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Question 1

This question is about the sampling distribution of the mean, and using X¯ as an estimator for the parameter

µ. Suppose X1, . . . , Xn are independent and identically distributed (i.i.d.) with the same distribution as X.

a. Recall that E[X¯] = E[X] and Var(X¯) = Var(X)

n

. Do we need to assume anything more about

X for X¯ to be an unbiased estimate of µ, and if so, what?

b. One way to write Chebyshev’s inequality is P(|Y – E[Y ]| = k

p

Var(Y )) = 1/k2

. Let Y = X¯

and write this inequality out in terms of X, E ¯ [X], k, Var(X), and n.

c. If we let k =

v

n, the answer from the previous part should simplify to P(|

p

X¯ – E[X]| =

Var(X)) = 1/n. If we want 99% probability that the sample mean is within one standard

deviation of X away from E[X], how large of a sample do we need, according to this formula?

d. Suppose X¯ is unbiased: E[X¯] = µ. What is the mean squared error of this estimate,

MSE(X¯)? Simplify as much as you can.

e. Assume now that X ~ Ber(p). For some constant c, cX¯ has a distribution that you know.

What is the constant, and what is the distribution?

1

Question 2

This question will guide you through a simulation study in R to understand the bias of a certain estimator. Suppose U1, . . . , Un are independent and identically distributed as U[0, ?], with ? = 1 and let

ˆ? = max{U1, . . . , Un}. Since we are generating this data ourselves, we know the true value of ? = 1, so we

can compute the bias of ˆ?. Our goal will be to study how this bias decreases as the sample size n increases.

The following code creates a function that you can use to generate observations of ˆ?.

theta_hat

You need to run this code once so that R learns the definition of theta_hat. After that, you can “call” the

function by running, for example, theta_hat(10) to generate one observation of a maximum of sample size

n = 10.

theta_hat(10)

## [1] 0.9809853

To generate a sample of many i.i.d. copies of ˆ?, we use the replicate function:

replicate(10, theta_hat(10))

## [1] 0.8791939 0.7226144 0.9647908 0.9541105 0.9452272 0.8321122 0.9678001

## [8] 0.9972075 0.9865613 0.9607960

Finally, we estimate the bias by generating many samples of ˆ?, taking their average, and subtracting ?:

mean(replicate(10000, theta_hat(10))) – 1

## [1] -0.09016838

a. Run this code again to estimate the bias when ˆ? is based on a sample of size n = 100, and

again for a sample of size n = 1000.

# write code here and delete this comment

b. Compute these answers to the answer I gave in class: compute (n – 1)/n – 1 = -1/n for

the same values of n. Are the simulation estimates of bias reasonably close to the exact

mathematical answer?

c. Use the sd function to estimate the standard deviation of ˆ? based on n = 10 and on n = 100.

# write code here and delete this comment

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