marginal distributions involved are in the same elliptical family. We study mixing and moments properties to identify the differences compared to the standard case.