B6. Suppose X?, X?, . . . , X? are independent and...

B6. Suppose X?, X?, . . . , X? are independent and identically distributed random variables with the common probability mass function (pmf): p(x) = ?(1 ? ?)??¹ for x = 1, 2, . . . and 0 < ? < 1. (i) Write down the likelihood function of ?. (ii) Find the maximum likelihood estimator of ?. (iii) Find the maximum likelihood estimator of ? = 1/?. (iv) Determine the bias, variance and the mean squared error of the maximum likelihood estimator of ?. (v) Is the maximum likelihood estimator of ? unbiased? Is it consistent?

Transcript

00:01 Hello students, according to the given question, we have to find the given bits.

00:06 So they have given the pmf, p of x will be equal to theta into 1 minus theta whole power x minus 1.

00:17 Here x is equal to 1, 2, so on and 0 less than theta less than 1.

00:23 In the first bit, we have to find the likelihood function of theta.

00:27 So l of theta will be equal to pi, i is equal to 1 to n, p of xi.

00:35 So which is equal to pi, i is equal to 1 to n, so by substituting theta into 1 minus theta whole power xi minus 1.

00:44 Which is equal to theta power n into 1 minus theta power summation x i is equal to 1 to n minus n minus n.

00:53 So in the second bit we have to find the maximum likelihood estimator of theta.

01:00 So l will be equal to log of l of theta.

01:06 So from the above by applying log we get n log theta plus summation xi is equal to 1 to n minus n into log of 1 minus theta.

01:21 So because mla requires minimization of l with respect to theta.

01:30 Now, do l by do theta will be equal to 0.

01:36 That implies n by theta minus summation xi minus n by 1 minus theta is equal to 0.

01:45 So n by theta will be equal to summation xi minus n by 1 minus theta.

01:51 So 1 minus theta by theta will be equal to 1 by n summation xi minus 1.

02:01 Now 1 by theta will be equal to x bar minus 1 plus 1.

02:08 So 1 by theta will be equal to x bar.

02:11 Therefore, mla of theta is theta cap will be equal to 1x bar.

02:26 In the third bit, we have to find the maximum likelihood estimator of zai will be equal to 1 by theta.

02:35 So from invariance property of mle, that is maximum likelihood estimator, so we know that theta be the mle of theta and phi of theta be any continuous function.

02:54 So then mle, that is maximum likelihood estimator of zhi will be the mlea, which is maximum likelihood estimator be equal to 1 by theta.

03:02 So that implies zy cap will be equal to 1 by theta cap so which is equal to x bar.

03:10 So in the fourth bit from the given pmf expectation of x will be equal to 1 by theta from the exponential...

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