Non-asymptotic Confidence Estimation of the Autoregressive Parameter in AR(1) Process by Noisy Observations.pdf (LSE), maximum likelihood and other have been developed. The properties of the estimators are studied usually in asymptotic when the number of observations tends to infinity. For an autoregressive process the asymptotic properties of the LSE have been studied in [1]. To overcome the problems of investigation the properties of the estimators obtained by a fixed number of observations the sequential methods were developed. Sequential procedures use the sampling schemes with random stopping times. It allows one to study the properties of the estimators. The sequential sampling scheme was proposed in works [2] and [3] to estimate the parameter of a first order autoregressive process xk =Qxk-i +Ч, k = 1,2,.... The estimator in [2] has the guaranteed mean square deviation. The problem of estimation of parameters in ARMA processes and in AR processes with noise was studied in [4-7]. A sequential procedure of identification parameters of autoregressive processes by noisy observations was proposed in [8]. This procedure uses the Yule-Walker estimators with guaranteed mean square deviation. A problem of confidence estimation of the mean in a sequence of independent identically distributed Gaussian variables with unknown variance was studied in [9]. A sequential procedure was proposed because no procedure with fixed sample size can guarantee the prescribed coverage probability. Recently a procedure for constructing a fixed-size confidence interval with any prescribed coverage probability for parameter in AR(1) process was proposed in [10]. The interval estimation for a first-order autoregressive process was constructed in [11]. The paper is organized as follows. In Section 1 we construct a sequential point estimator of unknown parameter in AR(1) process with noises. In section 2 we construct a confidence interval with any prescribed coverage probability for the parameter in AR(1) model with unknown noise variances. In section 3 the results of numerical simulation are presented. 1. Sequential point estimator Consider an unobservable AR(1) process described by equations xk = ӨЧ-і + sk, k = 1,2,., (1) which is observed with noises Ук = xk +4k, k = 1,2,-‘-. (2) Here Ө is unknown parameter, | Ө |< 1, {sk} and (qk} are the sequences of Gaussian independent random variables with zero means Esk = E qk = 0 and variances Es2 = a2; E^ = A2 respectively, initial value x0 and processes {sk}, {qk} are independent. It's assumed that a2 and A2 are unknown. The problem is to construct the non-asymptotic confidence interval for the parameter Ө using observations {yk } . Note that from (1) and (2), we obtain the equation for the observed process yk =^k-i +%k, (3) where $k =sk + %-^-ъ E^k = 0 and E^2 =a2(1 + ө2) + a2. First we obtain a point estimator of an unknown parameter of AR(1) process. The estimator is used later to construct the confidence interval. The scheme of estimating the parameter Ө follows the approach proposed in [10] and includes three stages. First we obtain the pilot Yule-Walker estimator of Ө by a fixed number of observations. On the second stage we construct an estimator of the variance of the random variable ^k . On the third stage we construct a sequential modification of the Yule-Walker estimator of the parameter Ө . For an integer n1 > 3 define the Yule-Walker estimator of the parameter Ө as ӨОі) = f І Ук-іУк-і 1 I Ук-гУк ■ (4) \\k=3 J k=3 84 Vorobeychikov S.E., Pupkov A. V. Non-asymptotic confidence estimation of the autoregressive parameter To compensate an unstable behavior of estimator (4) for small sample size we use the projection of the estimator into the interval [-1,1]: if№)i(\\,i 0 the random variables h 1 2^, (h) and h 1 2^2(/?) are standard Gaussian. In the next section we construct a non-asymptotic confidence interval for parameter Ө . 2. Non-asymptotic confidence interval The main result of the paper gives the following Theorem. Theorem 3. Let {sk} and (qk} in (3) be the sequences of Gaussian random variables with zero means and variances Es 1 = a2, Eq^ = A2, and the sequential point estimators Ө, (h), i = 1,2 are defined by formulas (9), (10), (11). Then for all h > 0 and z > 0 Pa Г S*(h)| ,^(h) + Ө 2(h) ^ .'s/niT 2 where Г is defined by (5) and s*(h) = min (s1(h), s2(h)) and a(h, z) = 4 J 1 -ф( z^JyhC(n2)) 0 ^ v y > 1 -a(h, z), (14) ,/2-1 /Jr(V2) Here C (n2) is defined by (6), Г(п2/2) is the Gamma-function and 1 x exp(-y)dy. ф( x) = J да ( - y-dy. V 2 Г (15) 86 Vorobeychikov S.E., Pupkov A. V. Non-asymptotic confidence estimation of the autoregressive parameter Proof: Using estimators (5) and (10) we obtain the inequality

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