Maximum Likelihood Estimator for Log Normal Distribution

 Let  $x_1, x_2, x_3,...........,x_n$ denote the observations taken from a $Uniform \\ Distribution$ with pdf:

`f(x, \theta) =   1 ; \theta - \frac{1}{2} \leq x \leq \theta + \frac{1}{2}, -\infty < \theta < \infty`

we need to compute the maximum likelihood estimator of $\theta$.

Considering the $\log$ function for the pdf,

`L(\theta ; x) = \prod_{i=1}^{n} f(x_{i}, \theta) = \prod_{i=1}^{n} I_{(0,\theta)}(x_{i})`

`L =L(\theta ; x_1, x_2,.....,x_n)=\begin{cases}  1 & \text{,}\theta - \frac{1}{2}\leq x_i \leq \theta + \frac{1}{2} \\0 & \text{, } otherwise \end{cases}`

id $x_{(1)},x_{(2)},x_{(3)},................x_{(n)}$ is the order sample, then 

`\theta - \frac{1}{2} \leq x_{(1)},x_{(2)},x_{(3)},................x_{(n)} \leq \theta + \frac{1}{2}`

Thus, $L$ will attain the maximum if 

`\theta - \frac{1}{2} \leq x_{(1)}  ,and,  x_{(n)} \leq \theta + \frac{1}{2}`

`\theta \leq x_{(n)} + \frac{1}{2} ,and,  x_{(n)} - \frac{1}{2} \leq \theta`

Hence every statistic $t = t \left( x_1, x_2, x_3,...........,x_n \right)$ such that

`x_{(n)} - \frac{1}{2} \leq  t \left( x_1, x_2, x_3,...........,x_n \right) \leq x_{(1)} + \frac{1}{2}`, provides an MLE for $\theta$  

Unifrom Density Plot

Uniform Distribution with $X \sim U[0, \theta]$

Let $X_{1},X_{2},X_{3},..............,X_{n}$ be a random sample drawn from $U(0 , \theta) , \theta > 0$ then the mle of  $\theta$ will be:

The likelihood function for $\theta$ is 

`L(\theta) = \prod_{i=1}^{n} f_{X}(x_{i}) \Rightarrow \prod_{i=1}^{n} \frac{1}{\theta^{n}}I_{(0, \theta)}(x_{n})`

Here the likelihood function is maximum when the value of $\theta$ is maximum where $x_{(n)}$ order statistic is maximum. So $L(0, \theta)$ is maximum for the value of  $x_{(n)}$. There for the MLE of $\theta$ is 

`\hat{\theta}_{MLE} = X_{(n)}`

Plot for $X \sim U[0, \theta]$

Enter θ (θ > 0):