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I just would like some reassurance on my interpretation of a few concepts in this paper Xiao, R. & Scott, L. (2011).

The question is about the likelihoods. Am I reading correctly the following expressions of $L$ when evaluated at $$\hat{\theta_0}=(\hat{\mu_0},\hat{\sigma}^2)$$ and $$\hat{\theta_1}=(\tilde{\pi},\tilde{\mu_0},\tilde{\alpha_R},\tilde{\sigma}^2).$$

In other words, are $L(\hat{\theta_0})$ and $L(\hat{\theta_1})$ given by the following expressions?

Likelihood under the null hypothesis: $$L(\hat{\theta_0}) = L(\hat{\theta_0};y_{i\in\{G_i=RR,rr\}}) = \Pi_{i\in\{G_i=RR,rr\}} f(y_i; \hat{\mu_0}, \hat{\sigma}^2)$$

Likelihood under the alternative hypothesis:

$$ L(\hat{\theta_1}) =L(\hat{\theta_1};y_{i\in\{G_i=RR,rr, Rr\}}) = \Pi_{i\in\{G_i=RR,rr\}} f(y_i; \tilde{\mu_0}, \tilde{\sigma}^2) \times \\ \Pi_{i\in\{G_i=Rr\}} \{ \tilde{\pi}\ f(y_i;\tilde{\mu_0}+\tilde{\alpha_R}, \tilde{\sigma}^2) + (1-\tilde{\pi})f(y_i;\tilde{\mu_0}-\tilde{\alpha_R}, \tilde{\sigma}^2)\} $$

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    $\begingroup$ Both likelihoods must use the same data, I would thus think $L(\hat{\theta}_0)$ is evaluated as a Normal likelihood for all $y_i$'s. $\endgroup$
    – Xi'an
    Mar 28, 2018 at 5:36

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