$X_1,\ldots,X_n$ are observations of a population with density
$f(x)=\frac{1}{2}\left\{\begin{matrix} \lambda e^{\lambda x}
\;\,\text{ if } x<0\\ \lambda e^{-\lambda x} \text{ if } x \geq 0
\end{matrix}\right.$where parameter $\lambda$ is unknown. Determine a maximum-likelihood estimation for $\lambda$.
In my last question I try to solve the problem with another method (method of moment estimation: Given is a density.. Determine a method of moment for $\lambda$) I don't know if I do it correct but this time I like to know how can you do it correct with maximum-likelihood? Because I write test soon and I want use a reliable method but not sure how you use maximum-likelihood for this example?
I think the method allow us to see the density as function of $\lambda$ which we are looking for, that why we can write as likelihood function
$$L(\lambda) = \prod_{i=1}^{n}f_{X_i}(x_i;\lambda)$$
Now need to maximize this based on $\lambda$ so we get maximum-likelihood estimation for $\lambda$ if I understand wikipedia article correct till here.
But I don't understand how we do this here and use the formula? We need derive for $\lambda$ and set it equal to zero but no idea how you can apply this here… : /
Best Answer
\begin{align} L(\lambda) & \propto \lambda^n \prod_{i=1}^n e^{-\lambda |x_i|} \\[10pt] \ell(\lambda) = \log L(\lambda) & = n\log\lambda - \lambda \sum_{i=1}^n |x_i| + \text{constant} \\[10pt] \ell\,'(\lambda) & = \frac n \lambda - \sum_{i=1}^n |x_i| \quad \begin{cases} >0 & \text{if } \lambda < \text{something}, \\ <0 & \text{if } \lambda > \text{something.} \end{cases} \end{align}