Given $n$ jointly-normal random variables $X_1, X_2, \dots, X_n$, with $$\mu_i=\mu\forall i \in\mathbb{N}^+$$ $$\sigma_i=\sigma\forall i$$ $$\rho_{i,j}=\rho\forall i,j \mbox{ with } i\neq j$$ what is their log-likelihood function?

My idea is the following: Be $\mathbf{x}=(X_1,X_2, \dots,X_n)^T$ the vector of observed values and $\mu=(\mu, \dots,\mu)^T$ a $n\times1$-Vektor. Then the $n\times n$ covariance matrix is given by $\Sigma$ with all diagonal elements being $\sigma^2$ and all other entries $\rho\sigma^2$.

The log-likelihood is given by: $$ L(\mu, \sigma, \rho) =\log [ exp(\frac{-1}{2}(\mathbf{x}-\mu)^T)\Sigma^{-1}(\mathbf{x}-\mu))]$$

Is this correct?

  • $\begingroup$ What do you mean by $\rho_i$? If you mean by that a correlation parameter, then it should have a double index (the correlation is really a measure of dependence between two variables). $\endgroup$ – Learner Mar 16 '13 at 15:46
  • $\begingroup$ @nullUser Please see my edit. $\endgroup$ – Stephen Mar 16 '13 at 15:54
  • $\begingroup$ @Learner My mistake, I edited the question. It should have a double index. $\endgroup$ – Stephen Mar 16 '13 at 15:57

Assuming $\boldsymbol{x}= \left(\begin{array}{c} x_1\\ \vdots\\ x_n \end{array}\right)$ (here, this means one observation of the $n$-dimensional multivariate normally distributed vector), then the log-likelihood is proportional to $$ - \frac{1}{2} \log ( \left| \boldsymbol{\Sigma} \right|) - \frac{1}{2} \left( \boldsymbol{x}-\boldsymbol{\mu} \right)^T \boldsymbol{\Sigma}^{- 1} \left( \boldsymbol{x}-\boldsymbol{\mu} \right) $$ $\left| \boldsymbol{\Sigma} \right|$ is the determinant of $\boldsymbol{\Sigma}$ and including it is essential.

Please note that in the typical setup, you would have say $N$ vectors $\boldsymbol{x}_1, \ldots, \boldsymbol{x}_N$.


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