On the use of the directional derivative in obtaining multivariate extreme values (Q1060514)

The author extends the concept of directional derivative to finite dimensional vector spaces, and uses this tool to solve a number of current problems in multivariate analysis. The main idea in this approach is to treat a vector (matrix) of parameters as a single variable entity. For maximum likelihood estimation it consists of taking a derivative of the likelihood function with respect to the variable vector (matrix), equating this derivative to zero and solving the resulting equation. Let x be a vector (matrix) and f a function of X. Let \(\epsilon\) be a positive scalar. The derivative of f with respect to x in the direction of the vector (matrix) \(\theta\), denoted by \(D_{\theta,x}f\) \(is\) D\({}_{\theta,x}f=\lim_{\epsilon \to 0}(\epsilon \| \theta \|)^{-1}[f(x+\epsilon \theta)-f(x)],\) provided this limit exists. Six formulas of \(D_{\theta,x}f\) are derived. Applications are made to classical maximum likelihood estimation in the case of the multivariate normal densities and to other multivariate problems involving stationary points. For example, to estimate the mean vector \(\mu\) of a multivariate normal distribution, we maximize \(\ell n[L]\) with respect to \(\mu\). Using \(D_{\Phi,\mu }\ln (L)\), where \(\Phi\) is an arbitrary vector, we have \[ D_{\Phi,\mu }\ln (L)=\Phi^ T\Sigma^{-1}\sum^{N}_{\alpha =1}(x_{\alpha}-\mu)=0. \] Solving this equation, we get the result, \({\hat \mu}=N^{-1}\sum_{\alpha}x_{\alpha}=\bar x\).