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Neyman and Scott (1948) gave the following example which shows that, when the number of parameters increases with the sample size, the MLE may not be consistent. Suppose that two observations are collected from m individuals. Each individual has its own (unknown) mean, say, is for the i th individual. Suppose that the owervations are independent and normally distributed with variance (7?. The problem of interest is to estimate :72. The model may be expressed as the following: ys'j = m + 5:155: where 615's are independent and distributed as N (0, a2) . Note that this may be viewed as a special case of the linear mixed model in which Z = 0. 1. Show that the MLE is inconsistent as the number of individuals increases. 2. Show that the MLE based on 2,; = y\" ya, 1' = 1, . . . ,m, that is the REML estimator of 0'2, (why?), is consistent as m increases. a Neyman, J. and Scott, E. {1948), Consistent estimates based on partially consistent obser- vations, Econometke 16, 1-32