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.     

Stereological formulas for stationary surface processes

In the paper surface processes are considered, i.e. sets of surfaces distributed at random in the space. For translation invariant surface processes the PALM distribution of the direction of the normal in a typical surface point is defined and related to two roses of intersection. Fibre and point processes are studied which result by intersections with planes and lines respectively. For these processes stereological formulas are proved which connect some quantities with such of the surface process. As a special ease motion invariant, i.e. translation invariant and isotropic, surface processes are treated, Furthermore, weighted surface processes are considered.     

Non-linear models 1

This is the first application of a new method for testing stationary random point processes. Consider the class of all stationary ergodic point processes on the real line with arbitrary dependences among the inter–point distances (spacing).The hypothesis is :The observed process φ is a homogeneous Poisson process or more (resp.less) regular than a Poisson process.The sample is the vector of the first n points t₁, …,t_n.There is a close relation between our method for testing and queueing theory: For finding an appropriate test statistic, we observe the behaviour of a single server queue with the input φ.A table of critical values is given.     

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     

Maximum likelihood estimation of the bimodal failure rate for censored and tied observations

DIMITROV, RACHEV and YAKOVLEV ( 1985 ) have obtained the isotonic maximum likelihood estimator for the bimodal failure rate function. The authors considered only the complete failure time data. The generalization of this estimator for the case of censored and tied observations is now proposed.     

A simple approximation for the doubly noncentral t-distribution

A simple approximation for the doubly noncentral t-distribution, based upon the Fieller-Geary Theorem (1930) and approximate normality of the square root of the noncentral chi-square variable observed by Patnaik (1949), is developed, This approximation and an Edgeworth series expansion associated with it are evaluated. The simple approximation is seen to be reasonably accurate for most practical purposes.     

A modified subset selection formulation with special reference to one-way and two-way layout experiments

The usual formulation of subset selection due to Gupta (1956) requires a minimum guaranteed probability of a correct selection. The modified formulation of the present paper includes an additional requirement that the expected number of the nonbest populations be bounded above by a specified constant when the best and the next best populations are ‘sufficiently’ apart. A class of procedures is defined and the determination of the minimum sample size required is discussed. The specific problems discussed for normal populations include selection in terms of means and variances, and selection in terms of treatment effects in a two-way layout.     

Minimum average risk estimators for coefficients in linear models

We consider the problem of estimating the coefficient vector β of a linear regression model with quadratic loss function. Some biased estimators which utilize the prior information about β are considered. Also studied is the problem of estimating the parameters of an over-identified structural equation from undersized samples.     

Multiple Time Series Modeling and Extended Sample Cross-Correlations

This article provides an expository account of the multivariate autoregressive moving average models and proposes an extended sample cross-correlation approach for practical model identification. An iterative model building procedure for applying these models to real data is discussed and demonstrated by analyzing the 5-series U.S. Hog Data.