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.

 

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