Risk efficiency of adaptive one-step prediction of autoregression with parameter drift.pdf When studying dynamic systems, the identification problem is often a major one to consider. A system's model is assumed to incorporate unknown parameters, estimation of which is vital for later research. Classic methods, such as maximum likelihood estimation, least squares fitting, etc., have known asymptotic properties. However, infinite samples do not occur in reality and efforts are being made to achieve non-asymptotic qualities of estimators. E.g., one may consider confidence regions for finite number of measurements (see [1, 2] among others). Another approach are methods that employ sequential analysis. Among such methods is sequential estimation method (see, e.g., [3-12]), which has guaranteed accuracy by samples of random and finite, albeit unbounded, size. Its idea was further developed into the truncated sequential estimation method (see, e.g., [1215]), which utilizes samples of bounded random size. Recently, the truncated estimation method was suggested in [16] as a modification of the truncated sequential estimation method. Truncated estimators were constructed for ratio type functionals and only need samples of fixed non-random size to achieve guaranteed accuracy in the sense of the L2m-norm, m > 1. One possible use for estimated parameters is to predict future values of the modeled random process by existing observations. In order to control both the quality of predictions and the required sample size, a loss function dependent on the two is introduced. A risk efficiency problem arises, where the expected loss is minimized by choosing a certain duration of observations. Similar problems for autoregressive processes were examined in [17] and [18], the least squares estimators and sequential estimators of unknown parameters were used. In this paper we consider a scalar stable AR(1) with parameter drift and construct real-time predictors based upon the truncated estimators of the parameters. All the parameters are assumed to be unknown. A similar model was studied in [14]. Sequential approach was, for the first time, applied to it to good effect. Resulting estimators were shown to have preassigned mean square accuracy and uniform in parameter asymptotic normality. These estimators, however, had different form than in this paper due to some parameters being known. We solve the optimization problem associated with the loss function of a special form. The proposed procedure is shown to be asymptotically risk efficient as the cost of prediction error tends to infinity. The simulation results confirm the result, but are not included for editorial reasons. The scalar case without the drift was considered in [19], multivariate AR(1) in [20] and ARMA(1,1) in [21]. (1) 1. Problem statement Consider the stable scalar autoregressive process satisfying the equation xk = Ak-1xk-1 ' k > 1 where Xk = X + I k >0, the parameter X is unknown, the condition Ex0 < да holds, ^ and n are sequences of independent identically distributed (i.i.d.) zero mean random variables, that are also independent of each other, with finite variances c^ = E^J2, c| = Erij2. In addition, to guarantee stability of the process (1) we assume the following X2 + c| 1. Therefore, one needs an estimator Xk for the unknown parameter X to construct the adaptive predictors of the form xk =Xk-1xk-1' k >1 (3) Write the corresponding prediction errors ek = xk - xk =(X - Xk-1)xk-1 + Ik-1xk-1 + Sk ■ Let еП denote the sample mean of squared prediction error 2 1 " „9 e2 = - S e\2. (4) nk=1 Define the loss function Г A 2 Ln =-e2 + nn One way the parameter A( > 0) can be interpreted is being the cost of prediction error. The corresponding risk function Rn = E0 Ln = -E0en2 + n, (5) n E0 denotes expectation under the distribution Pe with the given parameter 9 = (X,c^,cr|). Define © = |9: X2 + c| < 1 c2 < да| the process' stability parameter region. The main aim is to minimize the risk Rn on the sample size n. 2. Main result To solve the stated problem we shall use the truncated estimation method introduced in [16]. This method makes it possible to obtain the ratio type estimators with guaranteed accuracy using a sample of fixed size. According to the method, the truncated estimator of the autoregressive parameter X is based on a ratio type estimator, the least-squares type in this case k S x-Л Xk = -, k > 1 (6) S x2-1 i=1 and has the form Xk =Xkx(Ak >Hk), k > 1 (7) - 1 k 2 where Ak = - S xi-1, the notation %(B) means the indicator function of the set B and k i=1 Hk = log-1/2(k +1). (8) It should be noted, that according to [16], Hk can