Exercise Solution 4.3

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

   [s1]

   

   [s2]

   

   [s3]

   

   [s4]

   

   [s5]

   
2. 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]

   

   To obtain 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.

3. 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.