On ar(1) processes with exponential white noise

We consider the autoregressive model X_t= bX_t-1= Y_twhere 0 ≤ b < 1 and Y_tare independent random variables with an exponential distribution. The moments of the stationary distribution of X_tare calculated and the distribution of an approximation to the maximum likelihood estimator for b is derived. The result is used for a construction of a confidence interval for b.     

A conjugate family for ar(1) processes with exponential errors

In this article estimation of autoregressive processes AR(1) with exponential errors, denoted by ARE(1), is considered from a Bayesian perspective. For these processes a new family of conjugate distributions, denoted by GBTP, is shown to exist which follows for recursive estimation of the parameters in the model. Further extensions of the model are also considered.     

Bayesian analysis of non-negative ar(2) processes

A non-negative AR(2) process with exponentially distributed white noise is investigated in the paper. It is assumed that the autoregressive parameters are random variables with a vague prior density. They can be esto,ated by their posterior expectations. Explicit formulas for these estimators are derived and their strong consistency is proved. An approximation to the estimators is proposed which is easier for calculation. The results are illustrated in a simulation study     

Exact predictors for a generalized ar(1) process with an ar(1) parameter

The generalized AR(1) process y _t = a _t y _t-1+ v _t is considered, where the parameter a _t follows the AR(1) process a _t = Ga _t-1+ w _t.Assuming that V _t and w _t are Gaussian and independent, the first six exact predictors for future values of y _t are derived. These exact predictors are compared with Box-Jenkins -type approximations. MACSYMA, a computer algebra program, is utilized in the derivation of the predictors.     

Trimmed mean and bounded influence estimators for the parameters of the ar(1) process

Trimmed mean type estimators are proposcd for estimating the parameters of an AR(1) process. Thcsc definitions are then extended to bounded influence trimmed mcans in analogy to those in the regression case. The behaviour of the estimators are studied numerically under two

outlicr generating models.     

Goodness-of-fit tests for binomial AR(1) processes

The binomial AR(1) model describes a nonlinear process with a first-order autoregressive (AR(1)) structure and a binomial marginal distribution. To develop goodness-of-fit tests for the binomial AR(1) model, we investigate the observed marginal distribution of the binomial AR(1) process, and we tackle its autocorrelation structure. Motivated by the family of power-divergence statistics for handling discrete multivariate data, we derive the asymptotic distribution of certain categorized power-divergence statistics for the case of a binomial AR(1) process. Then we consider Bartlett's formula, which is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocorrelations, but which is not applicable when the underlying process is nonlinear. Hence, we derive a novel Bartlett-type formula for the asymptotic distribution of the sample autocorrelations of a binomial AR(1) process, which is then applied to develop tests concerning the autocorrelation structure. Simulation studies are carried out to evaluate the size and power of the proposed tests under diverse alternative process models. Several real examples are used to illustrate our methods and findings.     

Some inference results on pr(x < y) in the bivariate exponential model

This paper provides three different estimators for Pr(X < Y) when X and Y have a bivariate exponential distribution. The asymptotic variances of the three estimators are also derived. A test for the equality of the means of X and Y and confidence limits for the difference of the two means are presented. Our results are directly applicable in a reliability context with underlying bivariate exponential distribution.     

Strict stationarity of ar(p) processes generated by nonlinear random functions with additive perturbations

Let {X_n} be a generalized autoregressive process of order ρ defined by X_n=Φ_n(X_n-ρ,…,X_n-1)-η_m, where {φ_n} is a sequence of i.i.d. random maps taking values on H, and {η_n} is a sequence of i.i.d. random variables. Let H be a collection of Borel measurable functions on R_P to R. By considering the associated Markov process, we obtain sufficient conditions for stationarity, (geometric) ergodicity of {X_n}.     

Binomial AR(1) processes with innovational outliers

Abstract

Binomial integer-valued AR processes have been well studied in the literature, but there is little progress in modeling bounded integer-valued time series with outliers. In this paper, we first review some basic properties of the binomial integer-valued AR(1) process and then we introduce binomial integer-valued AR(1) processes with two classes of innovational outliers. We focus on the joint conditional least squares (CLS) and the joint conditional maximum likelihood (CML) estimates of models’ parameters and the probability of occurrence of the outlier. Their large-sample properties are illustrated by simulation studies. Artificial and real data examples are used to demonstrate good performances of the proposed models.     

New Section on Social Statistics

The analysis of a general k-factor factorial experiment having unequal numbers of observations per cell is complex. For the special case of a 2 ^k experiment with unequal numbers of observations per cell, the method of unweighted means provides a simple vehicle for analysis that requires no matrix inversion and can be used with existing software programs for the analysis of balanced data. All numerator sums of squares for testing main effects and interactions are χ² with one degree of freedom. In addition, for tests having one degree of freedom in any factorial experiment, the method of unweighted means may be modified to yield exact tests.     

