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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?

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