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Register a new userA Note on Taylors Expansion and Mean Value Theorem With Respect to a Random Variable
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Yifan Yang
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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We introduce a stochastic version of Taylors expansion and Mean Value Theorem, originally proved by Aliprantis and Border (1999), and extend them to a multivariate case. For a univariate case, the theorem asserts that suppose a real-valued function $f$ has a continuous derivative $f$ on a closed interval $I$ and $X$ is a random variable on a probability space $(Omega, mathcal{F}, P)$. Fix $a in I$, there exists a textit{random variable} $xi$ such that $xi(omega) in I$ for every $omega in Omega$ and $f(X(omega)) = f(a) + f(xi(omega))(X(omega) - a)$. The proof is not trivial. By applying these results in statistics, one may simplify some details in the proofs of the Delta method or the asymptotic properties for a maximum likelihood estimator. In particular, when mentioning there exists $theta ^ *$ between $hat{theta}$ (a maximum likelihood estimator) and $theta_0$ (the true value), a stochastic version of Mean Value Theorem guarantees $theta ^ *$ is a random variable (or a random vector).
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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.
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