Some Estimators of Finite Population Variance of Stratified Sample Mean

We consider a continuous-time branching random walk on Z ^d , where the particles are born and die at a single lattice point (the source of branching). The underlying random walk is assumed to be symmetric. Moreover, corresponding transition rates of the random walk have heavy tails. As a result, the variance of the jumps is infinite, and a random walk may be transient even on low-dimensional lattices (d = 1, 2). Conditions of transience for a random walk on Z ^d and limit theorems for the numbers of particles both at an arbitrary point of the lattice and on the entire lattice are obtained.     

On the Moments of a Semi-Markovian Random Walk with Gaussian Distribution of Summands

In this article, a semi-Markovian random walk with delay and a discrete interference of chance (X(t)) is considered. It is assumed that the random variables ζ _n , n = 1, 2,…, which describe the discrete interference of chance form an ergodic Markov chain with ergodic distribution which is a gamma distribution with parameters (α, λ). Under this assumption, the asymptotic expansions for the first four moments of the ergodic distribution of the process X(t) are derived, as λ → 0. Moreover, by using the Riemann zeta-function, the coefficients of these asymptotic expansions are expressed by means of numerical characteristics of the summands, when the process considered is a semi-Markovian Gaussian random walk with small drift β.     

Limit theory for autoregressive-parameter estimates in an infinite-variance random walk

We consider the asymptotic behaviour of least-squares and M-estimates of the autoregressive parameter when the process is an infinite-variance random walk. It is shown that certain M -estimates converge faster than least-squares estimates and that they are also asymptotically normal.     

On random walks on affine group

Diaconis' presumption that the number of steps required to get close to uniform for a random walk on the affine group A _pis c(p)p ²with c(p) →ã is verified. We also discuss the random number generation associated with the random walk on the affine group. The number of steps to force the generated number to become random is improved. A modified version of Diacohis-Shahshahani's upper bound lemma is given and applied     

Geometric versus arithmetic random walk the case of trended variables

The asymptotic behavior of the empirical means and variances for the geometric and arithmetic random walks are studied, when the underlying random walk is trended. Thus the effect of misspecifications can be described, and two tests are proposed. The first test uses a classical approach in model selection and is based on the comparison of estimated quasi likelihoods. The second one is obtained by estimating some nuisance parameters in a Neyman-Pearson test. Some bounds for the power functions are given, which suggest that the second test may be very powerful and better than the first one.     

Un cas élémentaire d'oscillations

The probability laws describing the numbers of oscillations of the continuous symmetric random walk express themselves through the numbers of Stirling of first kind. The asymptotic behavior can easily be derived using characteristic functions.     

Asymptotic distribution theory for the kalman filter state estimator

An asymptotic distribution theory for the state estimate from a Kalman filter in the absence of the usual Gaussian assumption is presented. It is found that the stability properties of the state transition matrix playa key role in the distribution theory. Specifically, when the state equation is neutrally stable (i.e., borderline stable-unstable) the state estimate is asymptotically normal when the random terms in the model have arbitrary distributions. This case includes the popular random walk state equation. However, when the state equation is either stable or unstable, at least some of the random terms in the model must be normally distributed beyond some finite time before the state estimate is asymptotically normal.     

Asymptotic efficiency of model selection criteria: the nonzero mean gaussian ar(∞) case

Motivated by Shibata’s (1980) asymptotic efficiency results this paper dis-cusses the asymptotic efficiency of the order selected by a selection procedure for an infinite order autoregressive process with nonzero mean and unob servable errors that constitute a sequence of independent Gaussian random variables with mean zero and variance σ² The asymptotic efficiency is established for AIC–type selection criteria such as AIC’, FPE, and S_n(k). In addition, some asymptotic results about the estimators of the parameters of the process and the error–sequence are presented.     

Distributional Studies and the Computer: An Analysis of Durbin's Rank Test

A study of the distribution of a statistic involves two major steps: (a) working out its asymptotic, large n, distribution, and (b) making the connection between the asymptotic results and the distribution of the statistic for the sample sizes used in practice. This crucial second step is not included in many studies. In this article, the second step is applied to Durbin's (1951) well-known rank test of treatment effects in balanced incomplete block designs (BIB's). We found that asymptotic, χ², distributions do not provide adequate approximations in most BIB's. Consequently, we feel that several of Durbin's recommendations should be altered.     

Optimal scaling of discrete approximations to Langevin diffusions

We consider the optimal scaling problem for proposal distributions in Hastings–Metropolis algorithms derived from Langevin diffusions. We prove an asymptotic diffusion limit theorem and show that the relative efficiency of the algorithm can be characterized by its overall acceptance rate, independently of the target distribution. The asymptotically optimal acceptance rate is 0.574. We show that, as a function of dimension n , the complexity of the algorithm is O ( n ^1/3), which compares favourably with the O ( n ) complexity of random walk Metropolis algorithms. We illustrate this comparison with some example simulations.     

Maximum likelihood estimation for branching migration process

The individuals of a population reproduce according to either the line arbirth process or the line arbirth- death process and simultaneously and independently migrate on the set of all integers according to a random walk in continuous time. The maximum likelihood estimators of the transition rates are obtained and their finite time and asymptotic properties are studied.     

