Guaranteed parameter estimation of stochastic linear regression by sample of fixed size.pdf Modern evolution of mathematical statistics is turned to development of data processing methods by dependent sample of finite size. One of such possibilities gives a well-known sequential estimation method, which was successfully applied to parametric and non-parametric problems. This approach for a scheme of independent observations has been primarily proposed in [1]. Then this idea has been applied to parameter estimation problem of dynamic systems in many papers and books (see [2-7, 12, 13, 15] among others). In particular, sequential estimators of the parameter AR(1) with unknown noise variance were proposed in [15]. To obtain sequential estimators with an arbitrary accuracy one needs to have a sample of unbounded size. However, in practice the observation time of a system is usually not only finite but fixed. One of the possibilities for finding estimators with the guaranteed accuracy of inference using a sample of fixed size is provided by the approach of truncated sequential estimation. The truncated sequential estimation method was developed in [8, 9] and others for parameter estimation problems in discrete-time dynamic models. In these papers, estimators of dynamic system parameters with known variance by sample of fixed size were constructed. Another but very similar approach was proposed in [10, 11]. It is known that nonlinear stochastic systems are being widely used for describing real processes in economics, technics, medicine etc. For simple models, for example, scalar first-order autoregression with discrete and continuous time, one-step sequential estimation procedure [2-5] can be constructed. In these cases one-step sequential estimators appear to be least squares estimators calculated in a special stopping time. These estimators are unbiased and simple for researching. In more complicated models such as auto-regressive processes of high order and multidimensional regressive processes we can apply two-step sequential estimation procedure [4-7] etc. At that, there is a set of multidimensional models that makes possible to construct one-step procedure of estimation the unknown parameters [2, 5, 12]. In this paper we consider models of this type. There is constructed truncated sequential parameter estimation procedure of general regression. As an example we analyze models of scalar processes ARARCH(1,1), AR(1) and two-dimensional autoregression of special type. 1. General regression model Let (Q, F, P) be an arbitrary but fixed probability space with filtration F - |Fn }n>o. It is supposed that the observable p-dimensional process {x(n)} satisfies the following equation: x(n) - A(n - 1)X + B(n -1)4 (n), n > 1, (1) where A(n), B(n) are Fn - adapted observable matrixes of size p x q, p x m respectively. Elements of these matrixes may depend on realizations of the process (x(n)). Noises \(n) form the sequence of Fn - adapted independent identically distributed (i.i.d.) random vectors with E\(n) - 0, E\(n)\'(n) - I; X - (Xj,..., X )' - vector of unknown parameters. Here and below the prime means transposition. The purpose is to construct the truncated sequential estimator of the parameter 9 - a' X, where a is a given constant vector. For construction the estimation procedure we introduce pseudoinverse matrixes A+ (n) - [ A '(n) A(n)]-1 A '(n) (assume all the inverse matrixes [ A '(n) A( n)]-1 are almost surely (a.s.) determined). Moreover, the Fn - adapted matrixes Z(n):- B (n) B (n) are supposed to be known or uniformly bounded in sense of square forms for all n > 0: X( n) H}, X c(n) > H, n-1 n-1 N N, X c(n) < H n-1 (5) LH ,N and the weights 1, n H, ZH, N -1 H - Z c(n) n=1 H ,N> Pn = a h = (h , N ) c Define 5H,N = PAI Z c (n)< H I and a2 = 1 if the process (B (n)) is observable and ||Z|| in the other n=1 case. As well as we denote EA the expectation under the distribution PA with the given parameter A. Theorem 1. Assume the process (1), matrix functions A(n) and B(n) are such that the condition (2) is fulfilled and EAс(n) 1 and H > 0 the estimator 9H N, defined in (4), possess the following property: Ea (9 H, N -9)2 < H2+e2-5 H n . Proof. To prove the theorem we find with (3) the deviation of truncated sequential estimator (4) -e-x 1 2H, N 9hN-e = - Z P„c(n)a'A+ (n - 1)C(n)-x H n=1 N Z c (n )> H n=1 N Z c (n )< H n=1 Estimate