The robustness of the quasilikelihood estimator

The quasilikelihood estimator is widely used in data analysis where a likelihood is not available. We illustrate that with a given variance function it is not only conservative, in minimizing a maximum risk, but also robust against a possible misspecification of either the likelihood or cumulants of the model. In examples it is compared with estimators based on maximum likelihood and quadratic estimating functions.     

Random-effect models with singular precision

We show that smoothing spline, intrinsic autoregression (IAR) and state-space model can be formulated as partially specified random-effect model with singular precision (SP). Various fitting methods have been suggested for the aforementioned models and this paper investigates the relationships among them, once the models have been placed under a single framework. Some methods have been previously shown to give the best linear unbiased predictors (BLUPs) under some random-effect models and here we show that they are in fact uniformly BLUPs (UBLUPs) under a class of models that are generated by the SP of random effects. We offer some new interpretations of the UBLUPs under models of SP and define BLUE and BLUP in these partially specified models without having to specify the covariance. We also show how the full likelihood inferences for random-effect models can be made for these models, so that the maximum likelihood (ML) and restricted maximum likelihood (REML) estimators can be used for the smoothing parameters in splines, etc.     

Improving TAIEX forecasting using fuzzy time series with Box–Cox power transformation

Box–Cox together with our newly proposed transformation were implemented in three different real world empirical problems to alleviate noisy and the volatility effect of them. Consequently, a new domain was constructed. Subsequently, universe of discourse for transformed data was established and an approach for calculating effective length of the intervals was then proposed. Considering the steps above, the initial forecasts were performed using frequently used fuzzy time series (FTS) methods on transformed data. Final forecasts were retrieved from initial forecasted values by proper inverse operation. Comparisons of the results demonstrate that the proposed method produced more accurate forecasts compared with existing FTS on original data.     

Bayes sequential estimation for a Poisson process under a LINEX loss function

In this paper, within the framework of a Bayesian model, we consider the problem of sequentially estimating the intensity parameter of a homogeneous Poisson process with a linear exponential (LINEX) loss function and a fixed cost per unit time. An asymptotically pointwise optimal (APO) rule is proposed. It is shown to be asymptotically optimal for the arbitrary priors and asymptotically non-deficient for the conjugate priors in a similar sense of Bickel and Yahav [Asymptotically pointwise optimal procedures in sequential analysis, in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, University of California Press, Berkeley, CA, 1967, pp. 401–413; Asymptotically optimal Bayes and minimax procedures in sequential estimation, Ann. Math. Statist. 39 (1968), pp. 442–456] and Woodroofe [A.P.O. rules are asymptotically non-deficient for estimation with squared error loss, Z. Wahrsch. verw. Gebiete 58 (1981), pp. 331–341], respectively. The proposed APO rule is illustrated using a real data set.     

Non-ergodic martingale estimating functions and related asymptotics

Well-known estimation methods such as conditional least squares, quasilikelihood and maximum likelihood (ML) can be unified via a single framework of martingale estimating functions (MEFs). Asymptotic distributions of estimates for ergodic processes use constant norm (e.g. square root of the sample size) for asymptotic normality. For certain non-ergodic-type applications, however, such as explosive autoregression and super-critical branching processes, one needs a random norm in order to get normal limit distributions. In this paper, we are concerned with non-ergodic processes and investigate limit distributions for a broad class of MEFs. Asymptotic optimality (within a certain class of non-ergodic MEFs) of the ML estimate is deduced via establishing a convolution theorem using a random norm. Applications to non-ergodic autoregressive processes, generalized autoregressive conditional heteroscedastic-type processes, and super-critical branching processes are discussed. Asymptotic optimality in terms of the maximum random limiting power regarding large sample tests is briefly discussed.     

Higher-order and non-stationary properties of lampard's stochastic reversible counter system

Summary. In this paper we discuss higher-order and non-stationary properties of LAM-PAED'S (1968) stochastic reversible counter model whose output is a stochastic point process with MABKOV dependent time intervals. Statistical properties of the primed counter system, are also considered. A characterization of the probability density function of the interval lengths and the probability distribution of the counts by means of a mixed POXSSOH process is then derived. The distribution of counts of the primed counter system is shown to be mathematically equivalent to the birth-and-death process with immigration. A generalization of the problem when initial counts and incremental counts have independent distributions is also discussed     

Comparison of conditional and unconditional confidence intervals for the double exponential distribution

Conditional confidence intervals for the location parameter of the double exponential distribution based on maximum likelihood estimators conditioned on a set of ancillary statistics and the corresponding unconditional confidence intervals based on the maximum likelihood estimators alone are compared in two ways. Monte Carlo techniques are used and the conditional approach appears to give slightly better results although agreement as n becomes larger is noted     

Amemiya‘s generalized least squares and tests of overidentification in simultaneous equation models with qualitative or limited dependent variables

Amemiya's generalized least squares method for the estimation of simultaneous equation modeis with qualitative or limited dependent variables is known to be efficient relative to many popular two stage estimators. This note points out that test statistics for overidentification restrictions can be obtained as by-products of Amerniya's generalized least squares procedure. Amemiya's procedure is shown to be a minimum chisquare method. The Amemiya procedure is valuable both for efficient estimation and for model evaluation of such models.     

New approximate confidence intervals for the difference between two Poisson means and comparison

The problem of estimating the difference between two Poisson means is considered. A new moment confidence interval (CI), and a fiducial CI for the difference between the means are proposed. The moment CI is simple to compute, and it specializes to the classical Wald CI when the sample sizes are equal. Numerical studies indicate that the moment CI offers improvement over the Wald CI when the sample sizes are different. Exact properties of the CIs based on the moment, fiducial and hybrid methods are evaluated numerically. Our numerical study indicates that the hybrid and fiducial CIs are in general comparable, and the moment CI seems to be the best when the expected total counts from both distributions are two or more. The interval estimation procedures are illustrated using two examples.     

Point estimation after early stopping in a repeated measures trial

When the individual measurements are statistically independent, the maximum likelihood estimator calculated at the end of a sequential procedure overestimates the underlying effect. There are many clinical trials in which we are interested in comparing changes in responses between two treatment groups sequentially. Lee and DeMets (1991, JASA 86, 757–762) proposed a group sequential method for comparing rates of change when a response variable is measured for eaeh patient at successive follow-up visits. They assumed that the response follows the linear mixed effects model and derived the asymptotic joint distribution of the sequentially computed statistics. In this article, we consider the maximum likelihood estimator (MLE), the median unbiased estimator (MUE) and the midpoint of a 100(1-α)% confidence interval as point estimators for the rate of change in the linear mixed effects model, and investigate their properties by Monte Carlo simulation.