Empirical Likelihood for First-order Autoregressive Error-in-variable of Models With Validation Data

In this article, we consider the empirical likelihood for the autoregressive error-in-explanatory variable models. With the help of validation, we first develop an empirical likelihood ratio test statistic for the parameters of interest, and prove that its asymptotic distribution is that of a weighted sum of independent standard χ²₁ random variables with unknown weights. Also, we propose an adjusted empirical likelihood and prove that its asymptotic distribution is a standard χ². Furthermore, an empirical likelihood-based confidence region is given. Simulation results indicate that the proposed method works well for practical situations.     

LOCAL POLYNOMIAL QUASI-LIKELIHOOD REGRESSION ON RANDOM FIELDS

In this paper, local quasi‐likelihood regression is considered for stationary random fields of dependent variables. In the case of independent data, local polynomial quasi‐likelihood regression is known to have several appealing features such as minimax efficiency, design adaptivity and good boundary behaviour. These properties are shown to carry over to the case of random fields. The asymptotic normality of the regression estimator is established and explicit formulae for its asymptotic bias and variance are derived for strongly mixing stationary random fields. The extension to multi‐dimensional covariates is also provided in full generality. Moreover, evaluation of the finite sample performance is made through a simulation study.     

Approximations to noncentral distributions

Two methods for approximating the distribution of a noncentral random variable by a central distribution in the same family are presented. The first consists of relating a stochastic expansion of a random variable to a corresponding asymptotic expansion for its distribution function. The second approximates the cumulant generating function and is used to provide central χ² and gamma approximations to the noncentral χ² and gamma distributions.     

Jackknifing two-sample statistics

In this paper, a new simple method for jackknifing two-sample statistics is proposed. The method is based on a two-step procedure. In the first step, the point estimator is calculated by leaving one X (or Y) out at a time. At the second step, the point estimator obtained in the first step is further jackknifed, leaving one Y (or X) out at a time, resulting in a simple formula for the proposed point estimator. It is shown that by using the two-step procedure, the bias of the point estimator is reduced in terms of asymptotic order, from O(n⁻¹) up to O(n⁻²), under certain regularity conditions. This conclusion is also confirmed empirically in terms of finite sample numerical examples via a small-scale simulation study. We also discuss the idea of asymptotic bias to obtain parallel results without imposing some conditions that may be difficult to check or too restrictive in practice.     

Missing data mechanisms and their implications on the analysis of categorical data

We review some issues related to the implications of different missing data mechanisms on statistical inference for contingency tables and consider simulation studies to compare the results obtained under such models to those where the units with missing data are disregarded. We confirm that although, in general, analyses under the correct missing at random and missing completely at random models are more efficient even for small sample sizes, there are exceptions where they may not improve the results obtained by ignoring the partially classified data. We show that under the missing not at random (MNAR) model, estimates on the boundary of the parameter space as well as lack of identifiability of the parameters of saturated models may be associated with undesirable asymptotic properties of maximum likelihood estimators and likelihood ratio tests; even in standard cases the bias of the estimators may be low only for very large samples. We also show that the probability of a boundary solution obtained under the correct MNAR model may be large even for large samples and that, consequently, we may not always conclude that a MNAR model is misspecified because the estimate is on the boundary of the parameter space.     

The X2-goodness-of-fit tests with moment type estimators

In the x²-goodness-of-fit test the underlying null hypothesis usually involves unknown parameters. In this article we study the asymptotic distribution of the Pearson statistic when the unknown parameters are estimated by a moment type estimator based on the ungrouped data. As is expected the usual Pearson statistic is no longer asymptotically x²-distributed in this situation. We propose a statistic [Qcirc] which under certain regularity conditions is asymptotically x²-distributed. We also propose a statistic Q? for the goodness-of-fit test when the class boundaries are random. The asymptotic powers of [Qcirc] and [Qcirc]? tests are discussed.     

On the rate of convergence of recursive kernel estimates of probability densities

Recursive estimates f_n^r(x)of the rth derivative f^r(x)(r=0,1)of the univariate probability density f(x) for strictly stationary processes {X_j,} are considered. The asymptotic variance-covariance of f_n^r(x)is established for stationary triangular arrays of random variables satisfying various asymptotic independence-uncorrelatedness conditions.     

Saddlepoint approximations for some models of circular data

In this article we provide saddlepoint approximations for some important models of circular data. The particularity of these saddlepoint approximations is that they do not require solving the saddlepoint equation iteratively, so their evaluation is immediate. We first give very accurate approximations to P-values, critical values and power functions for some optimal tests regarding the concentration parameter under wrapped symmetric α-stable and circular normal models. Then, we consider an approximation to the distribution of a projection of the two-dimensional Pearson random walk with exponential step sizes.     

A sequential procedure for testing the existence of a random walk model in finite samples

Given the random walk model, we show, for the traditional unrestricted regression used in testing stationarity, that no matter what the initial value of the random walk is or its drift or its error standard deviation, the sampling distributions of certain statistics remain unchanged. Using Monte Carlo simulations, we estimate, for different finite samples, the sampling distributions of these statistics. After smoothing the percentiles of the empirical sampling distributions, we come up with a new set of critical values for testing the existence of a random walk, if each statistic is being used on an individual base. Combining the new sets of critical values, we finally suggest a general methodology for testing for a random walk model.