Missing Mass of Rank-2 Markov Chains

Estimation of missing mass with the popular Good-Turing (GT) estimator is well-understood in the case where samples are independent and identically distributed (iid). In this article, we consider the same problem when the samples come from a stationary Markov chain with a rank-2 transition matrix, which is one of the simplest extensions of the iid case. We develop an upper bound on the absolute bias of the GT estimator in terms of the spectral gap of the chain and a tail bound on the occupancy of states. Borrowing tail bounds from known concentration results for Markov chains, we evaluate the bound using other parameters of the chain. The analysis, supported by simulations, suggests that, for rank-2 irreducible chains, the GT estimator has bias and mean-squared error falling with number of samples at a rate that depends loosely on the connectivity of the states in the chain.