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Nonlinear Renewal Theory in Sequential Analysis (CBMS-NSF by Michael Woodroofe PDF

By Michael Woodroofe

ISBN-10: 0898711800

ISBN-13: 9780898711806

The worldwide method of nonlinear renewal idea is built-in with the author's personal neighborhood process. either the idea and its functions are positioned in point of view by way of together with a dialogue of the linear renewal theorem and its purposes to the sequential chance ratio try out. purposes to repeated importance exams, to exams with energy one, and to sequential estimation also are incorporated. The monograph is self-contained for readers with a operating wisdom of measure-theoretic likelihood and intermediate statistical conception.

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Extra resources for Nonlinear Renewal Theory in Sequential Analysis (CBMS-NSF Regional Conference Series in Applied Mathematics)

Example text

6. p. 1, but E(T) = CO E(f). Proof. -, 0) = 1. p. p. 1. p. 1. 4. E(r] — 0, which is false. p. 1. The dual case may be handled similarly. 5 relates the distributions B and B of ST and Sf to F. Here THE FACTORIZATION THEOREM. F = B +B ~B*B. Proof. Observe that 0 and /3 are convergent when 5 = 1, in which case they are the characteristic functions of B and B. 12) yields and the theorem follows from the unicity of characteristic functions. 3. Exponential tails. If F has an exponential right tail, say 1 F(x) = C exp (-(ox) for all x ^ 0 for some positive C and w, then ST has a (possibly where b = defective) exponential distribution, 24 CHAPTER 2 P{T

Then Ra=Zt — a has a limiting distribution K as a -> oo through integers. The distribution K is determined by the conditions for / ^ O and Borel sets A c [0, 1), where G0 denotes the distribution of (g) and 46 CHAPTER 4 Hdenotes the asymptotic distribution ofSTa - a; that is, H{k} = (l/E(ST))P{ST ^ k k^l, where r denotes the first ladder height. 2 may be written in the alternative form: Ra converges in distribution to R0 + (^0)-l, where R0 and £0 are independent, R0 has distribution H, and £0 has distribution G.

1 for all a ^ 0, so Ra, a ^ 0, are well defined. These notations and assumptions are used throughout the chapter. We occasionally write t for ta to avoid second order subscripts. 3. p. p. 1 as n -> oo. If gn, n ^ 1, and £"n, n ^ 1, are two slowly changing sequences, then £„ = gn + £%, n ^ 1, defines another slowly changing sequence. 1. d. with a finite mean v and a finite, positive variance. _ (i) The sequence £„ = n(y n -^) 2 , rc^l, is slowly changing. p. 8. (ii) If g is a positive, twice continuously differentiable function on (—00, oo), then Zn = ng( Yn) may be written in the form Zn = Sn + £„, where and vn denotes an intermediate point between Yn and v, n^l.

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Nonlinear Renewal Theory in Sequential Analysis (CBMS-NSF Regional Conference Series in Applied Mathematics) by Michael Woodroofe


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