Asymptotic bootstrap validity for finite markov chains

Two methods of bootstrap, viz., standard, and conditional, are presented for estimating the transition probabilities of a finite state Markov chain. Asymptotic validity of the bootstrap estimates are established for both methods. An applica- tion to a bootstrapped statistic for testing independence is briefly discussed together with some simulation results.     

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.     

Computation of the stationary distribution of a markov chain

Eight algorithms are considered for the computation of the stationary distribution l´ of a finite Markov chain with associated probability transition matrix P. The recommended algorithm is based on solving l´(I—P+eú)=ú, where e is the column vector of ones and u´ is a row vector satisfying u´e ≠0.An error analysis is presented for any such u including the choices ú= e_jP and ú=e´_j where é_j is the jth row of the identity matrix. Computationalcomparisons between five of the algorithms are made based on twenty 8 x 8, twenty 20 x 20, and twenty 40 x 40 transition matrices. The matrix (I—P+eú)^?1 is shown to be a non-singular generalized inverse of I—P when the unit root of P is simple and úe ≠ 0. A simple closed form expression is obtained for the Moore-Penrose inverse of I—P whenI—P has nullity one     

A geometric approach to transdimensional markov chain monte carlo

The authors present theoretical results that show how one can simulate a mixture distribution whose components live in subspaces of different dimension by reformulating the problem in such a way that observations may be drawn from an auxiliary continuous distribution on the largest subspace and then transformed in an appropriate fashion. Motivated by the importance of enlarging the set of available Markov chain Monte Carlo (MCMC) techniques, the authors show how their results can be fruitfully employed in problems such as model selection (or averaging) of nested models, or regeneration of Markov chains for evaluating standard deviations of estimated expectations derived from MCMC simulations.     

Improved likelihood ratio tests and pearson chi-square tests for independence in two dimensional contingency tables

When an I×J contingency table has many cells having very small frequencies, the usual chi-square approximation to the upper tail of the likelihood ratio goodness-of-fit statistic, G² and Pearson chi-square statistic, X², for testing independence, are not satisfactory. In this paper we consider the problem of adjusting G² and X². Suitable adjustments are suggested on the basis of analytical investigation of asymptotic bias terms for G² and X². A Monte Carlo simulation is performed for several tables to assess the adjustments of G² and X² in order to obtain a closer approximation to the nominal level of significance.     

Convergence rates and powers of six power-divergence statistics for testing independence in 2 by 2 contingency table

Several tests for testing independence of 2 by 2 contingency tables have been proposed over the years. Cressie and Read (1984) identified several of these tests as members of a power-divergence family, and much of the characteristics for these tests are unified. However, the question of which test is best is still not fully understood. This paper provides algorithms for chi-square estimates and investigates the convergence rates and powers of these chi-square tests.     

Testing for markov process vs semi-markov process

Here we have obtained a non-parametric test for testing the null hypothesis H₀ that the given realization is from a Markov process against the alternative hypothesis H₁ that it is from a semi-Markov process with the transition rate monotone increasing (or decreasing). We have shown that the test criterion has normal distribution asymptotically, and the test is consistent and unbiased.     

Multiple Marginal Independence Testing for Pick Any/C Variables

Many survey questions allow respondents to pick any number out of c possible categorical responses or “items”. These kinds of survey questions often use the terminology “choose all that apply” or “pick any”. Often of interest is determining if the marginal response distributions of each item differ among r different groups of respondents. Agresti and Liu (1998, 1999) call this a test for multiple marginal independence (MMI). If respondents are allowed to pick only 1 out of c responses, the hypothesis test may be performed using the Pearson chi-square test of independence. However, since respondents may pick more or less than 1 response, the test's assumptions that responses are made independently of each other is violated. Recently, a few MMI testing methods have been proposed. Loughin and Scherer (1998) propose using a bootstrap method based on a modified version of the Pearson chi-square test statistic. Agresti and Liu (1998, 1999) propose using marginal logit models, quasisymmetric loglinear models, and a few methods based on Pearson chi-square test statistics. Decady and Thomas (1999) propose using a Rao-Scott adjusted chi-squared test statistic. There has not been a full investigation of these MMI testing methods. The purpose here is to evaluate the proposed methods and propose a few new methods. Recommendations are given to guide the practitioner in choosing which MMI testing methods to use.     

A bayesian method in determining the order of a finite state markov chain

In this paper, we use the Bayesian method in the application of hypothesis testing and model selection to determine the order of a Markov chain. The criteria used are based on Bayes factors with noninformative priors. Com¬parisons with the commonly used AIC and BIC criteria are made through an example and computer simulations. The results show that the proposed method is better than the AIC and BIC criteria, especially for Markov chains with higher orders and larger state spaces.     

