# Derive the Likelihood Ratio Test for multivariate normal and specific covariance matrix

Let $$X_1,\ldots, X_n$$ be i.i.d. $$N(µ, C)$$ random $$p$$-vectors. Derive the Likelihood Ratio Test for $$H_0: C = σ^2(1 − ρ)I_p + ρ1_p1_p^T$$, where $$1_p = \begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{pmatrix}_{p\times1}$$,
against the alternative hypothesis $$H_1$$ that the covariance matrix $$C$$ is not of this form. Here $$\mu$$ is unknown and $$\sigma$$, $$\rho$$ are free parameter.

I have tried to calculate the LRT directly: Let $$C_0 = σ^2(1 − ρ)I_p + ρ1_p1_p^T$$, then the LRT has the form $$\frac{sup_{H_0}L(C,\mu)}{supL(C,\mu)} = \frac{L(C_0,\hat{\mu}_{ML})}{L(\hat{C}_{ML},\hat{\mu}_{ML})} = |\frac{1}{n}A|^{n/2} |C_0|^{-n/2} exp^{-\frac{1}{2} (trace(C_0A^{-1}) -p^2)}$$ since $$\hat{C}_{ML} = \frac{1}{n} \sum_{i=1}^{n} (X_i - \hat{X})(X_i - \hat{X})^T =\frac{1}{n} A$$. Here i don't know how to continue the computations

I also wonder, if I know a LRT for spherical variance ($$H_0: C= \lambda I_p$$, for some $$\lambda \in R$$), can I use it somehow?