Finding the MLE of a parameter

Question

Let $X$ be a R.V. of some distribution, with PDF $f(x;a)=\frac{a\cdot 3^a}{x^{a+1}}$.

After 3 experiments, the results recieved were: $(e^6, e,e^2)$

We're asked to find the MLE for $a$

I'm not sure on how to approach this, all I know is that I'm looking for $L(a)=\prod$ of something, yet I'm not sure whether to replace $x$ with the results, and get $L(a)=\frac{a\cdot 3^a}{e^{6\cdot (a+1)}}\cdot \frac{a\cdot 3^a}{e^{a+1}}\cdot \frac{a\cdot 3^a}{e^{2\cdot (a+1)}}$ or to place the results instead of $a$ in the $\prod $ formula.

Answer 1

What you typically do is to study the log-likelihood instead and then solve for $\frac{d}{da}\log(L(a))=0$. In this case, you have (for $m$ samples)

\begin{equation} L(a) = \prod_{j=1}^m f(x;a) \Rightarrow \log(L(a)) = \sum_{j=1}^m \log(f(x;a)) \end{equation}

Then,

\begin{equation} \frac{d}{da}\log(L(a)) = \frac{d}{da}\sum_{j=1}^m(\log(a)+a\log(3)-(a+1)\log(x_j)) = \frac{m}{a} + m\log(3) - \sum_{j=1}^m \log(x_j) \end{equation}

from where you then obtain an estimate of $a$. Plug in the values for $x_j$, i.e. $e^6,e,e^2$, and you are finished.

Answer 2

Some crucial information is missing here:

• Are we assuming that these three observations are independent? Almost definitely yes, but it is good to be explicit about this.
• What is the support of $X$? Note that $f$ is undefined for some $x \le 0$, unless $a = -1$.
• What values for $a$ are allowed (i.e. what is the parameter space)? This, together with the support of $X$, determines whether $f$ is a valid PDF or not.

Regarding your actual question, the $\Pi$ symbol is simply used to denote multiplication of the elements of some sequence. In you case, you might index $(e^6, e, e^2)$ as $(x_1, x_2, x_3)$ and write $L(a) = \Pi_{i=1}^3 f(x_i,a)$. This is the exact same expression as you have. Then you'll want to the find the maximum of $L$ with respect to $a$. Differentiating is often but not always the right approach here; note that the maximum might lie on the boundary of the parameter space.