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We are learning matrix calculus in class. This is really new for me and I am trying to get extra practice by going through old assignments. One of the questions asks to find the derivative of the function below with respect to $\beta$ and if possible to find the value of the elements of $\beta$ that maximize the function. After taking logs to simplify the original function, I took the partial with respect to $\beta$. I thought my derivative was correct, but when I set the partial equal to zero, I am unable to solve for $\beta$. So either my derivative is wrong or I don't know enough to solve for $\beta$, or it's a trick question and there is no analytical expression for $\beta$. A more advanced student suggested that it can probably only be solved by numerical methods but I was wondering if anyone here can tell me if my derivative is correct and whether or not it's possible to solve for $\beta$?

$$ x_{1} = \begin{pmatrix} a\\ b\\ \end{pmatrix}\quad x_{2} = \begin{pmatrix} c\\ d\\ \end{pmatrix}\quad \beta = \begin{pmatrix} \beta_{1}\\ \beta_{2}\\ \end{pmatrix}\quad Y = \begin{pmatrix} y_{1}\\ y_{2}\\ \end{pmatrix}\quad $$

$$ f(x_{1},x_{2},y_{1},y_{2},\beta) = \frac{1}{x_{1}'\beta}\exp\left\{\frac{-y_{1}}{x_{1}'\beta}\right\}\frac{1}{x_{2}'\beta}\exp\left\{\frac{-y_{2}}{x_{2}'\beta}\right\} $$

My answer:

$$ \frac{\partial \ln f}{\partial\beta}=-\frac{1}{x_{1}'\beta}x_{1}-\frac{1}{x_{2}'\beta}x_{2}+\frac{x_{1}y_{1}}{x_{1}'\beta.x_{1}'\beta}+\frac{x_{2}y_{2}}{x_{2}'\beta.x_{2}'\beta}=0 $$

Is this correct and is it possible to solve for $\beta$? If so, how? Thanks!

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It seems that in the last equation you've already divided out common factors, so this isn't in fact $\partial f/\partial\beta$? – joriki Jan 17 '12 at 20:40
Well I'm not sure. That's why I'm asking for help. Taking logs of the original function eliminates the exp's - I'm under the impression this is OK since it's just a monotone transformation and so the Beta that maximizes the original function or its log will be the same. I just took the partial from there and got what I posted. I didn't try fiddling with it or try to cancel things out until I tried solving for Beta. So maybe my derivative is wrong? – Sean Jan 17 '12 at 20:59
I'm not sure I understand correctly. Are you saying that what you wrote as "$\partial f/\partial\beta$" is in fact your result for $\partial\log f/\partial\beta$? – joriki Jan 17 '12 at 21:08
Oh sorry, it was my fault I didn't indicate that I was taking the derivative of the transformed function as you pointed out. I made a correction in my original post. Thank you. – Sean Jan 17 '12 at 21:16
Two more small mistakes: You've got $y_1$ twice in $f$ where the second one should presumably be $y_2$, and the last two terms in the derivative have the wrong sign. – joriki Jan 17 '12 at 21:39

This function doesn't have a global maximum, at least not if $x_1$ and $x_2$ are linearly independent: You can choose $x_1'\beta$ such that the first exponential goes to $\infty$, and you can independently choose the sign of $x_2'\beta$ such that the product is positive and thus goes to $\infty$.

However, you can still look for the stationary points and see whether they are local maxima. Taking the logarithm before taking the derivative is a good approach, and the resulting equation can be solved with nothing but linear algebra.

If $x_1$ and $x_2$ are linearly independent, their coefficients in this equation have to vanish individually. That gives you one linear equation for $x_1'\beta$ and one for $x_2'\beta$. You can solve these, and then knowing the scalar products of $\beta$ with two linearly independent vectors, you can reconstruct $\beta$.

If $x_1$ and $x_2$ aren't linearly independent, you can substitute $x_2=\lambda x_1$ into the equation, and setting the coefficient of $x_1$ in the result to $0$ gives you a linear equation for $x_1'\beta$. In this case the stationarity condition only restricts $\beta$ in one direction, since the function is independent of the component of $\beta$ orthogonal to $x_1$ and $x_2$.

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