PROBLEM #3: Maximum likelihood estimation has been widely...

PROBLEM #3: Maximum likelihood estimation has been widely applied to the construction of a point estimator of an unknown parameter. Answer the following questions on maximum likelihood estimation. a. Let X be a discrete r.v. with the p.m.f. being given below x | 0 | 1 | 2 | 3 pX(x) | ?/3 | 2?/3 | 2(1 - ?)/3 | (1 - ?)/3 Suppose that we have 10 observations 2, 0, 0, 0, 2, 3, 1, 1, 2, 3 that are drawn from such a distribution. Find out the maximum likelihood estimate of the parameter ?. b. Suppose that X1, ?, Xn form a random sample and are drawn from a distribution with the pdf f(x; x0, ?) = ?x0^? x^-?-1, x ? x0, ? > 1. Let x0 > 0 be given. Find the maximum likelihood estimate of the parameter ?. c. Let X1, ?, Xn be a random sample with the pmf P?(X = x) = ?^x(1 - ?)^1-x, x = 0 or 1, 0 ? ? ? 0.5. Find the MLE of ? and calculate its mean squared error of the estimator.

Transcript

00:01 Hello everyone, we are going to solve a question.

00:03 In this question we are given that x is the discrete random variable.

00:09 We have x px.

00:11 When x0, px is theta divided by 3, x1, px is 2 theta divided by 3, x2, this is 2 multiplied by 1 minus theta divided by 3.

00:23 X3 then this is equal to 1 minus theta divided by 3.

00:28 So now we are given the points, 10 obligation, x1 is equals to 2, x2 is equals to 0, x3 is equals to 0, x4 is equals to 0, x5 is equals to 2, x6 is equals to 3, x7 is equals to 1, x8 is equals to 1 x9 is equal to 2 and x10 is equals to 3 so now in this first part we have to calculate to find maximum likelihood estimate maximum likelihood estimate of theta so now for solving this this is equals to is equal to p when x is equal to 2 multiplied with p when x is equal to 0 multiplied with p x is equal to 0 again multiplied with x equal p x is equal to 0 then p x is equal to 2 then p x is equal to 3 then p x is equal to 1 then again probability when x is equals to 1 then again probability when x is equal to 2 and the probability when x is equals to 3 now we put all the values here then this is equals to l theta is equals to 2 divided by 3 1 minus theta here the theta divided by 3 theta divided by 3 again theta divided by 3 again this is 2 divided by 3 1 minus theta this is 1 minus theta divided by 3 multiplied by 2 theta divided by 3 this multiplied word 2 divided by 3 1 minus theta and this multiplied with 1 minus theta divided by 3.

02:29 On solving this we get this is equals to l theta is equals to 32 divided by 3 raised to the power 10, theta multiplied with 5 and 1 minus theta raised to the power 5.

02:43 So now we can take log on both side to log likelihood is given by l theta is equals to log of l theta which is equal to log 32 plus 5 times log theta plus 5 times log 1 minus theta and minus 10 times or we can say log of 3 raised to the power 10.

03:11 So now this is equal to with differentiating this is l -dhast theta which is equals to 0 plus 5 divided by theta plus 5 divided by 1 minus theta multiplied with minus 1.

03:26 And here we have minus 0.

03:28 So we have, this is equals to l -dh -theta is 5 divided by theta minus 5 divided by 1 -teta.

03:36 Now, this is equals to, by putting l -dash -equals to 0, l -dh -theta is equals to 0.

03:44 We have 5 divided by theta minus 5 divided by 1 -teta equals to 0.

03:52 So this is equals to 1 divided by theta equals to 1 divided by 1 minus theta, 1 minus theta equals to theta.

04:00 So we have 2 theta equals to 1.

04:03 So from here we have theta is equals to 1 divided by 2.

04:07 So that means now we calculate l double dash theta, which is equals to minus 5 divided by theta square, minus 5 divided by 1 minus theta square.

04:21 And this is less than 0 as this is negative...

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