The widely orthant dependent (WOD) sequences are very weak dependent sequences of random variables. For the weighted sums of non-negative m-WOD random variables, we provide asymptotic expressions for their appropriate inverse moments which are easy to calculate. As applications, we also obtain asymptotic expressions for the moments of random ratios. It is pointed out that our random ratios can include some models such as change-point detection. Last, some simulations are illustrated to test our results.
In this paper, we investigate the CUSUM-type estimator of mean change-point models based on m-asymptotically almost negatively associated (m-AANA) sequences. The family of m-AANA sequences contains AANA, NA, m-NA, and independent sequences as special cases. Under some weak conditions, some convergence rates are obtained such as OP(n1/p−1), OP(n1/p−1log1/pn) and OP(nα−1), where 0≤α<1 and 1<p≤2. Our rates are better than the ones obtained by Kokoszka and Leipus (Stat. Probab. Lett., 1998, 40, 385–393). In order to illustrate our results, we do perform simulations based on m-AANA sequences. As important applications, we use the CUSUM-type estimator to do the change-point analysis based on three real data such as Quebec temperature, Nile flow, and stock returns for Tesla. Some potential applications to change-point models in finance and economics are also discussed in this paper.
Let {Zn, n≥1} be a sequence of nonnegative, weakly dependent random variables, and Xn=∑i=1nωniZi, where true{ωni,1≤i≤n,n≥1true} is an array of nonnegative weights. We show that E[]ftrue(Xntrue)prefix−1 can be asymptotically approximated by []ftrue(EXntrue)prefix−1 for a class of functions f(·) satisfying some mild conditions. Under some general conditions, we also prove that the expectation Etrue[Xnfalse/true(a+Yntrue) αtrue] approximates to EXnfalse/true(a+EYntrue) α with a certain convergence rate for any a > 0 and α>0, where Yn=∑i=1nZi. The results obtained in the article improve and extend some corresponding ones in the literature. Some numerical simulations and a real data example are also provided to support the theoretical results.
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