Louis (1982) presents a method for computing the observed information matrix and standard errors of maximum likelihood estimates obtained via the EM algorithm based on the complete-data log likelihood function. The problem illustrates the well-known method of Louis (1982) for a widely used qualitative response model in econometrics. The observed-data log likelihood function for the following model can, of course, be easily differentiated to obtain the observed information matrix; our objective is to illustrate the method and not to recommend its use for this model.

Consider the following ordered response model:

where y* is the underlying response variable, β is a (k × 1) vector of unknown parameters, x is a (k × 1) vector of known constants, and u is the residual. The term y* is not observed, but we know which of the m categories it belongs to. It belongs to the jth category if α_j−1 < y * < α_j (j = 1,2,…,m), α's being known constants and α₀ = −∞, α_m = ∞. Derive the observed information matrix for the model using the complete-data log likelihood function under the assumption that u_i are i.i.d. N(0,1).