2. Let X1, …, Xn be i.i.d. from Exp(θ). a. Find an unbiased...

2. Let $X_1, \dots, X_n$ be i.i.d. from $Exp(\theta)$. a. Find an unbiased estimator of $\theta$ based only on $X_{(1)}$ d. Use Rao-Blackwell to improve the estimator in (a) (b) Find a better estimator of $\theta$ which has smaller variance than the one in (a)

Transcript

00:01 Okay, so the pdf of the minimum order statistic is given by n times 1 minus f of x to n minus 1 times f of x, where the large f is the cdf of the exponential random variable in this case, and small f is the pdf of the exponential random variable in this case.

00:37 But this formula applies to any kind of iid random variables.

00:44 So let's calculate this one.

00:46 1 minus 1 minus e to negative x over theta to n minus 1 times theta times e to negative x over theta, or excuse me, 1 over theta here, so that we have n over theta times e to negative n minus 1 over theta times x times e to negative x over theta, which is equal to n over theta e to negative n over theta times x.

01:41 So this follows the exponential random variable with parameter parameter n over theta.

01:47 Let's calculate the expected value.

01:52 So since this x1 follows the exponential random variable with parameter n over theta, the mean is simply the inverse of the parameter, so that the estimator is n times x1, or the minimum order statistic, statistic, because n times theta over n equals theta, and it is unbiased.

02:34 So that's for a.

02:40 Okay, that's for a.

02:44 And let's calculate the variance here.

02:47 Variance of theta 1 equals variance of n times the minimum order statistic, which is n squared times variance of minimum order statistic.

03:04 So variance of the exponential random variable is the inverse of the parameter squared, n squared, times theta over n squared, which gives theta squared...

Question

2. Let $X_1, \dots, X_n$ be i.i.d. from $Exp(\theta)$. a. Find an unbiased estimator of $\theta$ based only on $X_{(1)}$ d. Use Rao-Blackwell to improve the estimator in (a) (b) Find a better estimator of $\theta$ which has smaller variance than the one in (a)

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