00258 ¿e59ae38ddf4d993b605efb46266481

260

Yander Wiel

dropping the subscript on U and the superscripts on Z_k because they are not necessary here.

Hawkins (1977) studied the problem of testing for a mean shift in iid normal variates which is the case of y = 0. For most testing problems -21n(LR) is asymptotically %² distributed. In this case, however, the necessary regularity conditions do not hoÅ‚d. The 3 theorems below extend Hawkins' results on the nuli distribution of U to the case where y > 0.

Theorem 1, proved in the Appendix, shows that |Z₀|,...,|Z,| is a Markov seÄ…uence. This property is the key to Theorem 2 which gives the distribution of U in terms of 2 sets of conditional probability functions F_k and F_k. Theorem 3 shows that the Fs can be computed recursively. The appendix gives intuitive arguments for Theorems 2 and 3. The proofs are straightforward but the details are omitted. We use <j>(-) for the jV(0, 1) density.

Theorem 1 Urtder H₀, Z₀,..., Z,is a Markov seÄ…uence with each Zt~N(0,1) and this implies that |Zo|,...,|Z_f| is a Markov seÄ…uence.

Theorem 2 Under Ho, U has density 2<j>(«)^^(M|«)Ą(«|w)

k-0

where 5(«|v) = P_r{\Z₉\Å›, tf,...,|Z₄_₁| ^ u given (|Z_k\= v)> F_k(u\v) = P_f{|Z_A+I|^ u,...,\Z_t\ Å› u given (|ZJ = v)}

Theorem 3 F_k andF_k (k = 0,...,t) can be computedrecursively from

F_k(u\v) =

Ä„(«|v) = £_oÄ„₊₁(w|5)g_Å‚+1(s|v)^

where F₀(u,v) = 1, F_t (u,v) = 1 and g_k(-\v) is the density of conditional on

l^Z*l ^{= v}-