Abstract
The paper considers the estimation problem of the autoregressive parameter in the first-order autoregressive process with Gaussian noises when the noise variance is unknown. We propose a non-asymptotic technique to compensate the unknown variance, and then, to construct a point estimator with any prescribed mean square accuracy. Also a fixed-width confidence interval with any prescribed coverage accuracy is proposed. The results of Monte-Carlo simulations are given.
Highlights
The problem of constructing a fixed-width confidence interval with any prescribed accuracy by a finite sample size might be complicated enough even for the process with independent observations
The problem of estimation with any prescribed accuracy of the first-order autoregressive process parameter was considered in Borisov and Konev (1977)
The sequential estimator of an unknown parameter of a diffusion-type process with any prescribed mean square accuracy was described in Novikov
Summary
The problem of constructing a fixed-width confidence interval with any prescribed accuracy by a finite sample size might be complicated enough even for the process with independent observations. In Konev and Vorobeychikov (2017), the sequential estimation procedure of Borisov and Konev (1977) was modified; it allows obtaining a point autoregressive parameter estimator with a non-asymptotic Gaussian distribution and constructing a fixed-width confidence interval with any prescribed probability of coverage. Unlike Dmitrienko and Konev (1994), we use an additional stage to obtain an estimator of an unknown autoregressive parameter and improve the upper bound of an unknown variance. It leads to the decrease of the estimation time as compared with Dmitrienko and Konev (1994). The presented results of the simulation demonstrate a good quality of the algorithm
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