SAMPLE MOMENTS 1.1. Moments about the origin (raw moments ...

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10 SAMPLE MOMENTSNow consider the third term on the right side of 42 (ignoring n 2 for now) which we can write as⎡E [ Ȳ 4 ] = 1 n∑n E ⎣4= 1 n 2 E ⎡⎣i =1n∑i =1Y in ∑j =1Y jn ∑k =1Y kn ∑l =1Y l⎤⎦Yi 4 + ∑∑ Y i 2 Yk 2 + ∑∑ Y i 2 Yj2 + ∑i ̸=k i ̸= ji ̸= jYi 2 Yj 2 + ···⎤⎦(45a)(45b)where for the first double sum (i = j ≠ k=l), for the second (i = k ≠ j=l), and for the last (i = l ≠ j=k) and ... indicates that all other terms include Y i in a non-squared form, the expected value of which willbe zero. Given that the Y i are independently and identically distributed, the expected value of each of thedouble sums is the same, which gives⎡E [ Ȳ 4] = 1 n∑n 4 E ⎣= 1 n 4 [ n∑i =1= 1 n 4 [ n∑i =1Yi 4 + ∑∑ Y i 2 Yk 2 + ∑∑ Y i 2 Yj2 + ∑i ̸= k i ̸= ji ̸= jEYi 4 +3 ∑∑ ]Y i 2 Yj 2 + terms containing EX ii ̸= ji =1EYi 4 +3 ∑∑ Y i 2 Yj2i ̸= j]Y 2i Y 2⎤j + ··· ⎦(46a)(46b)(46c)= 1 n 4 [nµ4 +3n (n − 1) ( µ 2 ) 2] (46d)= 1 n 4 [nµ4 +3n (n − 1) σ 4] (46e)= 1 n 3 [µ4 +3(n − 1) σ 4] (46f)Now combining the information in equations 44, 45, and 46 we obtain
SAMPLE MOMENTS 11⎡( ∑ nE ⎣i =1(Yi − Ȳ ) ) ⎤ ⎡2 (∑ n2⎦ =E ⎣i =1⎡( ∑ n=E ⎣i =1(Y2i − 2 Y i Ȳ + Ȳ 2)) 2 ⎤ ⎦ (47a)Y 2i) ⎤ 2 [⎦ − 2 nE[ 1=nµ 4 + n(n − 1)µ 2 2 − 2nnȲ 2n∑i =1Y 2i]+ n 2 E ( Ȳ 4 ) (47b)[µ4 +(n − 1)µ 2 ] ] [ 12 + n 2 [µ4 +3(n − 1)µ 2 ] ]n 3 2(47c)=nµ 4 + n (n − 1)µ 2 2 − 2 [ µ 4[+ (n − 1) µ 2 ] 1 [2 + µ4n+ 3(n − 1) µ 2 ] ]2(47d)= n2n µ 4 − 2 nn µ 4 + 1 n µ 4 + n2 (n − 1)nµ 2 2−2 n (n − 1)nµ 2 23(n − 1)+ µ 2 2 (47e)n= n2 − 2 n +1µ 4 + (n − 1) (n2 − 2 n +3)µ 2 2nn(47f)= n2 − 2 n +1µ 4 + (n − 1) (n2 − 2 n +3)σ 4nn(47g)Now rewrite equation 41 including 1 n 2as follows⎡( [ ( ) ]E M2 2n = 1 n 2 E ∑ n⎣i =1(Yi − Ȳ ) ) ⎤ 22⎦= 1 ( n 2 − 2 n +1n 2 µ 4 + (n − 1) )(n2 − 2 n +3)σ 4nn(48a)(48b)= n2 − 2 n +1n 3 µ 4 + (n − 1) (n2 − 2 n +3)n 3 σ 4 (48c)= ( n − 1)2n 3 µ 4 + (n − 1) (n2 − 2 n +3)n 3 σ 4 (48d)Now substitute equations 34 and 48 into equation 33 to obtainWe can simplify this asVar ( Mn2 ) [ (M ) ]=E2 2n − ( EMn2 ) 2(n − 1)2=n 3 µ 4 + (n − 1) (n2 − 2 n +3)n 3 σ 4 − ( n − (49)1)2n 2 σ 4
Page 1: SAMPLE MOMENTS1. POPULATION MOMENTS
Page 4 and 5: 4 SAMPLE MOMENTS3.2.2. Variance of
Page 6 and 7: 6 SAMPLE MOMENTSE ( [Mn2 ) n]1 =n E
Page 8 and 9: 8 SAMPLE MOMENTS[ (M )2 2]E n⎡(=E
Page 12 and 13: 12 SAMPLE MOMENTSVar ( Mn2 ) [ (M )
Page 14 and 15: 14 SAMPLE MOMENTS6. NORMAL POPULATI

SAMPLE MOMENTS 1.1. Moments about the origin (raw moments ...

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