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# Maximum Likelihood Estimation | ISI MStat 2017 PSB Problem 8 This problem based on Maximum Likelihood Estimation, gives a detailed solution to ISI M.Stat 2017 PSB Problem 8, with a tinge of simulation and code.

## Problem

Let be an unknown parameter, and be a random sample from the distribution with density. Find the maximum likelihood estimator of and its mean squared error.

### Prerequisites

• Proof algorithm to find the MLE of for U • Order Statistics • Mean Square Error

## Solution

Do you remember the method of finding the MLE of for U ? Just proceed along a similar line. Let's draw the diagram.

Thus, you can see that is maximized at .

Hence, .

#### MSE

Now, we need to find the distribution of .

For, that we need to find the distribution function of .

Observe   MSE( ) = E = = = = + - = Observe that .

### Let's add a computing dimension to it and verify it by simulation.

Let's take . MSE is expected to be around 0.002. You can change the and n and play around.

v = NULL
n = 15
theta = 1
for (i in 1:1000) {
r = runif(n, 0, theta)
s = theta*sqrt(r) #We use Inverse Transformation Method to generate the random variables from the distribution.
m = max(s)
v = c(v,m)
}
hist(v, freq = FALSE)
k = replicate(1000,1)
mse(v,k) =  0.001959095

You should also check out this link: Triangle Inequality Problems and Solutions

I hope that helps you. Stay tuned.

This problem based on Maximum Likelihood Estimation, gives a detailed solution to ISI M.Stat 2017 PSB Problem 8, with a tinge of simulation and code.

## Problem

Let be an unknown parameter, and be a random sample from the distribution with density. Find the maximum likelihood estimator of and its mean squared error.

### Prerequisites

• Proof algorithm to find the MLE of for U • Order Statistics • Mean Square Error

## Solution

Do you remember the method of finding the MLE of for U ? Just proceed along a similar line. Let's draw the diagram.

Thus, you can see that is maximized at .

Hence, .

#### MSE

Now, we need to find the distribution of .

For, that we need to find the distribution function of .

Observe   MSE( ) = E = = = = + - = Observe that .

### Let's add a computing dimension to it and verify it by simulation.

Let's take . MSE is expected to be around 0.002. You can change the and n and play around.

v = NULL
n = 15
theta = 1
for (i in 1:1000) {
r = runif(n, 0, theta)
s = theta*sqrt(r) #We use Inverse Transformation Method to generate the random variables from the distribution.
m = max(s)
v = c(v,m)
}
hist(v, freq = FALSE)
k = replicate(1000,1)
mse(v,k) =  0.001959095

You should also check out this link: Triangle Inequality Problems and Solutions

I hope that helps you. Stay tuned.

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