Multiplicative random walk Metropolis-Hastings on the real line

TL;DR

This paper introduces the random dive MH algorithm, a multiplicative random walk Metropolis-Hastings method for the real line, demonstrating its theoretical properties and superior performance over standard MH in simulations and real data.

Contribution

It proposes a novel multiplicative random walk MH algorithm, analyzes its convergence properties, and shows its advantages over existing methods through simulations and real data analysis.

Findings

Kernel is irreducible, aperiodic, Harris recurrent under mild conditions.

Kernel is geometrically ergodic for a broad class of target densities.

RDMH outperforms standard MH in mixing and convergence in simulations and real data.

Abstract

In this article we propose multiplication based random walk Metropolis Hastings (MH) algorithm on the real line. We call it the random dive MH (RDMH) algorithm. This algorithm, even if simple to apply, was not studied earlier in Markov chain Monte Carlo literature. The associated kernel is shown to have standard properties like irreducibility, aperiodicity and Harris recurrence under some mild assumptions. These ensure basic convergence (ergodicity) of the kernel. Further the kernel is shown to be geometric ergodic for a large class of target densities on $\mathbb{R}$. This class even contains realistic target densities for which random walk or Langevin MH are not geometrically ergodic. Three simulation studies are given to demonstrate the mixing property and superiority of RDMH to standard MH algorithms on real line. A share-price return data is also analyzed and the results are compared with those available in the literature.