Non‐negative Autoregressive Processes

Abstract

Consider a stationary autoregressive process given by XI = b,X,-, + . . . + b,X,-, + Y;, where the I; are independent identically distributed positive variables and b,, . . . , b, are non-negative parameters. Let the variables XI,. . . , X, be given. If p = 1 then it is known that b: = min(XJX1-,) is a strongly consistent estimator for b, under very general conditions. In this paper the case p = 2 is analysed in detail. It is proved that rnin(X,/X,-,) + b, almost surely (a.s.) and min(XJX,-,) + b, + b: a.s. as n + to. The convergence is very slow. Denote by b: and b: values of b, and b, respectively which maximize b, + b, under the conditions X, -b,X,-, -b,X,-2 2 0 for t = 3, . . . , n. We prove that b: + b, and b: + b, a.s. Simulations show that by and b:are better than the least-squares estimators of the autoregressive coefficients when the distribution of Y; is exponential.

Robust bayesian analysis of an autoregressive model with exponential innovations

“…Let us define the modified maximum likelihood estimator (MLE) of ρ introduced by Andẽl (1988Andẽl ( , 1989 as follows…”

Estimation for autoregressive processes with positive innovations

“…For the case of p > 1, a straightforward generalization of (1.2) does not seem to perform well- Andel (1989). Of course, one may ignore the special nature of the innovations and use the Yule-Walker estimators (see, for example, Brockwell and Davis(l991)), but we show that one can sometimes do better than the n1I2 rate of convergence.…”

“…This estimator is obtained by solving equations which turn out to be examples of generalized martingale estimating equations as described in Feigin(1991). Andel (1989) found by simulation that this estimator converges at a faster rate than the Yule-Walker estimator.…”

“…Of course, one may ignore the special nature of the innovations and use the Yule-Walker estimators (see, for example, Brockwell and Davis(l991)), but we show that one can sometimes do better than the n1I2 rate of convergence. Andel (1989) considered the case p = 2 and suggested two estimators of (dl, $2). One is based on a maximum likelihood argument and is the one we consider here.…”

An estimator for parameters of a nonlinear nonnegative multidimensional AR(1) process

“…For the case of AR(p) model with p > 1 a straightforward generalization of (1.1) does not perform well. Anděl [5] suggested another estimator based on a maximum likelihood argument. His method was modified and generalized by An [1] and by An and Huang [2].…”