Fitting DNA sequences through log-linear modelling with linear constraints

For some discrete state series, such as DNA sequences, it can often be postulated that its probabilistic behaviour is given by a Markov chain. For making the decision on whether or not an uncharacterized piece of DNA is part of the coding region of a gene, under the Markovian assumption, there are two statistical tools that are essential to be considered: the hypothesis testing of the order in a Markov chain and the estimators of transition probabilities. In order to improve the traditional statistical procedures for both of them when stationarity assumption can be considered, a new version for understanding the homogeneity hypothesis is proposed so that log-linear modelling is applied for conditional independence jointly with homogeneity restrictions on the expected means of transition counts in the sequence. In addition we can consider a variety of test-statistics and estimators by using φ-divergence measures. As special case of them the well-known likelihood ratio test-statistics and maximum-likelihood estimators are obtained.     

Reversible jump and the label switching problem in hidden Markov models

Reversible jump Markov chain Monte Carlo (RJMCMC) algorithms can be efficiently applied in Bayesian inference for hidden Markov models (HMMs), when the number of latent regimes is unknown. As for finite mixture models, when priors are invariant to the relabelling of the regimes, HMMs are unidentifiable in data fitting, because multiple ways to label the regimes can alternate during the MCMC iterations; this is the so-called label switching problem. HMMs with an unknown number of regimes are considered here and the goal of this paper is the comparison, both applied and theoretical, of five methods used for tackling label switching within a RJMCMC algorithm; they are: post-processing, partial reordering, permutation sampling, sampling from a Markov prior and rejection sampling. The five strategies we compare have been proposed mostly in the literature of finite mixture models and only two of them, i.e. rejection sampling and partial reordering, have been presented in RJMCMC algorithms for HMMs. We consider RJMCMC algorithms in which the parameters are updated by Gibbs sampling and the dimension of the model changes in split-and-merge and birth-and-death moves. Finally, an example illustrates and compares the five different methodologies.     

On the natural restrictions in the singular Gauss–Markov model

A Gauss–Markov model is said to be singular if the covariance matrix of the observable random vector in the model is singular. In such a case, there exist some natural restrictions associated with the observable random vector and the unknown parameter vector in the model. In this paper, we derive through the matrix rank method a necessary and sufficient condition for a vector of parametric functions to be estimable, and necessary and sufficient conditions for a linear estimator to be unbiased in the singular Gauss–Markov model. In addition, we give some necessary and sufficient conditions for the ordinary least-square estimator (OLSE) and the best linear unbiased estimator (BLUE) under the model to satisfy the natural restrictions.      

Testing exogeneity in overidentified models

Summary In this paper we analyse the consequences of model overidentification on testing exogeneity, when maximum likelihood techniques for estimation and inference are used. This situation is viewed as a particular case of the more general problem of considering how restrictions on nuisance parameters could help in making inference on the parameters of interest. At first a general model is considered. A suitable likelihood function factorization is used which allows a simple derivation of the information matrix and others tools useful for building up joint tests of exogeneity and overidentifying restrictions both of Wald and Lagrange Multiplier type. The asymptotic local power of the exogeneity test in the justidentified model is compared with that in the overidentified one, when we assume that the latter is the true model. Then the pseudo-likelihood framework is used to derive the consequences of working with a model where overidentifying restrictions are erroneously imposed. The inconsistency introduced by imposing false restrictions is analysed and the consequences of the misspecification on the exogeneity test are carefully examined.     

BAYESIAN DETECTION AND ANALYSIS OF CHANGING TRANSITION MATRICES OF STATIONARY MARKOV CHAINS

This paper considers the detection of abrupt changes in the transition matrix of a Markov chain from a Bayesian viewpoint. It derives Bayes factors and posterior probabilities for unknown numbers of change‐points, as well as the positions of the change‐points, assuming non‐informative but proper priors on the parameters and fixed upper bound. The Markov chain Monte Carlo approach proposed by Chib in 1998 for estimating multiple change‐points models is adapted for the Markov chain model. It is especially useful when there are many possible change‐points. The method can be applied in a wide variety of disciplines and is particularly relevant in the social and behavioural sciences, for analysing the effects of events on the attitudes of people.     

