Question 7 we are given independent identically distributed...

Name: question 7 we are given independent identically distributed iid observations x1 x2 dots xn which are uniformly distributed on the interval 0 theta a find the maximum likelihood estimate o 24754
Uploaded: 2024-01-17T07:16:02-08:00
Duration: 5 min 56 s
Channel: Mengchun Cai
Description: question 7 we are given independent identically distributed iid observations x1 x2 dots xn which are uniformly distributed on the interval 0 theta a find the maximum likelihood estimate o 24754

Question 7 We are given independent, identically distributed (iid) observations $X_1, X_2, \dots, X_n$ which are uniformly distributed on the interval $[0, \theta]$. (a) Find the maximum likelihood estimate of $\theta$. (b) Is the estimate you found biased? If it is biased, can you propose an unbiased estimator? (c) Is the estimate you found in part (a) consistent?

Transcript

00:01 Okay, given distribution, i mean given distribution x, which is a uniform distribution from 0 to theta, and theta is our parameter.

00:13 And to estimate this parameter theta, suppose we have n iid samples from x.

00:27 Okay, that means for each i, xi is a uniform distribution on 0 theta.

00:33 From this condition or equivalently for each xi, we know the density function for xi is equal to 1 over theta, when x is greater or equal to 0, that's equal to theta, is equal to 0 or else.

00:55 So by this property, we can write down the so -called joint likelihood function p.

01:02 P, we just use the vector x to represent x1, x2 to xn.

01:08 Okay, theta by the definition, which is just, this guy is just the multiplication of those density functions because they are independent.

01:19 So we have one over theta to the power n.

01:23 Okay, in principle, i can just write it in this way, but don't forget the most important term, the restriction, the indicative function, because this function is non -zero on this interval.

01:36 So this is the function for any i, x i is greater or equal to zero and less or equal to theta.

01:45 Okay, this is our joint likelihood function.

01:51 So to find the mle for theta, we are required to find some theta such that we find some theta hat such that p x bar, the vector x, theta hat, attains its maximum.

02:20 Okay, this is our requirement for theta hat.

02:24 Now let's consider our density function.

02:27 Function, to carefully consider this restriction here, we can write it as the first term is unchanged...

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