Likelihood ratio type unit root tests for ar(1)models with nonconsecutive observations

For the nonconsecutively observed or missing data situation likelihood ratio type unit root tests in AR(1)models containing an intercept or both an intercept and a time trend are proposed and are shown to have the same limiting distributions as the likelihood ratio tests for the complete data case as tabulated by Dickey and Fuller(1981). Some simulation results on our tests in finite samples under A–B sampling schemes are also presented.     

Fully observed INAR(1) processes

The innovations of an INAR(1) process (integer-valued autoregressive) are usually assumed to be unobservable. There are, however, situations in practice, where also the innovations can be uncovered, i.e. where we are concerned with a fully observed INAR(1) process. We analyze stochastic properties of such a fully observed INAR(1) process and explore the relation between the INAR(1) model and certain metapopulation models. We show how the additional knowledge about the innovations can be used for parameter estimation, for model diagnostics, and for forecasting. Our findings are illustrated with two real-data examples.     

Power and Sample Size Determination When the Alternative Hypotheses are Given in Quantiles

The power of normal-theory tests about means depends on a noncentrality parameter which is a function of the unknown parameter σ. In order to calculate power and to solve sample-size problems based on power, differences between hypothesized and alternative values of the means are frequently selected as a multiple of σ, a choice which eliminates σ from the noncentrality parameter and permits a solution. Perhaps a more natural (but equivalent) way to express alternatives is to give one or more means as the quantile of order p (say Q_p ) of a distribution with another mean. As we will demonstrate, this kind of alternative also eliminates σ from the problem.     

Nonlinear semiparametric AR(1) model with skew-symmetric innovations

In this paper, we expand a first-order nonlinear autoregressive (AR) model with skew normal innovations. A semiparametric method is proposed to estimate a nonlinear part of model by using the conditional least squares method for parametric estimation and the nonparametric kernel approach for the AR adjustment estimation. Then computational techniques for parameter estimation are carried out by the maximum likelihood (ML) approach using Expectation-Maximization (EM) type optimization and the explicit iterative form for the ML estimators are obtained. Furthermore, in a simulation study and a real application, the accuracy of the proposed methods is verified.     

Slow-explosive AR(1) processes converging to random walk

Abstract

This article investigates slow-explosive AR(1) processes, which converge to a random walk (RW) process with logarithm rates, to fill the gap between nearly non-stationary AR(1) and moderately deviated AR(1) processes, and derives the asymptotics of the least squares estimator using central limit theorems for (reduced) U-statistic. We successfully establish the smooth link between the nearly non-stationary AR(1) and the moderately deviated AR(1) processes. Some novel results are reported, which include the convergence of the least squares estimator to a biased fractional Brownian motion.     

The sample autocorrelation function of I(1) processes

In case of a random walk the theoretical autocorrelations tend to one asymptotically. The sample autocorrelations, however, may decline rather fast even with large samples. We will explain this observation by deriving the asymptotic distribution that turns out to be closely related to the Dickey-Fuller (1979) distribution. Moreover we discuss the behaviour of the sample autocorrelations of integrated MA(1) and AR(1) processes. In order to prove our results we consider more general I(1) processes and apply the functional central limit theorem injected to time series analysis by Phillips (1987). We obtain unit root tests that are based on autocorrelation estimators of higher lags. We discuss their finite sample behaviour experimentally.     

A fitted test for the behrens-fisher problem

In 1975, Lee and Gurland proposed a solution to the Behrens-Fisher problem. It had excellent control of size and power and was relatively simple to use. However it requires extensive special tables. This article proposes a modification of this approach. It replaces the tables with easily computed functions of the sample sizes and a standard t table. Control of size and power are equivalent to that obtained by Lee and Gurland. Furthermore, the test is also compared with the Welch's approximate t test and shows better control of size, with similar power curves when sample sizes are at least four from each of the two normal populations.     

Tests of symmetry based on transformed empirical processes

A probability distribution function F is said to be symmetric when 1 ‐ F(x) ‐ F(‐x) = 0 for all x∈ R. Given a sequence of alternatives contiguous to a certain symmetric F₀, the authors are concerned with testing for the null hypothesis of symmetry. The proposed tests are consistent against any nonsymmetric alternative, and their power with respect to the given sequence can easily be optimized. The tests are constructed by means of transformed empirical processes with an adequate selection of the underlying isometry, and the optimum power is obtained by suitably choosing the score functions. The test statistics are very easy to compute and their asymptotic distributions are simple.