Inferential properties of an individual-based survival model using release-recapture data: Sample size,validity and power

A model for analyzing release-recapture data is presented that generalizes a previously existing individual covariate model to include multiple groups of animals. As in the previous model, the generalized version includes selection parameters that relate individual covariates to survival potential. Significance of the selection parameters was equivalent to significance of the individual covariates. Simulation studies were conducted to investigate three inferential properties with respect to the selection parameters: (1) sample size requirements, (2) validity of the likelihood ratio test (LRT) and (3) power of the LRT. When the survival and capture probabilities ranged from 0.5 to 1.0, a total sample size of 300 was necessary to achieve a power of 0.80 at a significance level of 0.1 when testing the significance of the selection parameters. However, only half that (a total of 150) was necessary for the distribution of the maximum likelihood estimators of the selection parameters to approximate their asymptotic distributions. In general, as the survival and capture probabilities decreased, the sample size requirements increased. The validity of the LRT for testing the significance of the selection parameters was confirmed because the LRT statistic was distributed as theoretically expected under the null hypothesis, i.e. like a chi 2 random variable. When the baseline survival model was fully parameterized with population and interval effects, the LRT was also valid in the presence of unaccounted for random variation. The power of the LRT for testing the selection parameters was unaffected by over-parameterization of the baseline survival and capture models. The simulation studies showed that for testing the significance of individual covariates to survival the LRT was remarkably robust to assumption violations.     

Inferential properties of an individual-based survival model using release-recapture data: sample size, validity and power

A model for analyzing release-recapture data is presented that generalizes a previously existing individual covariate model to include multiple groups of animals. As in the previous model, the generalized version includes selection parameters that relate individual covariates to survival potential. Significance of the selection parameters was equivalent to significance of the individual covariates. Simulation studies were conducted to investigate three inferential properties with respect to the selection parameters: (1) sample size requirements, (2) validity of the likelihood ratio test (LRT) and (3) power of the LRT. When the survival and capture probabilities ranged from 0.5 to 1.0, a total sample size of 300 was necessary to achieve a power of 0.80 at a significance level of 0.1 when testing the significance of the selection parameters. However, only half that (a total of 150) was necessary for the distribution of the maximum likelihood estimators of the selection parameters to approximate their asymptotic distributions. In general, as the survival and capture probabilities decreased, the sample size requirements increased. The validity of the LRT for testing the significance of the selection parameters was confirmed because the LRT statistic was distributed as theoretically expected under the null hypothesis, i.e. like a chi 2 random variable. When the baseline survival model was fully parameterized with population and interval effects, the LRT was also valid in the presence of unaccounted for random variation. The power of the LRT for testing the selection parameters was unaffected by over-parameterization of the baseline survival and capture models. The simulation studies showed that for testing the significance of individual covariates to survival the LRT was remarkably robust to assumption violations.     

Stability of a characterization by the identical distribution of linear forms

Hidden Markov models (HMMs) have during the last decade become a widely spread tool for modelling sequences of dependent random variables. Inference for HMMs has been considered by several authors, but so far no work has been done on estimating their order. In this paper we propose a penalized likelihood estimator for this purpose. This estimator is based on the m-dimensional distribution of HMM, and it is shown that in the limit it does not underestimate the order.     

Testing the effect of treatment on survival time with an immediate intermediate event

In this paper, we consider testing the effects of treatment on survival time when a subject experiences an immediate intermediate event (IE) prior to death or predetermined endpoint. A two-stage model incorporating both (i) the effects of the covariates on the immediate IE and (ii) survival regression with the immediate IE and other covariates is presented. We study the likelihood ratio test (LRT) for testing the treatment effect based on the proposed two stage model. We propose two procedures: an asymptotic-based procedure and a resampling-based procedure, to approximate the null distribution of the LRT. We numerically show the advantages of the two stage modeling over the existing single stage survival model with interactions between the covariates and the immediate IE. In addition, an illustrative empirical example is provided.     

