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Define $\Theta = \left\{\left(\mu_x,\mu_y,\sigma_x^2,\sigma_y^2,\sigma_{xy}\right)\in\mathbb{R}^5\right\}$ to be the set of possible parametrizations of (possibly) correlated bivariate normal distributions. Let $\Theta_0\subset\Theta$ be some subset of those parameters satisfying the property, $$ \Theta_0 = \left\{\left(\mu_x,\mu_y,\sigma_x^2,\sigma_y^2\right)\in\mathbb{R}^4 : \mu_x \leq 0, \sigma_{xy}=0\right\}, $$ that is to say, the set of uncorrelated bivariate normals with horizontal mean less than or equal to zero.

I am developing a hypothesis test of the form, $$ H_0 : \theta \in \Theta_0 ~~~~~~~~~~~~~ H_1: \theta\not\in\Theta_0. $$ Let $\mathbf{X}^n$ be a set of $n$ i.i.d. data points drawn from a bivariate normal. Then usually I would use a likelihood ratio test statistic for the purposes of evaluating the test. In particular, $$ \Lambda\left(\mathbf{X}^n\right) = \frac{\sup_{\theta \in\Theta_0} \mathcal{L}\left(\mathbf{X}^n;\theta\right)}{\sup_{\theta \in\Theta} \mathcal{L}\left(\mathbf{X}^n;\theta\right)} $$

At this point a common question concerns the distribution of $\log \Lambda\left(\mathbf{X}^n\right)$.

Let us suppose that the null hypothesis is true. Suppose furthermore that $$ \theta^\star = \arg\sup_{\theta\in\Theta}\mathcal{L}\left(\mathbf{X}^n;\theta\right) $$ is an element of $\Theta_0$. That is to say, that the maximum likelihood parameter configuration is an element of the null hypothesis set. When this is true, it is fairly easy for me to verify that $-2\log\Lambda \sim\chi^2_1$. This is apparent because if the maximum likelihood estimate of $\mu_x$ is less than or equal to zero, then the full parameter space has an additional degree of freedom in $\sigma_{xy}$. But when $\theta^\star \not\in\Theta_0$, the distribution will be most messy and undesirable.

What I would like to show is that the asymptotic distribution of $\log\Lambda\left(\mathbf{X}^n\right)$ is still $\chi^2_1$ under the null. My "proof" of this would be quite straightforward since, due to the consistency of the maximum likelihood estimate, $$ \mathbb{P}\left[\lim_{n\to\infty} \theta^\star = \theta\right] = 1, $$ and under the null we have that $\theta\in\Theta_0$ and hence almost surely $\theta^\star\in\Theta_0$. This brings us back to the case where the denominator in the likelihood ratio statistic has a single extra degree of freedom and therefore the statistic goes to a $\chi^2_1$.

I would like to know if this reasoning regarding the asymptotic distribution of $\log\Lambda\left(\mathbf{X}^n\right)$ is correct.

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