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.
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.
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.