ON A BERRY-ESSEEN TYPE BOUND FOR THE MAXIMUM LIKELIHOOD ESTIMATOR OF A PARAMETER FOR SOME STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Maximum likelihood estimation of a parameter appearing linearly in some stochastic partial differential equations (SPDEs) has been considered by Hübner et al. [3]. Detailed discussion of these SPDEs and some interesting phenomena arising out of the parameter estimation have been considered by them in two examples. In this paper, we study the rate of convergence of the distribution of the maximum likelihood estimator (MLE) θ̂N , of the parameter θ occurring linearly in such SPDEs. Bounds on the difference |θ̂N , − θ0|, where θ0 is the true value of the parameter, can be obtained using these results as in Mishra and Prakasa Rao [6]. In Section 2, we describe a SPDE with parameter θ such that the corresponding stochastic process u generates measures {P θ , θ ∈Θ} which are mutually absolutely continuous, and the main results pertaining to this section have been described in Section 3. In Section 4, we describe a SPDE with parameter θ such that the corresponding stochastic process u generates measures which form a family of probability measures {P θ , θ ∈Θ} which are singular with respect to each other, and this section also contains the main results connected to this problem. Comprehensive surveys on statistical inference for such classes of SPDEs are given by Prakasa Rao [7, 8]. Throughout the paper, we will denote by C a positive constant different at different places of occurrence, possibly dependent on the initial conditions of the SPDEs.


Introduction
Maximum likelihood estimation of a parameter appearing linearly in some stochastic partial differential equations (SPDEs) has been considered by Hübner et al. [3]. Detailed discussion of these SPDEs and some interesting phenomena arising out of the parameter estimation have been considered by them in two examples. In this paper, we study the rate of convergence of the distribution of the maximum likelihood estimator (MLE)θ N, of the parameter θ occurring linearly in such SPDEs. Bounds on the difference |θ N, − θ 0 |, where θ 0 is the true value of the parameter, can be obtained using these results as in Mishra and Prakasa Rao [6]. In Section 2, we describe a SPDE with parameter θ such that the corresponding stochastic process u generates measures {P θ , θ ∈ Θ} which are mutually absolutely continuous, and the main results pertaining to this section have been described in Section 3. In Section 4, we describe a SPDE with parameter θ such that the corresponding stochastic process u generates measures which form a family of probability measures {P θ , θ ∈ Θ} which are singular with respect to each other, and this section also contains the main results connected to this problem. Comprehensive surveys on statistical inference for such classes of SPDEs are given by Prakasa Rao [7,8]. Throughout the paper, we will denote by C a positive constant different at different places of occurrence, possibly dependent on the initial conditions of the SPDEs.
with the initial and boundary conditions given by Here, Q is a nuclear covariance operator for the Wiener process W Q (t,x) taking values in L 2 [0,1] so that W Q (t,x) = Q 1/2 W(t,x) and W(t,x) is a cylindrical Brownian motion in L 2 [0,1]. Then it is known that (cf. Rozovskiȋ [9]) We define a solution u (t,x) of (2.1) as a formal sum: (cf. Rozovskiȋ [9]). It is known that the Fourier coefficients u i (t) satisfy the stochastic differential equation with the initial conditions It is further known that the function u (t,x) as defined above belongs to L 2 ([0,T] × Ω;L 2 [0,1]) together with its derivative in t. Furthermore, u (t,x) is the only solution of (2.1) under the boundary conditions (2.2) and (2.3). Let P θ be the measure generated by u on C[0,T] when θ is the true parameter. It has been shown by Hübner et al. [3] that the family of measures {P ( ) θ , θ ∈ Θ} is mutually absolutely continuous and (2.8) The projection of the solution u (t,x) onto the subspace π N spanned by {e 1 ,e 2 ,...,e N } (see Liptser and Shiryayev [4]) is given by u N (t,x) = N i=1 u i (t)e i (x). Let P ,N θ be the probability measure generated by the process u N (t,x) on C[0,T] when θ is the true parameter.
Then the measure P ,N θ is absolutely continuous with respect to the measure P ,N θ0 and (2.9) The MLE of the parameter θ is given bŷ

