Intractable maximum likelihood problems can sometimes be finessed with a Monte Carlo implementation of the E M algorithm. However, there appears to be little theory governing when Monte Carlo E M (MC E M) sequences converge. Consequently, in some applications, convergence is assumed rather than proved. Motivated by this problem in the context of modeling market penetration of new products and services over time, we develop (i) high-level conditions for rates of almost-sure convergence and convergence in distribution of any MC E M sequence and (ii) primitive conditions for almost-sure monotonicity and almostsure convergence of an MC E M sequence when Monte Carlo integration is carried out using independent Gibbs runs. We verify the main primitive conditions for the Bass product diffusion model and apply the methodology to data on wireless telecommunication services.
We propose numerical and graphical methods for outlier detection in hierarchical Bayes modeling and analyses of repeated measures regression data from multiple subjects; data from a single subject are generically called a "curve." The first-stage of our model has curve-specific regression coefficients with possibly autoregressive errors of a prespecified order. The first-stage regression vectors for different curves are linked in a second-stage modeling step, possibly involving additional regression variables. Detection of the stage at which the curve appears to be an outlier and the magnitude and specific component of the violation at that stage is accomplished by embedding the null model into a larger parametric model that can accommodate such unusual observations.As a first diagnostic, we examine the posterior probabilities of first-stage and second-stage anomalies relative to the modeling assumptions for each curve. For curves where there is evidence of a model violation at either stage, we propose additional numerical and graphical diagnostics. For first-stage violations, the diagnostics identify the specific measurements within a curve that are anomalous. For second-stage violations, the diagnostics identify the curve parameters that are unusual relative to the pattern of parameter values for the majority of the curves. We give two examples to illustrate the diagnostics, develop a BUGS program to compute them using MCMC techniques, and examine the sensitivity of the conclusions to the prior modeling assumptions.2
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