Joint Probability Mass Function Estimation From Asynchronous Samples

A common approach to study the relationship between different signals is to model them as random processes and estimate their joint probability distribution from the observed data. When synchronous samples of the random processes are available, then the empirical distribution gives a reliable estimate. However, in several situations, such as sensors spread over a vast region or a software probing smart phone sensors for readings, synchronous samples are either not available, or are difficult/costly to obtain. In such cases, we have to depend on non-periodic, asynchronous samples to obtain good estimates of the joint distribution. In this paper, we consider independent Poisson sampling of the individual random processes and we propose a kernel based estimate of the joint probability mass function. We prove that our estimate is consistent (in the mean-square sense) for strong mixing processes, which is a wide class of random processes including Markov processes. We also provide expressions for the asymptotic mean-square error (MSE), study the bias-variance tradeoff, and discuss the choice of the kernel bandwidth. By appropriately choosing the kernel, we show that we can obtain an asymptotic rate of T^-4/5 for the MSE, where T is the interval of observation. We also present several numerical results to discuss the accuracy of our asymptotic approximations for finite T.