Asymptotic efficiency of M.L.E. using prior survey in multinomial distributions

TL;DR

This paper analyzes how incorporating prior survey information affects the asymptotic efficiency of maximum likelihood estimation in multinomial distributions, revealing conditions under which it reduces or increases risk.

Contribution

It provides a detailed asymptotic risk expansion showing when prior survey data improves or worsens estimation risk in multinomial models.

Findings

Prior information reduces risk with large sample sizes.

Using prior data can increase risk when current sample size is small.

Guidelines for when to incorporate prior survey results.

Abstract

Incorporating information from a prior survey is generally supposed to decrease the estimation risk of the present survey. This paper aims to show how the risk changes by incorporating the information of a prior survey through watching the first and the second-order terms of the asymptotic expansion of the risk. We recognize that the prior information is of some help for risk reduction when we can acquire samples of a sufficient size for both surveys. Interestingly, when the sample size of the present survey is small, the use of the prior survey can increase the risk. In other words, blending information from both surveys can have a negative effect on the risk. Based on these observations, we give some suggestions on whether or not to use the results of the prior survey and the sample size to use in the surveys for a reliable estimation.