Finding Consistent estimator

Question

Let $X_1,X_2,......X_n$ be a random sample from a population density function

$ f_\theta(x) = \left\{ \begin{array}{ll} \frac{4}{\theta}x^3e^{\frac{-x^4}{\theta}}, \quad x>0, \\ \\ 0, \quad \text{otherwise} \end{array} \right.$

where $\theta >0$ is unknown. Then , a consistent estimator for $\theta$ is

I know to find consistency of a given estimator. Since there is no estimator given in the question should we find a consistency estimator? if so how do we find it ?

Answer 1

As per MLE properties, $\hat{\theta}_{ML}$ is consistent

So, simply let's define

$$L(\theta)\propto \frac{1}{\theta^n}e^{-\frac{\sum x^4}{\theta}}$$

$$log L=-nlog\theta-\frac{\sum x^4}{\theta}$$

$$\frac{\partial}{\partial{\theta}}log L=-\frac{n}{\theta}+\frac{\sum x^4}{\theta^2}$$

$$\hat{\theta}_{ML}=\frac{\sum x^4}{n}$$

which is the same solution as the above topic

Answer 2

If you would try to compute the fourth moment of the random variable $X$, you will find,

\begin{align} \mathbb{E}[X^4] &= \int_{0}^{\infty}x^4f(x)dx \\ &=\int_{0}^{\infty}\tfrac{4}{\theta}x^7e^{\tfrac{x^4}{\theta}}dx. \\ &= \theta\int_{0}^{\infty}y \cdot e^{-y}dx \hspace{3mm} (\text{taking transformation }y =\tfrac{x^4}{\theta} )\\ & = \theta \cdot \mathbb{E}[Y] \\ &= \theta \hspace{3mm}(\text{since } Y \sim \text{Exp}(1)) \end{align}

So a natural estimator would be the sample analogue of the fourth moment, i.e. $\hat{\theta}_n = \frac{1}{n}\sum_{i=1}^{n}X_i^4$. By the weak law of large number we then have: \begin{align} \hat{\theta}_n = \frac{1}{n}\sum_{i=1}^{n}X_i^4 \rightarrow \mathbb{E}[X^4] = \theta \end{align}