REGRESSION ASYMPTOTICS USING MARTINGALE CONVERGENCE METHODS

Testing the martingale difference hypothesis using integrated regression functions

This paper proposes an omnibus test for testing a generalized version of the martingale difference hypothesis (MDH). The generalized hypothesis includes the usual MDH or testing for conditional moments constancy such as conditional homoscedasticity (ARCH effects). Here we propose a unified approach for dealing with all of them. These hypotheses are long standing problems in econometric time ser...

Martingale Methods in Statistics

Asymptotics for Hazard Regression

In hazard regression (HARE), the logarithm of the conditional hazard function of a survival time given a covariate is modeled by a sum of polynomial splines and their tensor products. Under appropriate conditions, it has been shown that the (nonadaptive) HARE estimate of the conditional log-hazard function possesses an optimal L 2 rate of convergence. The current paper considers the L oo rates ...

Martingale approach to subexponential asymptotics for random walks∗

Consider the random walk Sn = ξ1 + · · · + ξn with independent and identically distributed increments and negative mean Eξ = −m < 0. Let M = sup0≤i Si be the supremum of the random walk. In this note we present derivation of asymptotics for P(M > x), x → ∞ for long-tailed distributions. This derivation is based on the martingale arguments and does not require any prior knowledge of the theory o...

Polya’s Urn and the Martingale Convergence Theorem

This paper is about Polya’s Urn and the Martingale Convergence Theorem. I will start with the formal definition, followed by a simple example of martingale and the basic properties of martingale. Then I will explain the Polya’s Urn model and how it contributes to proving the Martingale Convergence Theorem. Finally, I will give a full proof of the Martingale Convergence Theorem.