Corrected likelihood-ratio tests in logistic regression using small-sample data

Likelihood-ratio tests (LRTs) are often used for inferences on one or more logistic regression coefficients. Conventionally, for given parameters of interest, the nuisance parameters of the likelihood function are replaced by their maximum likelihood estimates. The new function created is called the profile likelihood function, and is used for inference from LRT. In small samples, LRT based on the profile likelihood does not follow χ² distribution. Several corrections have been proposed to improve LRT when used with small-sample data. Additionally, complete or quasi-complete separation is a common geometric feature for small-sample binary data. In this article, for small-sample binary data, we have derived explicitly the correction factors of LRT for models with and without separation, and proposed an algorithm to construct confidence intervals. We have investigated the performances of different LRT corrections, and the corresponding confidence intervals through simulations. Based on the simulation results, we propose an empirical rule of thumb on the use of these methods. Our simulation findings are also supported by real-world data.     

A stationarity test on Markov chain models based on marginal distribution

A stationarity test on Markov chain models is proposed in this paper. Most of the previous test procedures for the Markov chain models have been done based on the conditional probabilities of a transition matrix. The likelihood ratio and Pearson type chi-square tests have been used for testing stationarity and order of Markov chains. This paper uses the efficient score test, an extension of the test developed by Tsiatis (1980) [18], for testing the stationarity of Markov chain models based on the marginal distribution as obtained by Azzalini (1994) [2]. For testing the suitability of the proposed method, a numerical example of real life data and simulation studies for comparison with an alternative test procedure are given.     

LONG-RUN STRUCTURAL MODELLING

The paper develops a general framework for identification, estimation, and hypothesis testing in cointegrated systems when the cointegrating coefficients are subject to (possibly) non-linear and cross-equation restrictions, obtained from economic theory or other relevant a priori information. It provides a proof of the consistency of the quasi maximum likelihood estimators (QMLE), establishes the relative rates of convergence of the QMLE of the short-run and the long-run parameters, and derives their asymptotic distributions; thus generalizing the results already available in the literature for the linear case. The paper also develops tests of the over-identifying (possibly) non-linear restrictions on the cointegrating vectors. The estimation and hypothesis testing procedures are applied to an Almost Ideal Demand System estimated on U.K. quarterly observations. Unlike many other studies of consumer demand this application does not treat relative prices and real per capita expenditures as exogenously given.     

Comment on “recent developments in bootstrapping time series”

The paper develops a general framework for identification, estimation, and hypothesis testing in cointegrated systems when the cointegrating coefficients are subject to (possibly) non-linear and cross-equation restrictions, obtained from economic theory or other relevant a priori information. It provides a proof of the consistency of the quasi maximum likelihood estimators (QMLE), establishes the relative rates of convergence of the QMLE of the short-run and the long-run parameters, and derives their asymptotic distributions; thus generalizing the results already available in the literature for the linear case. The paper also develops tests of the over-identifying (possibly) non-linear restrictions on the cointegrating vectors. The estimation and hypothesis testing procedures are applied to an Almost Ideal Demand System estimated on U.K. quarterly observations. Unlike many other studies of consumer demand this application does not treat relative prices and real per capita expenditures as exogenously given.     

The equivalence between likelihood ratio test and F-test for testing variance component in a balanced one-way random effects model

In mixed linear models, it is frequently of interest to test hypotheses on the variance components. F-test and likelihood ratio test (LRT) are commonly used for such purposes. Current LRTs available in literature are based on limiting distribution theory. With the development of finite sample distribution theory, it becomes possible to derive the exact test for likelihood ratio statistic. In this paper, we consider the problem of testing null hypotheses on the variance component in a one-way balanced random effects model. We use the exact test for the likelihood ratio statistic and compare the performance of F-test and LRT. Simulations provide strong support of the equivalence between these two tests. Furthermore, we prove the equivalence between these two tests mathematically.     

Phase-shifting common cycles and common trends

In this paper, we outline a framework for modelling and analysing economic fluctuations and dynamics. It is assumed that there may exist common trends and common cycles in the time series to be analysed It is further generalised that common cycles may have non-coincident, or phase-shifting attributes These attributes are examined via the Markov transition matrix in a VAR system, revealing the way in which the phase-shifting works with the reduced rank Markov transition matrix. The links with the structural common trend model are also presented.     

