Bayesian Identification of Multivariate Autoregressive Processes

Identification is one of the most important stages of a time series analysis. This paper develops a direct Bayesian technique to identify the order of multivariate autoregressive processes. By employing the conditional likelihood function and a matrix normal-Wishart prior density, or Jeffrey' vague prior, the proposed identification technique is based on deriving the exact posterior probability mass function of the model order in a convenient form. Then one may easily evaluate the posterior probabilities of the model order and choose the order that maximizes the posterior mass function to be the suitable order of the time series data being analyzed. Assuming the bivariate autoregressive processes, a numerical study, with different prior mass functions, is carried out to assess the efficiency of the proposed technique. The analysis of the numerical results supports the adequacy of the proposed technique in identifying the orders of multivariate autoregressive processes.     

Conditional simulation for moving average processes with discrete or continuous values

A conditional simulation technique has previously been presented for variance reduction when estimating tail probabilities, particularly extreme ones, for a wide class of moving-average processes. Here, we generalize the technique from continuous to discrete random variables. Two distinct approaches to this generalization are presented and compared. We describe some of the empirical properties of the preferred method in simple examples, and present some more general examples including autoregressive moving-average processes in one and two dimensions. We show that the technique performs well for processes with a wide range of structures, provided the tail probability to be estimated is not too large. We discuss briefly the application of this technique in investigating volatility in financial models of, for example, asset prices.     

Ruin Probabilities of a Dual Markov-Modulated Risk Model

This article is devoted to studying a dual Markov-modulated risk model, which can properly represent, to some extent, surplus processes of companies that pay costs continuously and have occasional gains. We consider both the finite and infnite horizon ruin probabilities under this dual model. Upper and lower bounds of Lundberg type are derived for these ruin probabilities. We also obtain a time-dependent version of Lundberg type inequalities.     

Development of Specific hypothesis tests for estimated markov chains

We develop and evaluate the validity and power of two specific tests for the transition probabilities in a Markov chain estimated from aggregate frequency data. The two null hypotheses considered are (1) constancy of the diagonal elements of the one-step transition probability matrix and (2) an arbitrarily chosen transition probability’s being equal to a specific value. The formation of tests uses a general framework for statistical inference on estimated Markov processes; we also indicate how this framework can be used to form tests for a variety of other hypotheses. The validity and power performance of the two tests formed in this paper are examined in factorially designed Monte Carlo experiments. The results indicate that the proposed tests lead to type I error probabilities which are close to the desired levels and to high power against even small deviations from the null hypotheses considered.     

Bayesian Identification of Moving Average Models

This study approaches the Bayesian identification of moving average processes using an approximate likelihood function and a normal gamma prior density. The marginal posterior probability mass function of the model order is developed in a convenient form. Then one may investigate the posterior probabilities over the grid of the order and choose the order with the highest probability to solve the identification problem. A comprehensive simulation study is carried out to demonstrate the performance of the proposed procedure and check its adequacy in handling the identification problem. In addition, the proposed Bayesian procedure is compared with some non Bayesian automatic techniques and another Bayesian technique. The numerical results support the adequacy of using the proposed procedure in solving the identification problem of moving average processes.     

Computing probabilites of rectangles in case of multinormal distribution

In several stochastic programming models and statistical problems the computation of probabilities of n-dimensional rectangles is required in case of n-dimensional normal distribution. A recent simulation technique presented by the author for computing values of the distribution function can be modified to yield appropriate procedure for computing probabilities of rectangles. Some numerical work is provided to illustrate the use of the new algorithm.     

An efficient non-rejective implementation of the πps sampling designs

Poisson sampling is a method for unequal probabilities sampling with random sample size. There exist several implementations of the Poisson sampling design, with fixed sample size, which almost all are rejective methods, that is, the sample is not always accepted. Thus, the existing methods can be time-consuming or even infeasible in some situations. In this paper, a fast and non-rejective method, which is efficient even for large populations, is proposed and studied. The method is a new design for selecting a sample of fixed size with unequal inclusion probabilities. For the population of large size, the proposed design is very close to the strict πps sampling which is similar to the conditional Poisson (CP) sampling design, but the implementation of the design is much more efficient than the CP sampling. And the inclusion probabilities can be calculated recursively.     

