I am trying to compute the likelihood function for the test: $H_0:\mu\mu^t=1$ where my sample is of one element ($n=1$) of a $N_p(\mu,\sigma^2I)$ being $\sigma^2$ known.
So I should minimize $L(x,\mu,\Sigma)$ having in mind that $\mu\mu^t=1$.
Following the proof of the contrast of linear restrictions I have done as following:
Consider the function $a^+=a-n\lambda'(\mu\mu^t-1)$ where $\lambda$ is a Lagrange multiplicator, now differentiating with respect to $\mu$ I got:
$\dfrac{\partial{a^+}}{\partial{\mu}}=-2n\lambda'\mu+n(\bar{x}-\mu)\Sigma^{-1}=0$ this implies that: $\bar{x}-\mu=2n\lambda\mu$ $(1)$.
Using that $\mu\mu^t=1$ we get:
$(\bar{x}-\mu)\mu^t=\bar{x}\mu^t-1=2\lambda\mu\Sigma\mu^t \Longrightarrow \lambda=(\bar{x}\mu^t-1)(\mu\Sigma\mu^t)'$
Now, substituting $\lambda$ in $(1)$:
$\hat{\mu}=\bar{x}-2(\bar{x}\mu^t-1)(\mu^t\Sigma\mu)^{-1}\mu\Sigma$
Are these steps correct? I have had no lessons about Lagrange multiplicators so I am afraid I have done something wrong as the final result is not as compact as I thought it would be.
Any help would be appreciated