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This paper presents a nonlinear state space model with considering a first-order autoregressive model for measurement noises. A recursive method using Taylor series based approximations for filtering, prediction and smoothing problem of hidden states from the noisy observations is designed. Also, an expectation-maximization algorithm for calculating the maximum likelihood estimators of parameters is presented. The closed form solutions are obtained for estimating of the hidden states and the unknown parameters. Finally, the performance of the designed methods are verified in a simulation study. [ABSTRACT FROM AUTHOR]

The Danish fire insurance data have recently been modeled by composite distributions, i.e., distributions made up by piecing together two or more distributions. Here, we introduce a new non composite distribution that performs well with respect to the Danish fire insurance data. It fits better than almost all of the commonly known heavy-tailed distributions and some of the composite distributions. [ABSTRACT FROM AUTHOR]

It is demonstrated that the confidence intervals (CIs) for the probability of eventual extinction and other parameters of a Galton–Watson branching process based upon the maximum likelihood estimators can often have substantially lower coverage when compared to the desired nominal confidence coefficient, especially in small, more realistic sample sizes. The same conclusion holds for the traditional bootstrap CIs. We propose several adjustments to these CIs, which greatly improves coverage in most cases. We also make a correction in an asymptotic variance formula given in Stigler (1971). The focus here is on implementation of the CIs which have good coverage, in a wide variety of cases. We also consider expected CI lengths. Some recommendations are made. [ABSTRACT FROM PUBLISHER]

We develop an improved empirical Bayes estimation methodology for the analysis of two-state Markov chains observed from heterogeneous individuals. First, the two transition probabilities corresponding to each chain are assumed to be drawn from a common, bivariate distribution that has beta marginals. Second, randomly missing observations are incorporated into the likelihood for the hyperparameters by efficiently summing over all possible values for the missing observations. A likelihood ratio test is used to test for dependence between the transition probabilities. Posterior distributions for the transition probabilities are also derived, as is an approximation for the equilibrium probabilities. The proposed procedures are illustrated in a numerical example and in an analysis of longitudinal store display data. [ABSTRACT FROM AUTHOR]

The standard contagious distributions (see Douglas 1980) have been used in such varied fields as biology and automobile insurance, often to model various physical phenomena as well as provide a good fit to count data when other models are inadequate. Unfortunately, the parameterizations often used when working with these distributions normally lead to extremely high correlations of the maximum likelihood estimators (MLE's). This tends to lead to mathematical complexities, and causes difficulty or even errors in their interpretation. Furthermore, numerical difficulties may arise when using numerical procedures to locate the estimates. Some of these difficulties were discussed by Douglas (1980, pp. 171, 204-205), who suggested that a reparameterization to reduce or even eliminate such correlation is desirable. If the MLE's are asymptotically uncorrelated, the parameterization is orthogonal. Philpot (1964) derived an orthogonal parameterization for the Neyman Type A distribution; Stein, Zucchini, and Juritz (1987) derived for the Poisson mixture by the inverse Gaussian distribution. Parameter orthogonality has several attractive features in the present context. Since there is no correlation asymptotically, the estimates (with their standard errors) provide a simpler summary of the data than in the absence of such orthogonality. The use of a parameterization where the MLEs are highly correlated can lead to a misleading analysis, or at best a more complicated analysis that would be necessary if an orthogonal parameterization had been used. To the extent that a high correlation exists, the parameters involved tend to measure similar quantities, and orthogonality separates information about the parameters from each other. This article gives an... [ABSTRACT FROM AUTHOR]

When maximum likelihood estimates of the coefficients in a nonlinear model such as the probit model are obtained there are a number of asymptotically equivalent covariance matrix estimators that can be used. These covariance matrix estimators are typically associated with different computer algorithms. For example, with the Newton-Raphson algorithm the inverse of the negative of the Hessian matrix from the log-likelihood function is used; with the method of scoring the inverse of the information matrix is used; and with a procedure proposed by Berndt, Hall, Hall, and Hausman (1974), the inverse of the outer product of the first derivatives of the log-likelihood function is used. Although these three estimators are asymptotically equivalent, their performance can vary in finite samples. The main objective of this article is to use a Monte Carlo experiment to investigate the finite sample properties of the three covariance matrix estimators, in the context of maximum likelihood estimation of the probit model. Related questions concerning the empirical distributions of test statistics and the properties of a preliminary test estimator are also examined, under varying degrees of multicollinearity. We find that, on average, the Hessian matrix and the information matrix give almost identical results and lead to more accurate estimates of the asymptotic covariance matrix than does the estimator based on first derivatives. The finite sample mean squared error of the maximum likelihood estimator, however, is considerably greater than the asymptotic covariance matrix, and the estimator based on first derivatives provides a better estimate of finite sample mean squared error. All three estimators lead to empirical distributions that can be approximated by an asymptotic normal distribution. The pretest estimator formed by testing for the... [ABSTRACT FROM AUTHOR]

We examine the forecasting problem for a finite series of observations from either a nonseasonal or a seasonal autoregressive-moving average process. Four sources of bias in the usual estimator of forecast mean squared error are identified and analyzed for particular models using maximum likelihood and least squares parameter estimates. The usual estimator is found to be biased downwards, especially near the boundary of a stationary or invertible region, and the bias is severe for least squares estimators. An alternative estimator is proposed for the maximum likelihood case, which is shown generally to have reduced bias. [ABSTRACT FROM AUTHOR]

A model with dichotomous indicators of latent variables is developed. The latent variables are related to each other and to a set of exogenous variables in a system of structural relations. Identification and maximum likelihood estimation of the model are treated. A sociological application is presented in which a theoretical construct (an attitude) is related to a set of background variables. The construct is not measured directly, but is indicated by the answers to a pair of questionnaire statements. [ABSTRACT FROM AUTHOR]

A bivariate failure model is proposed in which the residual lifetime of one component is dependent on the working status of the other. General properties of the model are discussed, and the maximum likelihood estimates of the parameters are found in a bivariate exponential-life special case. [ABSTRACT FROM AUTHOR]

An efficient method of estimating the linear expenditure system (LES) by maximum likelihood is introduced. A test for insuring that parameter estimates obtained by numerical maximization satisfy the first-order conditions is given. In some experiments the new method of estimating the LES is compared to the more conventional method of estimation and is found to use significantly less computer time than the conventional method. [ABSTRACT FROM AUTHOR]