SPDE with linear drift (absolutely continuous case): Berry-Esseen type bound
We now prove two theorems leading to a Berry-Esseen type bound for the MLEθ N, . It can be checked that E θ0 We assume that θ 0 < π 2 , where θ 0 is the true parameter. Let Φ(·) denote the standard normal distribution function and define Theorem 3.2. Let N ≥ 1 be fixed. Then there exists a constant C depending on θ 0 , f , and T such that, for any 0 < δ ≤ 1 and 0 < < 1, We first state two lemmas needed in the sequel. Then, for any δ > 0, Proof. See Michel and Pfanzagl [5].
be a standard Wiener process and let Z be a nonnegative random variable. Then, for every x ∈ R and δ > 0, Proof. See Hall and Heyde [2, page 85].
Proof of Theorem 3.1. It follows from (2.10) that Now, for any y ∈ R, where W(·) is an independent standard Wiener process by using Theorem 2.3 in Feigin [1] (due to Kunita-Watanabe) and the fact that Hence By theÎto formula, we have or, equivalently, (3.13) We know, from (3.11), that (3.14) From (3.11) and (3.13), we obtain that Next, In addition, 116 Berry-Esseen type bound for SPDE Observe that for large N ≥ N 0 depending on θ and T and for all 0 < < 1. Choosing δ = 1−r , for some 0 < r < 1, we get that the bound in Theorem 3.2 is of order As a consequence of Theorems 3.1 and 3.2, we have the following main result giving a Berry-Esseen type bound for the MLEθ N, .

SPDE with linear drift (singular case): estimation and Berry-Esseen type bound
Let (Ω,Ᏺ,P) be a probability space and consider the process u i (t,x), 0 ≤ x ≤ 1, 0 ≤ t ≤ T, governed by the SPDE where θ > 0 satisfies the initial and boundary conditions Here, I is the identity operator, ∆ = ∂ 2 /∂x 2 as defined in Section 3, and the process W(t,x) is the cylindrical Brownian motion in L 2 [0,1]. In analogy with the discussion following the stochastic differential equation given by (2.6), it can be checked that the Fourier coefficients u i (t) satisfy the stochastic differential equation where conditions (2.7) hold. Let P θ be the measure generated by the process u on C[0,T] when θ is the true parameter. It can be shown that the family of measures {P θ , θ ∈ Θ} does not form a family of equivalent probability measures. In fact, P θ is singular with respect to P θ when θ = θ in Θ (cf. Hübner et al. [3]). Let u (N) (t,x) be the projection of u (t,x) onto the subspace spanned by {e 1 ,e 2 ,...,e N } in L 2 [0,1]. In other words, (4.5) It can be checked that the MLEθ N, of θ based on u (N) satisfies the likelihood equation when θ 0 is the true parameter, From (4.6), we obtain that It can be checked that Theorem 4.1. For any 0 < δ < 1, (4.12) We can prove Theorem 4.1 using Lemmas 3.3 and 3.4 and following the method in the proof of Theorem 3.1.
Theorem 4.2. Let 0 < < 1 be fixed. Then there exists a constant C depending on θ 0 , f 2 , and T such that, for any 0 < δ < 1 and N ≥ 1, (4.13) Proof. By theÎto formula, we get that or, equivalently, (4.16) Again, by theÎto formula, it follows that or (4.18) From (4.16) and (4.18), we get that (4.19) Hence Therefore, , 120 Berry-Esseen type bound for SPDE Observe that for some k 1 depending on , θ, and T, and hence for N ≥ N 0 depending on , θ, and T. Therefore, for N ≥ N 0 depending on , θ, and T. Choosing δ = N −γ , for some γ > 0, we get that the bound is of order (4.26) As a consequence of Theorems 4.1 and 4.2, we have the following result which gives a Berry-Esseen type bound for the MLEθ N, for any fixed 0 < < 1.  . In such a case, the bound can be obtained to be of order O(N −2/3 ) by choosing γ = 4/3. We can obtain the rate of convergence for the case when N is fixed but varies over the interval (0,1) by arguments similar to those given above. We omit the details.