Estimation of the variance of partial sums of dependent processes

TL;DR

This paper develops and proves the consistency of subsampling estimators for the variance of partial sums in dependent processes, with applications to rank tests and practical simulation validation.

Contribution

It introduces a non-overlapping block resampling method for variance estimation in dependent processes and extends to estimators involving the distribution function, with proven $L_2$-consistency.

Findings

Establishes $L_2$-consistency of the proposed estimators.

Simulations demonstrate estimator effectiveness and optimal block length selection.

Applicable to functionals of strongly mixing processes.

Abstract

We study subsampling estimators for the limit variance \[ \sigma^2=Var(X_1)+2 \sum_{k=2}^\infty Cov(X_1,X_k) \] of partial sums of a stationary stochastic process $(X_k)_{k\geq 1}$. We establish $L_2$-consistency of a non-overlapping block resampling method. Our results apply to processes that can be represented as functionals of strongly mixing processes. Motivated by recent applications to rank tests, we also study estimators for the series $Var(F(X_1))+2 \sum_{k=2}^\infty Cov(F(X_1),F(X_k))$, where $F$ is the distribution function of $X_1$. Simulations illustrate the usefulness of the proposed estimators and of a mean squared error optimal rule for the choice of the block length.