Value-at-Risk

Chapter 4, Page 170
Exercise 3

In this exercise, you will demonstrate that sample variance estimator [4.5] is biased but that alternative estimator [4.27] is unbiased.

a. First prove a technical result that will be needed for the derivations. Prove that, given any set of numbers {x^[1], x^[2], ... , x^[m]} whose average is :

[4.25]

b. Derive a formula for the bias of estimator [4.5].

c. Modify your derivation from (b) to obtain a formula for the bias of [4.27].

Solution

a. We first prove the technical result. This involves no random quantities. It is just algebra:

	[s1]
	[s2]
	[s3]
	[s4]
	[s5]

b. We next use our technical result to determine the bias of the sample variance estimator

[4.5]

The derivation is

[s6]

[s7]

[s8]

[s9]

[s10]

For the next step in the derivation, we apply the result of Exercise 3.15 twice:

	[s11]
	[s12]
	[s13]

We conclude that sample estimator [4.5] is biased.

c. Finally, we determine the bias of the alternative estimator of variance:

[4.27]

which we can denote

[s14]

The derivation largely parallels that of part (b):

	[s15]
	[s16]
	[s17]
	[s18]
	[s19]
	[s20]
	[s21]
	[s22]
	[s23]

Estimator [4.27] is unbiased.