The evaluation of trivariate normal probabilities defined by linear inequalities

This paper considers the evaluation of probabilities which are defined by a set of linear inequalities of a trivariate normal distribution. It is shown that these probabilities can be evaluated by a one-dimensional numerical integration. The trivariate normal distribution can have any covariance matrix and any mean vector, and the probability can be defined by any number of one-sided and two-sided linear inequalities. This affords a practical and efficient method for the calculation of these probabilities which is superior to basic simulation methods. An application of this method to the analysis of pairwise comparisons of four treatment effects is discussed.     

Is a pure jump process fitting the high frequency data better than a jump-diffusion process?

Two families of processes: pure jump processes and jump-diffusion processes are widely used in literatures. Recently, empirical findings demonstrate that the underlying processes of high frequency data sets are pure-jump processes of infinite variation in many situations. Statistical tests are also proposed to make the empirical findings theoretically grounded. In this paper, we extend the work of Jing et al. (2012) in two aspects: (1) the jump process in the null hypothesis and the alternative hypothesis could be different; (2) the null hypothesis covers more flexible processes which are more relevant in finance when considering models for asset prices or nominal interest rates. Theoretically, the test is proven to be very powerful and can control the type I error probabilities well under the nominal level.     

On the use of secondary capture-recapture samples to estimate temporary emigration and breeding proportions

The use of the Cormack-Jolly-Seber model under a standard sampling scheme of one sample per time period, when the Jolly-Seber assumption that all emigration is permanent does not hold, leads to the confounding of temporary emigration probabilities with capture probabilities. This biases the estimates of capture probability when temporary emigration is a completely random process, and both capture and survival probabilities when there is a temporary trap response in temporary emigration, or it is Markovian. The use of secondary capture samples over a shorter interval within each period, during which the population is assumed to be closed (Pollock's robust design), provides a second source of information on capture probabilities. This solves the confounding problem, and thus temporary emigration probabilities can be estimated. This process can be accomplished in an ad hoc fashion for completely random temporary emigration and to some extent in the temporary trap response case, but modelling the complete sampling process provides more flexibility and permits direct estimation of variances. For the case of Markovian temporary emigration, a full likelihood is required.     

On the use of secondary capture-recapture samples to estimate temporary emigration and breeding proportions

The use of the Cormack-Jolly-Seber model under a standard sampling scheme of one sample per time period, when the Jolly-Seber assumption that all emigration is permanent does not hold, leads to the confounding of temporary emigration probabilities with capture probabilities. This biases the estimates of capture probability when temporary emigration is a completely random process, and both capture and survival probabilities when there is a temporary trap response in temporary emigration, or it is Markovian. The use of secondary capture samples over a shorter interval within each period, during which the population is assumed to be closed (Pollock's robust design), provides a second source of information on capture probabilities. This solves the confounding problem, and thus temporary emigration probabilities can be estimated. This process can be accomplished in an ad hoc fashion for completely random temporary emigration and to some extent in the temporary trap response case, but modelling the complete sampling process provides more flexibility and permits direct estimation of variances. For the case of Markovian temporary emigration, a full likelihood is required.     

SVM-like decision theoretical classification of high-dimensional vectors

In this paper, we consider the classification of high-dimensional vectors based on a small number of training samples from each class. The proposed method follows the Bayesian paradigm, and it is based on a small vector which can be viewed as the regression of the new observation on the space spanned by the training samples. The classification method provides posterior probabilities that the new vector belongs to each of the classes, hence it adapts naturally to any number of classes. Furthermore, we show a direct similarity between the proposed method and the multicategory linear support vector machine introduced in Lee et al. [2004. Multicategory support vector machines: theory and applications to the classification of microarray data and satellite radiance data. Journal of the American Statistical Association 99 (465), 67–81]. We compare the performance of the technique proposed in this paper with the SVM classifier using real-life military and microarray datasets. The study shows that the misclassification errors of both methods are very similar, and that the posterior probabilities assigned to each class are fairly accurate.     

Monte Carlo Methods For Confidence Interval Estimation Using The Bootstrap Technique

An extension of Monte Carlo methods to confidence interval estimation, using the bootstrap technique, is investigated. The approach may have considerable potential for parameters that have estimators with complicated analytic properties but with probability distribution that can be simulated. Potential fields of application include ratio estimation, compound distribution and estimation of probabilities.     