Evaluating the power of the mann-whitney test using the bootstrap method

In this paper we evaluate the power of the Mann-Whitney test in the shift model G(x) = F (x+θ) for all x , where the distribution of G is obtained by shifting F by an amount of θ.

The bootstrap method was used to evaluate the power of the Mann-Whitney test . A comparison among the bootstrap power , the asymptotic power of the Mann-Whitney test and the t-test power proved that the bootstrap is a better technique , because , it does not require the assumption of normality.     

Quotas on runs of successes and failures in a multi-state markov chain

For the time-homogeneous multi-state Markov chain {X_n,n≧0} with states labeled as "0" (success) and "f"(failure), f=1,2,… the waiting time problems to be discussed arise by setting quotas on runs of success and failures. Some particular cases are considered.     

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.     

A simulation study of a test for the equality of the coefficients of variation

Simulation study results are given for the size and power of a test for the equality of the coefficients of variation from r normal populations. Independent samples of equal and unequal size from the normal and three other distributions were used. The size and power of the test compare favorably to two tests developed by Doornbos and Dijkstra and the test statistic is simpler to compute.     

Asymptotic properties of the score test for varying dispersion in exponential family nonlinear models

Wei et al. [B.C. Wei, J.Q. Shi, W.K. Fung, and Y.Q. Hu, Testing for varying dispersion in exponential family nonlinear models, Ann. Inst. Statist. Math. 50 (1998), pp. 277–294.] developed the score diagnostics for varying dispersion in exponential family nonlinear models, such as the normal, inverse Gaussian, and gamma models, and investigated the powers of these tests through Monte Carlo simulations. In this paper, the asymptotic behaviours, including asymptotic chi-square and approximate powers under local alternatives of the score tests, are studied and examined by Monte Carlo simulations. The methods to estimate local powers of the score tests are illustrated with Grass yield data [P. McCullagh, and J.A. Nelder, Generalized Linear Models, Chapman and Hall, London (1989).].     

The multidimensional markov chain with prespecified asymptotic means,and (auto-)covariances

In this paper a method is presented to construct random time series which, starting from their present values, converge to stationary time series with a priori specified mean values, standard deviations, correlations and autocorrelations. The method is applied to simulate time series of price-inflation, wage-inflation, and interest rates, whose mean values, standard deviations, correlations and autocorrelations converge to the values which are estimated from historical data This application is a circular part of a Decision Support System which assists management of pension funds in analysing new methods of calculating pension premiums.     

A structural model for the markov chain of change in vote intention

Assuming a first-order Markov chain, we propose a structural model for the transition probabilities in vote intention. The proposed model utilizes the ordering among the categories representing vote intentions and carries the flavor of distance models. It also allows a stochastic ordering among distributions reflecting the extent of change. The model is easy to fit and provides a nice interpretation of the data. The model is applied to a panel study of vote intention acquired through six successive interviews before the 1940 Presidential election in Erie County, Ohio.     

Applying a Markov chain for the stock pricing of a novel forecasting model

In this article, a stock-forecasting model is developed to analyze a company's stock price variation related to the Taiwanese company HTC. The main difference to previous articles is that this study uses the data of the HTC in recent ten years to build a Markov transition matrix. Instead of trying to predict the stock price variation through the traditional approach to the HTC stock problem, we integrate two types of Markov chain that are used in different ways. One is a regular Markov chain, and the other is an absorbing Markov chain. Through a regular Markov chain, we can obtain important information such as what happens in the long run or whether the distribution of the states tends to stabilize over time in an efficient way. Next, we used an artificial variable technique to create an absorbing Markov chain. Thus, we used an absorbing Markov chain to provide information about the period between the increases before arriving at the decreasing state of the HTC stock. We provide investors with information on how long the HTC stock will keep increasing before its price begins to fall, which is extremely important information to them.     

Markow chain markov field dynamics:models and statistics

This study deals with time dynamics of Markov fields defined on a finite set of sites with State Space E, focussing on Markow Chain Markow Field (MCMF) evolution. Such a model is characterized by two families of potentials:the instantaneous interaction potentials, and the time delay potentials. Four models are specified:auto-exponential dynamics (E=R⁺), auto-normal dynamics (E = R), auto-Poissonian dynamics (E = N) and auto-logistic dynamics (E qualitative and finite). Sufficient conditions ensuring ergodicity and strong law of large numbers are given by using a Lyapunov criterion of stability, and the conditional pseudo-likelihood statistics are summarized. We discuss the identification procedure of the two Markovian graphs and look for validation tests using martingale central limit theorems. An application to meteorological data illustrates such a modelling.     

A martingale-based test for independence of time to failure and cause of failure for competing risks models

In this article, we introduce a class of tests, using a martingale approach, for testing independence of failure time and cause of failure for competing risks data. Asymptotic distribution of the proposed test statistic is derived. The procedure is illustrated with a real-life data. A simulation study is carried out to assess the level and power of the test.