[Paper Review] Self-normalized Cramér Type Moderate Deviations under Dependence

Xiaohong Chen, Qi-Man Shao+1 more|arXiv (Cornell University)|Sep 12, 2014

Statistical Methods and Inference24 references3 citations

TL;DR

This paper establishes Cramér-type moderate deviation results for self-normalized sums under weak dependence, using two block-based schemes—big-block-small-block and interlacing—under weak moment conditions. It shows that self-normalization extends the range of Gaussian approximation compared to non-self-normalized versions, even under dependence, with the interlacing scheme offering superior finite-sample performance.

ABSTRACT

We establish a Cramér-type moderate deviation result for self-normalized sums of weakly dependent random variables, where the moment requirement is much weaker than the non-self-normalized counterpart. The range of the moderate deviation is shown to depend on the moment condition and the degree of dependence of the underlying processes. We consider two types of self-normalization: the big-block-small-block scheme and the interlacing or equal-block scheme. Simulation study shows that the latter can have a better finite-sample performance. Our result is applied to multiple testing and construction of simultaneous confidence intervals for high-dimensional time series mean vectors.

Motivation & Objective

• Address the lack of moderate deviation theory for self-normalized sums in weakly dependent, fat-tailed time series.
• Extend self-normalized limit theory from independent to weakly dependent processes under minimal moment conditions.
• Develop two block-based self-normalization schemes (big-block-small-block and interlacing) suitable for dependent data.
• Demonstrate that self-normalization yields a wider range of Gaussian approximation than non-self-normalized counterparts under same moment and dependence conditions.
• Provide theoretical justification for using self-normalized statistics in high-dimensional inference under dependence.

Proposed method

• Propose two self-normalized sum statistics based on block sampling: big-block-small-block and interlacing (equal-block) schemes.
• Use functional dependence measures and β-mixing coefficients to quantify weak dependence in non-linear time series.
• Establish moderate deviation results via martingale approximation and exponential tail bounds, particularly leveraging Lemma 6.3 for negative deviation probabilities.
• Apply Burkholder’s inequality and projection techniques to control the moments of block sums and their conditional expectations.
• Derive bounds on the ratio of tail probabilities to standard normal tail probabilities, showing asymptotic equivalence up to a controlled error term.
• Use the interlacing scheme to improve finite-sample performance, as validated by simulation.

Experimental results

Research questions

• RQ1 Can self-normalized sums maintain a wide range of Gaussian approximation under weak dependence and weak moment conditions?
• RQ2 How does the choice of block scheme (big-block-small-block vs. interlacing) affect the finite-sample performance of self-normalized moderate deviation approximations?
• RQ3 What is the impact of dependence and moment conditions on the range of moderate deviations for self-normalized statistics?
• RQ4 Can self-normalized statistics be reliably used in high-dimensional inference for dependent time series?
• RQ5 How do functional dependence measures and β-mixing coefficients affect the theoretical bounds in moderate deviation results?

Key findings

• The self-normalized sum based on the interlacing scheme exhibits better finite-sample performance than the big-block-small-block scheme in simulations.
• Under weak dependence and moment conditions, the moderate deviation range for self-normalized sums is wider than for non-self-normalized sums under the same moment constraints.
• For i.i.d. sequences with $ E|X_i|^{2+ u} < ty $, the moderate deviation range for self-normalized sums is $ O(n^{1/2}) $, matching the i.i.d. case.
• The range of moderate deviation is shown to depend on both the moment condition $ u $ and the degree of dependence, with the interlacing scheme achieving a larger effective range.
• The ratio $ P(T_n o x) / (1 - ar{\Phi}(x)) $ is bounded by $ \exp(O(1)(1+x)^{2+ u}/d_{n, u}^{2+ u}) $ for $ x $ up to $ O(n^{1/2}) $, under weak moment and dependence conditions.
• Theoretical results are extended to multiple testing and simultaneous confidence intervals for high-dimensional time series mean vectors, generalizing prior i.i.d. methods to dependent settings.