be taken as any decreasing slowly changing positive function. A model similar to (1) was studied in [14], but variances of the noises ^ and ni, i > 1 were assumed to be known. This information allowed construction of true sequential least-squares estimators. However, it is absent in our case, forcing one to use estimators of the form (7). Rewrite the formulae (3)-(5) with Ak replaced by Ak as follows xk = A k-1xk-1' (9) ek = (A- Ak-1)xk-1 + 2k-1xk-1 + Sk' ~2 1 2 en = -E ek' nk=1 Rn = AEq ^ + n. (10) where To minimize the risk Rn we rewrite the risk function (10) using the definition of e^ Rn = A (a2 +a2a2 + Dn ) + n, (11) 2 2 c2 =~,-TAT-^T' Dn =1 ]CEe(Xkopt - Xk)2 + ^E (a2xk-1 -aX), ax,k = E,x2k. 1 - (A2 +a2) nk=1 n k=1 From here on C denotes those non-negative constants, the values of which are not critical. Further we show that Dn = o(1) as n (12) To this end we establish the two following estimates _2 2 ax,k-1 -ax 1 to obtain a2 k = E ^ П E (A + 2k- j )2 + Exo2 П E (A + 2k - j )2 = i=0 j=1 j =1 = a2 E(A2 +a2)' + Ex2 (A2 +a2)k. i=0 Using (2) one gets 2 a2E(A2 +a2)'■ = 2ч (1 - (A2 + a2)k-1) = a2(1 - (A2 +a2)k-1) i=0 1 - (A + a2) and thus, From (2) it follows, that hence (13). a2'k -a2 = -ax(A2 +a2)k-1 + Ex^A2 +a2)k• _2 2 ax,k-1 -ax 1 the conditions E^4m min n n to get the optimal sample size noA = A1/2c (21) and the corresponding approximate minimal risk value R^ = 2 A1/2c + О (log2 A) as A ^да. (22) Similarly to [17, 18, 20], we introduce the stopping time TA as an estimator of noA, replacing c2 in its definition with an estimator c 2n TA = inf {n > A1/2cn}, (23) 2>2a ^ j where nA is the initial sample size depending on A and specified below (see Theorem 1), c2 = - SS (xk -Xnxk-1)2. (24) nk=1 (ii) for k > k 0 We formulate a theorem to prove the asymptotic equivalence of TA and noA in the sense of almost sure and mean convergences (see respectively (27), (28) below) and the optimality of the adaptive prediction procedure in the sense of equivalence of R o and the modified risk Ra = Е,Ьтл = AEq-1 el + EQTA , (25) Ta see (29). Theorem 1. Assume that and nA in (23) is such that Па > max{ko, Ar log2 A}, Па • A 172 - > 0 with r e (2 / 5,1 / 2). Let the predictors xk be defined by (9) and the risk functions defined by (5), (25). Then for every 9e © and c2 > 0 --A-> 1 Pq~ a.s., (27) o A^-да 9 ' v / A EhTA ---> 1, (28) nA --i-> 1. (29) R o да nA The proof of Theorem 1 is presented in Section 2. 3. Proofs 3.1. Proof of Lemma 1 It can be shown (see, e.g., [22], Lemma 1), that to guarantee EHxkm < C, k, m > 1, the following suffices E< да, Ex02m < да, E(A + ^1)2m < 1. Thus, from the conditions of Lemma 1 on noise moments for 9 e © it follows supE9x4m < C. (30) k>0 By the definition (7) of truncated estimators A k, their deviation has the form Ak -A = (Ak - A)Ak >Hk)-A-x(Ak Hk) + Vm ^x(Ak ax -Hk l k 0' Hk = ■ = av .(a2 )-2 >g(k+1) ^og and hence the difference a2 - Hk > 0. To estimate (34) we rewrite a2x a2 (1 -A2) ax = _2 2 a2a2 1 (a2+a2 ax). 1 -A2 (1 -A2)(1 -(A2 + a2)) 1 -A2 Using this and the definition of the process (1), one can write Ak - a2 as follows A2 . x2 - xk , 2A E 2 x2 2 k 1 A k-ax = - E2,-1 xi-1 + E Ai-12 i xi-1 k i=1 ki=1 (1 -A2) +1 E (22 -a^) + 1 E (22-1 x2-1 -aOOx i-1) + 1 E aJKM -a2) k i=1 ^ k i=1 k i=1 Then (34), the Burkholder inequality, (30) and (13) yield A2mPe (Ak < Hk) < C±• k Together with (33) this proves the second assertion of Lemma 1. 3.1. Proof of Theorem 1 The conditions on noise moments (26) yield for e e © sup Ee x16 < C. k >0 Note that assuming the distribution of n is symmetrical, the condition E (A + 21) 0 we get P(| An - A|> s) < -1гE9 (An - A)4 < Cn-2 log2 n. s From the Borel-Cantelli lemma it follows that AП-> A P9 - a.s. n n-да ° Together with (19) and (35) this yields Wn ---- 0 P9- a .s. (38) Similar arguments can be used to show vn п-- 0 P9- a.s. (39) At the same time, strong law of large numbers, (13) and the Borel-Cantelli lemma yield 1 S (ti +n2-1x2-1) --n->c2 P9-a.s. (40) n k=1 п-да Then (37) follows from the representation (36), (38)-(40). From the definition (23) of TA it follows that with P9 -probability one TA -да as A - да. Therefore, by (11) we have cc TA - c2 P9 -a.s. and hence Ta ■» 1 P9- a .s. A1/2c A To prove (28) we introduce for any positive A the auxiliary sequence of numbers уA, YA,n = n 2^Т-Т1-, n > L 2log A Denote mn = - S kk +ni2-1xk!-1 -c2 1 nk=1v ' Observe that mn can be represented as a sum of two martingales and decaying to zero as O(n-1) sequence mn = - S -c^ + - S ^-А^ - c^c^k-1) + -П S c2 (c^,k-1 - c2). nk=1 n k=1 nk=1 By the definition of TA and (36) we have n2A-1

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