Functional Estimation of a Density Under a New Weak Dependence Condition

The purpose of this paper is to prove, through the analysis of the behaviour of a standard kernel density estimator, that the notion of weak dependence defined in a previous paper (cf. Doukhan & Louhichi, 1999) has sufficiently sharp properties to be used in various situations. More precisely we investigate the asymptotics of high order losses, asymptotic distributions and uniform almost sure behaviour of kernel density estimates. We prove that they are the same as for independent samples (with some restrictions for a.s. behaviours). Recall finally that this weak dependence condition extends on the previously defined ones such as mixing, association and it allows considerations of new classes such as weak shifts processes based on independent sequences as well as some non-mixing Markov processes.     

Testing for two states in a hidden Markov model

The authors consider hidden Markov models (HMMs) whose latent process has m ≥ 2 states and whose state‐dependent distributions arise from a general one‐parameter family. They propose a test of the hypothesis m = 2. Their procedure is an extension to HMMs of the modified likelihood ratio statistic proposed by Chen, Chen & Kalbfleisch (2004) for testing two states in a finite mixture. The authors determine the asymptotic distribution of their test under the hypothesis m = 2 and investigate its finite‐sample properties in a simulation study. Their test is based on inference for the marginal mixture distribution of the HMM. In order to illustrate the additional difficulties due to the dependence structure of the HMM, they show how to test general regular hypotheses on the marginal mixture of HMMs via a quasi‐modified likelihood ratio. They also discuss two applications.     

Using hidden Markov models with raw,triaxial wrist accelerometry data to determine sleep stages

Accelerometry is a low‐cost and noninvasive method that has been used to discriminate sleep from wake, however, its utility to detect sleep stages is unclear. We detail the development and comparison of methods which utilise raw, triaxial accelerometry data to classify varying stages of sleep, ranging from sleep/wake detection to discriminating rapid eye movement sleep, stage one sleep, stage two sleep, deep sleep and wake. First‐ and second‐order hidden Markov models (HMMs) with time‐homogeneous and time‐varying transition probability matrices, along with continuous acceleration observations in the form of a Gaussian‐observation HMM and K‐means classified acceleration in a discrete‐observation HMM were explored. In addition, generalised linear mixed models (GLMMs) with binary and multinomial responses and logit link functions were considered as was whether incorporating adjoining acceleration information into the models improved prediction. Model predictions were compared to the reference‐standard in sleep detection (polysomnography) and outcome accuracies were calculated. Consistently, HMMs yielded greater sleep stage detection than GLMMs but there was little difference between first‐ and second‐order HMMs. Varying degrees of difference were observed when comparing Gaussian‐observation HMMs to discrete‐observation HMMs, and time‐varying HMMs yielded greater discrimination than time‐homogeneous HMMs, as did models which considered adjoining acceleration information. These results suggest that wrist‐worn accelerometry data may be able to detect sleep stages but that further investigation is required to optimise classification accuracy.     

Economic and economic statistical designs for three-level control charts

In some situations, an appropriate quality measure uses three or more discrete levels to classify a product characteristic. For these situations, some control charts have been developed based on statistical criteria regardless of economic considerations. In this paper, we develop economic and economic statistical designs (ESD) for 3-level control charts. We apply the cost model proposed by Costa and Rahim.[Economic design of X charts with variable parameters: the Markov chain approach, J Appl Stat 28 (2001), 875–885] Furthermore, we assume that the length of time that the process remains in control is exponentially distributed which allows us to apply the Markov chain approach for developing the cost model. We apply a genetic algorithm to determine the optimal values of model parameters by minimizing the cost function. A numerical example is provided to illustrate the performance of the proposed models and to compare the cost of the pure economic and ESD for three-level control charts. A sensitivity analysis is also conducted in this numerical example.     

Selection of Multivariate Stochastic Volatility Models via Bayesian Stochastic Search

We propose a Bayesian stochastic search approach to selecting restrictions on multivariate regression models where the errors exhibit deterministic or stochastic conditional volatilities. We develop a Markov chain Monte Carlo (MCMC) algorithm that generates posterior restrictions on the regression coefficients and Cholesky decompositions of the covariance matrix of the errors. Numerical simulations with artificially generated data show that the proposed method is effective in selecting the data-generating model restrictions and improving the forecasting performance of the model. Applying the method to daily foreign exchange rate data, we conduct stochastic search on a VAR model with stochastic conditional volatilities.