Abstract
Consider a first-order autoregressive processes , where the innovations are nonnegative random variables with regular variation at both the right endpoint infinity and the unknown left endpoint θ. We propose estimates for the autocorrelation parameter f and the unknown location parameter θ by taking the ratio of two sample values chosen with respect to an extreme value criteria for f and by taking the minimum of over the observed series, where represents our estimate for f. The joint limit distribution of the proposed estimators is derived using point process techniques. A simulation study is provided to examine the small sample size behavior of these estimates.
Highlights
In many applications, the desire to model the phenomena under study by non-negative dependent processes has increased
We propose estimates for the autocorrelation parameter and the unknown location parameter θ by taking the ratio of two sample values chosen with respect to an extreme value criteria for and by taking the minimum of Xtn Xt 1 over the observed series, wheren represents our estimate for
In this paper we examine the behavior of traditional estimates under conditions leading to non-Gaussian limits
Summary
The desire to model the phenomena under study by non-negative dependent processes has increased. The realization of this estimator was the stepping stone for the work done in this paper along with Davis and McCormick [4] which first considered this alternative estimator and used a point process approach to obtain the asymptotic distribution of the natural estimator ˆn This was done in the context that the innovations distribution F varies regularly at 0, the left endpoint, and satisfy some moment condition. This naturally motivates a comparison between the estimation procedure presented in this paper and the standard linear programming estimates mentioned above, since within a nonnegative AR(1) model the linear programming estimate reduces to the estimate proposed, namely, min t n Xt Xt 1 , where Xt denotes the AR(1) process
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