The evaluation of general non-centred orthant probabilities

Summary. The evaluation of the cumulative distribution function of a multivariate normal distribution is considered. The multivariate normal distribution can have any positive definite correlation matrix and any mean vector. The approach taken has two stages. In the first stage, it is shown how non-centred orthoscheme probabilities can be evaluated by using a recursive integration method. In the second stage, some ideas of Schläfli and Abrahamson are extended to show that any non-centred orthant probability can be expressed as differences between at most ( m −1)! non-centred orthoscheme probabilities. This approach allows an accurate evaluation of many multivariate normal probabilities which have important applications in statistical practice.     

Lower and Upper Bounds for Rectangular Probabilities

This article presents some results showing how rectangular probabilities can be studied using copula theory. These results lead us to develop new lower and upper bounds for rectangular probabilities which can be computed efficiently. The new bounds are compared with the ones obtained from the generalized Fréchet–Hoeffding bounds and Bonferroni-type inequalities.     

A bayesian analysis of interrelated bernoulli processes with an application for clinical trials

Prior information regarding the interrelation of two Bernoulli processes may justify a clinical trial designed to corroborate this information. Antelman (1973) has studied the Dirichlet-beta which permits the expression of the prior knowledge of such interrelation. However, use of this prior distribution leads to complicated and intractable analyses. Alternately, such prior information regarding the interrelation of the processes may be adequately summarized by a simple Dirichlet distribution. Procedures for testing hypotheses regarding a priori interrelations of the success probabilities of the processes are given. Exact expressions for the posterior probabi1ities of these hypotheses are shown to be approximately equal to weighted p-values or 1ikelihood ratios.     

Evaluation of probabilities for the noncentral distributions and the difference of two T-variables with a desk calculator

It is demonstrated that integrals of the noncentral chi-square, noncentral F and noncentral T distributions can be evaluated on desk calculators. The same procedure can be used to compute probabilities for the distribution of the difference of two T-variables with equal degrees of freedom. The proposed method of computation can be used with any computer which yields probabilities for the chi-square and F distributions.     

Single-index modelling of conditional probabilities in two-way contingency tables

When analysing a contingency table, it is often worth relating the probabilities that a given individual falls into different cells from a set of predictors. These conditional probabilities are usually estimated using appropriate regression techniques. In particular, in this paper, a semiparametric model is developed. Essentially, it is only assumed that the effect of the vector of covariates on the probabilities can entirely be captured by a single index, which is a linear combination of the initial covariates. The estimation is then twofold: the coefficients of the linear combination and the functions linking this index to the related conditional probabilities have to be estimated. Inspired by the estimation procedures already proposed in the literature for single-index regression models, four estimators of the index coefficients are proposed and compared, from a theoretical point-of-view, but also practically, with the aid of simulations. Estimation of the link functions is also addressed.     

Analyzing Quantitative Trait Loci for the Arabidopsis thaliana using Markov Chain Monte Carlo Model Composition with restricted and unrestricted model spaces

Quantitative Trait Loci (QTL) mapping is a growing field in statistical genetics. However, dealing with this type of data from a statistical perspective is often perilous. In this paper we extend and apply a Markov Chain Monte Carlo Model Composition (MC³) technique to a data set of the Arabidopsis thaliana plant for locating the QTL mapping associated with cotyledon opening. The posterior model probabilities as well as the marginal posterior probabilities of each locus belonging to the model are presented. Furthermore, we show how the MC³ method can be used to deal with the situation where the sample size is less than the number of parameters in a model using a restricted model space approach.     

A kernel smoothing method of adjusting for unit non‐response in sample surveys

Non‐response is a common problem in survey sampling and this phenomenon can only be ignored at the risk of invalidating inferences from a survey. In order to adjust for unit non‐response, the authors propose a weighting method in which kernel regression is used to estimate the response probabilities. They show that the adjusted estimator is consistent and they derive its asymptotic distribution. They also suggest a means of estimating its variance through a replication‐based technique. Furthermore, a Monte Carlo study allows them to illustrate the properties of the non‐response adjustment and its variance estimator.