hw1 - ˆ θ and known covariance matrix Σ Assume further...

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STAT 760, September 23, 2011 Homework 1, Due 9/28/11 In the system under study we are able to observe p - dimensional random vectors Y 1 , Y 2 , . . . , Y T . We model them as follows: Y t = t + V t (1) for t = 1 , 2 , . . . , T , where F is a known p × q matrix, p - dimensional random vectors { V t } are independent and identically normally distributed with mean vector 0 and known p × p covariance matrix Σ V , and θ t is a q - dimensional latent random vector governed by the following equation: θ t = t - 1 + W t (2) where G is a known q × q matrix and { W t } are independent and identically normally distributed random vectors with mean 0 and known q × q covariance matrix Σ W . Finally, { W t } and { V t } are independent, and θ 0 is governed by a normal distribution with known mean vector
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Unformatted text preview: ˆ θ and known covariance matrix Σ . Assume further that all the covariance matrices are positive definite. Define the estimator ˆ θ t of the latent state θ t , based on data Y 1 ,Y 2 ,...,Y t , as ˆ θ t = E ( θ t | Y 1 ,Y 2 ,...,Y t ) . Your task is to establish the following recursive formula for this estimator: ˆ θ t = G ˆ θ t-1 + R t F T ± Σ V + FR t F T ²-1 ( Y t-ˆ Y t ) where ˆ Y t = FG ˆ θ t-1 , R t = Σ W + G Σ t-1 G T and, for any t = 1 , 2 ,...,T , Σ t = R t-R t F T (Σ V + FR t F T )-1 FR t ....
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