APPM 5720 Problem Set Three (Due Wednesday, September 27)

2 1. Let X1,X2,...,Xn be a random from any distribution with µ and σ . From class, we know thatµ ˆ = X is an unbiased estimator of µ. Suppose that we want to estimate the variance. The interpretation of the variance (σ2 = E[(X −µ)2]) as the mean squared deviation from the mean µ leads us to the natural estimator that involves averaging the squared deviations in the sample from the sample mean:

Pn (X − X)2 i=1 i . n

Some books/people refer to this quantity as the sample variance. However, many other books/people define the sample variance to be

Pn 2 i=1(Xi − X) n − 1 because it is an unbiased estimator for σ2. Let Pn (X − X)2 Pn (X − X)2 S2 := i=1 i and S2 = i=1 i . 1 n 2 n − 1 (a) Show that n n n X 2 X 2 X 2 (Xi − X) = Xi − ( Xi) /n. i=1 i=1 i=1 2 2 (b) Show that S2 is an unbiased estimator of σ . 2 (c) Use part (a) to quickly find the of S1 .(Hint: Don’t make this too much work! Use part (b)!)

[Note: Throughout this course, we will be using the second form of the sample variance– the one with the n − 1 in the denominator. We will drop the subscript and denote it by S2.]

iid 2. Let X1,X2,...,Xn ∼ unif(0, θ). It is natural to want to estimate θ, the right endpoint of the support of this distribution, with the maximum value seen in the sample. It is also reasonable to expect that the expected value of the maximum will not quite achieve the value of θ.

(a) Show that X(n) is a biased estimator of θ and that its expected value is below θ.

(b) Find an unbiased estimator of θ based on X(n). (c) Find an unbiased estimator of θ based on the sample mean X. (d) Compare the of your two estimators. Which do you prefer?

(e) Show that the biased estimator X(n) at least converges in to θ. 3. Suppose that {Xn} is a sequence of random variables and that X is another such that h 2i lim E (Xn − X) = 0. n→∞ P Show that Xn → X.

4. Let X be a random variable with finite mean and let g be a convex function. Show that

E[g(X)] ≥ g(E[X]).

2 2 5. (a) Suppose that X1 ∼ χ (n1) and X2 ∼ χ (n2) are independent random variables. Let Y = X1 + X2. What is the distribution of Y ? (Name it.)

(b) Suppose that X1 and X2 are independent random variables, that Y = X1 + X2 and that 2 2 Y ∼ χ (n) and X1 ∼ χ (n1). What is the distribution of X2? (Name it.) 2 2 (c) Suppose that X1 ∼ χ (n1) and X2 ∼ χ (n2) for n1 > n2 and further suppose that X1 2 and X2 are independent. Define Y = X1 − X2. Is Y ∼ χ (n1 − n2)? Explain.