By P. Sint (auth.), T. Havránek, Z. à idák, M. Novák (eds.)

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**Additional resources for Compstat 1984: Proceedings in Computational Statistics**

**Example text**

6) L( k) a lower triangular matrix with l's on the diagonal. ( ii) We can write L(k) = L(O) + 11k L(-U(k) A(k) where = kA(l) (4. 8) on k, and the are uniformly bounded in absolute value. ( iii) Let be and i-th diagonal elements of A(l) ,A(O) and A(-ll(k) >. ~-1){k). Then >. 3 allows us to define the likelihood via the following theorem. 3. 0 ] by T-d T-d A(O) )-1/2 exp -1/2 I;' £2/ A(O) t f.. t t j=l j j=l j j (II independent of k and the para- e. 4). 11). be Kalman computed recursively with the modified Kalman filter of Section 3.

For the purposes of this paper it is simply stated that when n consists of band matrices whose elements decrease monotonically away from the diagonal. and var(Sk) is a positive semi-definite quadratic form of the least-squares residuals. the problem reduces to alternately (a) solving a large linear program (LP) and (b) finding a vector such that a large quadratic form is negative. (a) The LP is minimize cTx with respect to ~, subject to ~ ~ (~ and x) 0 , where ~ and E are vectors of length t (where t takes values between 1 and 30), b is a vector of length m (where m depends on the choice of n but lies between nand n(n+l)/2).

1. n where F belongs to the domain of attraction of the Gumbel distribution (fact that we shall denote by F~~(A». e •• P{M ~a x+b } • Fm(a x+b ) m m m m m + ur- A(x). ~1. a >0. b E JR. 's replica of Y. In other words. Fm(a x+b ) ~ A( x ). m ~ m m and in practical situations we often identify Fm(x). F usually unknown. with A«x-b)/a) for some estimated a>O and b •. even for moderate values of m. f. , and our study had to be 2. conducted in the light of stochastic simulation. f. 's covered in this paper are the exponential.