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We consider a one-dimensional random walk in random environment. We prove that the logarithm of the local time can be used as an estimator of the random environment. We give a constructive method allowing us to locally built, up to a translation, the random potential associated to the environment from a single trajectory of the random walk.
In this paper we discuss inferential aspects and the local influence analysis of the multivariate null intercept measurement error model where the unobserved value of the covariate (latent variable) follows a skew-normal distribution. In order to develop the hypotheses testing of interest and the local influence diagnostics, closed-form expressions of the marginal likelihood, the score function and the observed information matrix are presented. Additionally, an EM-type algorithm for evaluating the unrestricted and restricted maximum likelihood estimates of the parameters under equality constraints on the regression coefficients is examined. Also, we derive the appropriate matrices to assess the local influence on the parameters estimate under different perturbation schemes. The results and methods are applied to a dental clinical trial presented in Hadgu and Koch [Journal of Biopharmaceutical Statistic9 (1999) 161–178].
We develop a design-based prediction approach to estimate the finite population mean in a simple setting where some responses are missing. The approach is based on indicator sampling random variables that operate on labeled units (subjects). We define missing data mechanisms that may depend on a subject, or on a selection (such as when the study design assigns groups of selected subjects to different interviewers). Using an approach usually reserved for model-based inference, we develop a predictor that equals the sample total divided by the expected sample size. The methods are based on best linear unbiased prediction in finite population mixed models. When the probability of missing is estimated from the sample, the empirical estimator simplifies to the mean of the realized nonmissing responses. The different missing data mechanisms are revealed by the notation that accounts for the labels and sample selections. The mean squared error (MSE) of the empirical estimator, counterintuitively, is smaller than the MSE if the probability of missing is known.
In this article we consider the problem of analysing the interoccurrence times between ozone peaks. These interoccurrence times are assumed to have an exponential distribution with some rate λ>0 (which may have different values for different interoccurrence times). We consider four parametric forms for λ. These parametric forms depend on some parameters that will be estimated by using Bayesian inference through Markov Chain Monte Carlo (MCMC) methods. In particular, we use a Gibbs sampling algorithm internally implemented in the software WinBugs. We also present an analysis to detect the possible presence of change points. This is performed using the 95% credible interval of the difference between two consecutive means. Results are applied to the maximum daily ozone measurements provided by the monitoring network of Mexico City. An analysis in terms of the number of possible change points present in the model in terms of different years and seasons of the year is also presented.
This article addresses the problem of estimating the population mean of the study variable y using information on two auxiliary variables x and z in presence of nonresponse. Two classes of combined regression and ratio estimators are defined in two different situations along with their properties. An empirical study is carried out to judge the merits of the suggested estimators over usual unbiased estimator, ratio estimator and regression estimators. Both theoretical and empirical results are encouraging.
Beta-binomial/Poisson models have been used by many authors to model multivariate count data. Lora and Singer [Stat. Med.27 (2008) 3366–3381] extended such models to accommodate repeated multivariate count data with overdipersion in the binomial component. To overcome some of the limitations of that model, we consider a beta-binomial/gamma-Poisson alternative that also allows for both overdispersion and different covariances between the Poisson counts. We obtain maximum likelihood estimates for the parameters using a Newton–Raphson algorithm and compare both models in a practical example.