the mean square deviation of 9H N. Second moment of the first summand can be estimated similar to, e.g. [3], considering the definition of %,N, c(n) and properties Pn < 1 XH ,N Ea(9hn -9)2 -Ea Z P2c2(n)a'A+ (n- 1)Z(n)Z'(n)(A+ (n-1))'a + 92-Sh,n < H < ^^ - EA1Z P2c2 (n)a 'A+ (n - 1)E+ (n -1)(A+ (n -1))'a + 92 - 5h,n < H2 n=1 2 ^h N 2 0 be a scalar first order ARARCH process: xn = A - xn-1 WC0i +C12x«-1 , n > 1 (6) with the initial zero mean variable x0 having the eighth moment; } is a sequence of i.i.d. zero mean random variables having density which is an even function, does not increase as module of argument grows and EE,n2 = 1, in addition, x0 and } are mutually independent. The parameters a2 and aj2 are supposed to be known and a2 > 0. Note that the volatility coefficients B(n) = -\/a2 +a2x^ in (6) are observable. n=1 Put in the truncated sequential plan (4), (5) weights w(n) = 1 and H - HN - Р-ц1 • N, where P e (0,1). Then the estimator (4) and the stopping time (5) in this case will be defined as M1 - E 1 + с\2 follows: X - 1 v p«x« • x«-1 „ N ~ JJ S 2 22 'X HN n-1 Cq + C X«-1 N Sln-1 (7) .2 , _2„2 HN n-1 Cq +C1 X„ -1 inf \k e[1, N ]: S ln-1 2 2 2 -1 C0 +C1 Xn-1 N, n-1 2 , 2 2 - HN, 1 Cq +C1 Xn-1 2 n-1 2 , 2 2 < HN, X, N S n-1 C0 +C1 Xn-1 I N >Hn \, Sx: where the weights Pn are defined as: 1, 1 < n < I N N I n - In , S- Xn-1 < H 2 2 2 N> n-1 Cq +C1 Xn-1 N X2 E n-1 ^ ц 2 2 2 N> n-1 C0 +C1 Xn-1 Pn - Л xn-1 Hn - S 4T 1 1N-1 S-1 2 2 2 n-1 C0 +C1 Xn-1 2 2 2' C0 +C1 XIN-1 Theorem 2. Assume model (6). Then for every 0 < L HN, am >(log m) n=m+1 Denote for every N such that log m (N) >a the function S N = N2 - m2 8(1+ 5)2B4E(E12 -a2)4 where C21 = 1 - h (1 + 5) v Burkholder inequality. N 2 C22 = V C0 hN^m 21 2 /, \-1\4' v m (a -(logm) ) 2C22 -(logm) 2C21 C1/4 -(logm)3/2 2C -(logm) h4 - C (1+ 5)6 , B4 is the coefficient from the -h(1 + 5)) According to Assumption 1 S n = oV n у as n ^да. The following theorem contains the main result of the section. Theorem 3. Assume the model (8) with the parameter |A| < 1. Then the truncated sequential estimator (13) has the following property: S N; 1) E,(An-A)2 < NT + JV ' N - h if in addition the noises En and x0 for some positive integer s have moments of the order 8s then there exist the numbers C (s) such that as N ^да; 2) E,( n-A)2s < CNQ+o \ N. ]. Proof. The proof of the first assertion of the theorem is based on the following representation of the estimator's deviation: ~ a xn A N -A = ^- Z Pnxn-1^n -X -A- H N n=m+1 N Z xn2-1 < HN n=m+1 N 2 2 / Z xn-1 > HN, am >(log m) ,n=m+1 (log m) 1 (14) a2 > Л = I1 +12 +13. According to this formula we have E,(xN -x) < E^1° + 2E^22 + 2E^3°. (15) Consider separately the summands in (15). By the definition of the stopping time in using (9) and the technique proposed in [8, 9, 15] we can estimate E Л2 - с2 E. < •x V S Pnxn2-1lF V n-m+1 у \ * U 2 * HN ( 2 1 xn S P«X«-1| Fm V n-m+1 •x 1 log m , TT2 ~Ц HN -L E -ChN цс2 "X _2 2 cm - с C Cq-(N - m) N 1N hdN >с2 - < p. < Cq-(N - m) hN (cm-с2) Cq-(N - m ) S (n-с2) N - m n-m+1 1 N hN < P. 1- >c Cq-(N - m) // S ((-с2) N - m n-m+1 i4 N 4 E,(am-a2 ) 8B4E,(2 -a2) (am >(logm)1). Using (9) and the Chebyshev inequality for N large enough we get C -(logm)2 ( 1 (log m ) 1) 222 a2-a2 >a2 - (19) 4 = o V N E,I,2 < P, P,( >(log m) ) = m2 (a2 - (log m) 1 j The first assertion of Theorem 3 follows from the obtained inequalities (15), (16), (18), (19). The second assertion can be proved analogously to the first one. Theorem 3 is proved. Similar results for the sequential estimators of A were presented in [15]. N 3b. Efficiency of X In this section we consider a little bit more complicated modification of the estimator (13) and prove its optimality in the sense of some risk function defined below. - 2 a„ Put in the definition (13) the threshold HN = h N N1 -Am -(N - m ), hN = 1 - (log N) 1 and m = m (N) is a sequence of integer numbers satisfying the following. Assumption 2. a) m(n) = o(N), m(N) ^ да as N ^да; log m (N) b) as N ^да; m ( = o (N) V^N - log2 N m (N ) (logm) n - m < 0 Then the inequality (21) follows from (14), (19), (22) and (23). From (21) it follows that the truncated estimator {AN } is optimal (see [11, 13]) in the asymptotic minimax sense lim RrN (An )> lim inf RrN (AN )= lim RrN (AN ) = 1, N ^да N^да AN where Rr,N (AN ) = suP suP 1 (^ f )N - E,(AN -A)2 p |A

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