Download e-book for kindle: Asymptotic statistical methods for stochastic processes by Linkov YU. N.

By Linkov YU. N.

ISBN-10: 0821811835

ISBN-13: 9780821811832

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3. 1. For any a G [0,1] there exists a test S^a^'£^ of level a, where (c (a ),e (a )) is some solution of the equation a(££,e) = a with respect to (c,e). 2. 1. A test with e(0) = 1 is called the NeymanPearson test of level a for distinguishing between hypotheses H1 and H l (in what follows this test is denoted by 5*'a). 1. 1. It is not difficult to show that the functions c(a) and e(a) can be chosen to be a = 0, oo, ( 2. 11) c(a) = < c(a), , 0, 0 < a < a t, e(a) = 0, 1, a e(a), 0 < a < at, = e(a )f at < a < l , at < a < 1, where c(a) is the least c satisfying the inequalities P*(zt > c) < a < P \ zt > c), a - P \ zt > c(a )) Pt(zt = c(a )) ’ e(a) .

38 ASYMPTOTIC DISTINGUISHING P r o o f . 13). 1. 12), we have ||P‘ A P ‘ ||= E‘ x ( 2t > l ) + E*x(2t < l ) = EQitx(zt > 1) + Eq 3x (zi < 1) = Eq (3j A lt). This equality explains the notation ||P* AP*||. 2. 11) among them to prove the following theorem, which yields the characteri­ zation of complete asymptotic distinguishability (if*) A (if*) in various terms. 1. oo p(P\ p t) = 1. P r o o f , a) =$• b). Let tn t oo, n -* oo, and let Stn 6 £ ‘ n be such that a (h n) —> 0 and 0{Stn) -> 0. Putting An = x(dtn > j)> where 0 < 7 < 1, in view of the Chebyshev inequality we have P ^ (A n) = P ‘ "( 7 ) < 7 -1 « ( * J - 0.

1) y* = M* + Cf* * (#** - i/*) , where M l G AIfoc 0(Ft, P t), /z* is an F*-optional integral-valued random measure on R + x Ro for which the measure satisfies the property of P (F* )-oo. 22 1. 1. For fixed s g (0, oo) and any e G (0,1] let the following conditions be fulfilled: 1) P M im ^ oo x(|^| > e) 117*1 * v\t = 0; 2) P t~\imt^ 00[(M t)st + x(|17£| < e )|[/£|2 * i/*t] = c(s) G [0,oo), where c(s ) zs some nonrandom number dependent on s.

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Asymptotic statistical methods for stochastic processes by Linkov YU. N.

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