Sample Size Requirements for Zero-Order Models

This paper investigates the sensitivity of maximum likelihood estimates with a view to finding out how many individuals are needed and how many pur-chases are required for each individual to accurately estimate parameters for zero order models. Our results reveal that the estimation of the typically formulated original parameters requires about 2000 individuals with 5 purchases per consumer. In many zero order applications, however, knowledge of market share and loyalty index, which are both functions of the original parameters, should be adequate. Reduced sample sizes of about 400 with 5 purchase records per household are shown to be sufficient to estimate the transformed parameters, the market share and the loyalty index. Our numerical results use the beta distribution as the mixing distribution for the individual p values; however, the spirit of our results holds for arbitrary mixtures. Namely, much smaller sample sizes are required if we only need to know the location (market share) and shape (loyalty index) of the mixing distribution than if we need detailed knowledge of the original parameters. n^5(o3^ 1-