By B.L.S. Prakasa Rao

ISBN-10: 3034802390

ISBN-13: 9783034802390

ISBN-10: 3034802404

ISBN-13: 9783034802406

This ebook offers a entire evaluate of effects for linked sequences and demimartingales constructed thus far, with precise emphasis on demimartingales and similar approaches. Probabilistic homes of linked sequences, demimartingales and similar procedures are mentioned within the first six chapters. functions of a few of those effects to a few difficulties in nonparametric statistical inference for such approaches are investigated within the final 3 chapters.

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**Extra resources for Associated Sequences, Demimartingales and Nonparametric Inference**

**Example text**

15) where Sk− = {s = (s1 , . . , sk ) ∈ Sk : for some i and j {1, . . , k}, si > 0 and sj < 0}. Let X = (X1 , . . , Xk ) be an inﬁnitely divisible random vector with the characteristic function φX (t) = exp{ Rk −{0} (ei(t,x) − 1 − i I(||x|| ≤ 1)(t, x))ν(dx) + i(t, μ)}. 16) Here ν is called the Levy measure of X and μ = (μ1 , . . , μk ) ∈ Rk . Resnick (1988) has proved that a suﬃcient condition for (X1 , . . , Xk ) to be associated is that ν{x = (x1 , . . , xk ) : xi xj < 0 for some i = j, 1 ≤ i, j ≤ k} = 0.

Xn ) is said to be smaller than another random vector Y = (Y1 , . . , Yn ) in the supermodular order, denoted by X sm Y, if E[f (X)] ≤ E[f (Y)] for all supermodular functions f for which the expectations exist. The following results are due to Christoﬁdes and Vaggelatou (2004). 32. Let X and Y be two n-dimensional random vectors with the same marginal distributions. If E[f (x)] ≤ E[f (Y)] for all f twice diﬀerentiable increasing supermodular functions, then X sm Y. 3. 33. Let the set {Xi , 1 ≤ i ≤ n} be a set of associated random variables and {Xi∗ , 1 ≤ i ≤ n} be another set of independent random variables independent of the sequence {Xi , 1 ≤ i ≤ n} such that the distributions of Xi and Xi∗ are the same for i = 1, .

Sj∗ )] E[(Sj+1 = E[(Sj+1 − Sj )I[τ ≥j+1] f (S1 , . . , Sj )] = E[(Sj+1 − Sj )uj (S1 , . . 10) for j ≥ 1. Hence the sequence {Sj∗ , j ≥ 1} is a demisubmartingale. We now obtain some consequences of this theorem. 4. Let the sequence {Sn , n ≥ 1} be a demisubmartingale. and τ be a positive integer-valued random variable. Furthermore suppose that the indicator function I[τ ≤j] = hj (S1 , . . , Sj ) is a componentwise nonincreasing function of S1 , . . , Sj for j ≥ 1. 11) Suppose the sequence {Sn , n ≥ 1} is a demimartingale and the indicator function I[τ ≤j] = hj (S1 , .

### Associated Sequences, Demimartingales and Nonparametric Inference by B.L.S. Prakasa Rao

by Robert

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