Find a Maximum Likelihood Estimator for $\theta$

Question

I am trying to solve the following problem:

$X$ is a random variable with logistic logarithmic distribution:

$$ f(x,\theta) = 3x^2\theta(1 + \theta x^3)^{-2} ; x\in \mathbb{R} $$

$(x_1, x_2, ..., x_n)$ Simple Random Sample

Find a maximum likelihood estimator for $\theta$.

So based on what I understood, I started resolving the problem:

$$ L = f(x_1, x_2, ..., x_n) = \prod_{j=1}^{n}f(x_j, \theta) \\ = \prod_{j=1}^{n}3{x_j}^2\theta(1 + \theta {x_j}^3)^{-2} \\ = \prod_{j=1}^{n}\frac{(2)3{x_j}^2\theta}{(1 + \theta {x_j}^3)} = \prod_{j=1}^{n}\frac{6{x_j}^2\theta}{(1 + \theta {x_j}^3)} \\ = (6\theta)^n\prod_{j=1}^{n}\frac{{x_j}^2}{(1 + \theta {x_j}^3)} $$

Do let me know if I did this correctly. From here I am not sure how to proceed. There is one condition if I remember correctly which says that the natural logarithm can be applied if the constant $e$ (Euler) is used in the equation but this does not seem to apply here.

How do I proceed in solving this problem?

Answer 1

As StubbornAtom has allready commented $L(\theta)=(3\theta)^n\prod_{j=1}^n \left(\frac{x_j}{1+\theta x_j^3}\right)^2$. Now we are taking logs. Here we have the rule

$\ln\left(\prod f \right)=\sum \ln(f)$. That means in your case

$$\ln\left(L(\theta)\right)=\ln\left((3\theta)^n\right)+\ln\left(\prod_{j=1}^n \left(\frac{x_j}{1+\theta x_j^3}\right)^2 \right)$$

$$=n\cdot\ln\left(3\theta\right)+\sum_{j=1}^n \ln\left(\left(\frac{x_j}{1+\theta x_j^3}\right)^2 \right)$$

$$=n\cdot\ln\left(3\right)+n\cdot\ln\left(\theta\right)+2\cdot \sum_{j=1}^n \ln\left(\frac{x_j}{1+\theta x_j^3} \right)$$

$$=n\cdot\ln\left(3\right)+n\cdot\ln\left(\theta\right)+2\cdot \sum_{j=1}^n \ln\left({x_j}\right)- 2\cdot \sum_{j=1}^n\ln\left(1+\theta x_j^3 \right)$$

At the next step we calculate the derivative w.r.t $\theta$ and set it equal to zero.

$$\frac{\partial \ln\left(L(\theta)\right)}{\partial \theta}=\frac{n}{\theta}-2\cdot \sum_{j=1}^n \frac{x_j^3}{1+\theta x_j^3}=0$$

Now we have the problem that this equation cannot solved in general for $\theta$. At the end we have to look if it is possible to estimate $\theta$ for specific values of the $x_j$'s