Estimating parameters in a regression model with dependent noises

Let on the probability space (Q, F,P) the observations be described by the equation Y = 0 +v|, (1) where 0e© c M^d is a vector of unknown parameters, v is a known positive number, | is the vector of first d values of the AR(p)/ARCH(q) process which satisfies the equation It = в0 + £ PaIi-1 + >0 + £ ajlj-1 St. (2) i=1 V j=1 We suppose that the noise | has a conditionally Gaussian distribution with respect to some a-algebra f with a zero mean and the conditional covariance matrix D(f) such that trDf) -^(Df)) >K(d ) >0 and EXmax(Df)) O,'. Let |₀ be a random variable with a zero mean and variance s . The matrix Df) may depend on v,P_i, a j, s². The coefficients a₀,..., a_k are assumed to be nonnegative. The noise (s_t) >₀ in (2) is a sequence of i.i.d. random variables with a finite mean and constant variance ст² [13]. The nuisance parameters (p.)_1£i< , (a j )_1£j£q , and s² of the noise are unknown. The problem is to estimate the vector of unknown parameters 0 = (0₁,..., 0_d) in the model (1) by observations Y. It is known that, in the class of linear unbiased estimators, the best one is the least-squares estimator (LSE) 0 = Y . (3) However, for example, in the case of pulse-type disturbances, such an estimate may have a low accuracy. In [8-12], special modifications of this estimate were developed for discrete and continuous models with dependent conditionally Gaussian noises. Following this approach, this paper proposes the following shrinkage procedure for estimating the parameter 0: ⁰* =('^- n>; ⁽⁴⁾ where с = v²K(d)Sd, 5d-(p + fiV^^Ц²¹) • P = sup{101}. The main result of this paper is the following theorem.

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