Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRefined Cramer Type Moderate Deviation Theorems for General Self-normalized Sums with Applications to Dependent Random Variables and Winsorized Mean
94
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let {(X_i,Y_i)}_{i=1}^n be a sequence of independent bivariate random vectors. In this paper, we establish a refined Cramer type moderate deviation theorem for the general self-normalized sum sum_{i=1}^n X_i/(sum_{i=1}^n Y_i^2)^{1/2}, which unifies and extends the classical Cramer (1938) theorem and the self-normalized Cramer type moderate deviation theorems by Jing, Shao and Wang (2003) as well as the further refined version by Wang (2011). The advantage of our result is evidenced through successful applications to weakly dependent random variables and self-normalized winsorized mean. Specifically, by applying our new framework on general self-normalized sum, we significantly improve Cramer type moderate deviation theorems for one-dependent random variables, geometrically beta-mixing random variables and causal processes under geometrical moment contraction. As an additional application, we also derive the Cramer type moderate deviation theorems for self-normalized winsorized mean.
rate research
Read More
A Cramer-type moderate deviation theorem quantifies the relative error of the tail probability approximation. It provides theoretical justification when the limiting tail probability can be used to estimate the tail probability under study. Chen Fang and Shao (2013) obtained a general Cramer-type moderate result using Steins method when the limiting was a normal distribution. In this paper, Cramer-type moderate deviation theorems are established for nonnormal approximation under a general Stein identity, which is satisfied via the exchangeable pair approach and Steins coupling. In particular, a Cramer-type moderate deviation theorem is obtained for the general Curie--Weiss model and the imitative monomer-dimer mean-field model.
Let $(X _i)_{igeq1}$ be a stationary sequence. Denote $m=lfloor n^alpha rfloor, 0< alpha < 1,$ and $ k=lfloor n/m rfloor,$ where $lfloor a rfloor$ stands for the integer part of $a.$ Set $S_{j}^circ = sum_{i=1}^m X_{m(j-1)+i}, 1leq j leq k,$ and $ (V_k^circ)^2 = sum_{j=1}^k (S_{j}^circ)^2.$ We prove a Cramer type moderate deviation expansion for $mathbb{P}( sum_{j=1}^k S_{j}^circ /V_k^circ geq x)$ as $nto infty.$ Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed.
Let $(xi_i,mathcal{F}_i)_{igeq1}$ be a sequence of martingale differences. Set $S_n=sum_{i=1}^nxi_i $ and $[ S]_n=sum_{i=1}^n xi_i^2.$ We prove a Cramer type moderate deviation expansion for $mathbf{P}(S_n/sqrt{[ S]_n} geq x)$ as $nto+infty.$ Our results partly extend the earlier work of [Jing, Shao and Wang, 2003] for independent random variables.
In this article, we are interested in the normal approximation of the self-normalized random vector $Big(frac{sum_{i=1}^{n}X_{i1}}{sqrt{sum_{i=1}^{n}X_{i1}^2}},dots,frac{sum_{i=1}^{n}X_{ip}}{sqrt{sum_{i=1}^{n}X_{ip}^2}}Big)$ in $mathcal{R}^p$ uniformly over the class of hyper-rectangles $mathcal{A}^{re}={prod_{j=1}^{p}[a_j,b_j]capmathcal{R}:-inftyleq a_jleq b_jleq infty, j=1,ldots,p}$, where $X_1,dots,X_n$ are non-degenerate independent $p-$dimensional random vectors with each having independent and identically distributed (iid) components. We investigate the optimal cut-off rate of $log p$ in the uniform central limit theorem (UCLT) under variety of moment conditions. When $X_{ij}$s have $(2+delta)$th absolute moment for some $0< deltaleq 1$, the optimal rate of $log p$ is $obig(n^{delta/(2+delta)}big)$. When $X_{ij}$s are independent and identically distributed (iid) across $(i,j)$, even $(2+delta)$th absolute moment of $X_{11}$ is not needed. Only under the condition that $X_{11}$ is in the domain of attraction of the normal distribution, the growth rate of $log p$ can be made to be $o(eta_n)$ for some $eta_nrightarrow 0$ as $nrightarrow infty$. We also establish that the rate of $log p$ can be pushed to $log p =o(n^{1/2})$ if we assume the existence of fourth moment of $X_{ij}$s. By an example, it is shown however that the rate of growth of $log p$ can not further be improved from $n^{1/2}$ as a power of $n$. As an application, we found respecti
Our purpose is to prove central limit theorem for countable nonhomogeneous Markov chain under the condition of uniform convergence of transition probability matrices for countable nonhomogeneous Markov chain in Ces`aro sense. Furthermore, we obtain a corresponding moderate deviation theorem for countable nonhomogeneous Markov chain by Gartner-Ellis theorem and exponential equivalent method.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
