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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.

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.

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

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, .

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 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.

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.

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

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, .

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 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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