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Comparison of equations (1.2), (1.4), and (1.5) indicates the qualitative difference between the exogenous and choice-based sampling likelihoods and the nature of both of these relative to the general stratified expression. In exogenous sampling, when the likelihood is considered a function of the unknown parameters 0*, the kernel is the choice probability function P(i | z, 0), 0 e regardless of the manner in which Z is stratified or the probability measure H imposed. In choice-based samples, on the other hand, the kernel is P (i | z, Q)/Q (b 10), since the marginal distribution Q is dependent on 0*,20 In general stratified sampling the kernel is the expression P(i | z, 0)/5(b 10), where S(b 10) = XAbP(j \ y, 0)/>(y).

We note for later use the special cases in which exogenous and choice-based processes yield random samples from C x Z. The exogenous

19. This likelihood form is the same as would be obtained in a stimulus response experimental setting in which the analyst presents subjects with choice sets and observes their responses. In this context the distribution g(z) characterizes the experimental design.

20. If the relation between Q and 6* is ignored, the choice-based sampling kernel reduces to the exogenous sampling one. It might be thought that ignoring this relation would lower the efficiency of estimators for 0* but not affect their consistency. In fact recognition of the relation turns out to be generally necessary for consistency, and the choice-based sampling kernel cannot be reduced to the exogenous sampling one. See in particular section 1.5.

sampling likelihood takes the form (1.3) if B = Z and H{z) = p (z). In choice-based samples we require H(b) = Q(b\Q*) for each beB. It is important to recognize that, while the true likelihood of exogenous and choice-based sampling observations are identical when the above conditions are met, the respective likelihood function kernels remain distinct.

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