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

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4 SAMPLE MOMENTS3.2.2. Variance of C r n. First consider the case where we have a sample X 1 ,X 2 , ... ,X n .(Var( Cn r 1n∑ ()=Var Xi − µ ′ i,1ni=1If the X’s are independently distributed, thenVar( C r n )= 1 n 2n ∑i=1If the X’s are independent and identically distributed, then) r)= 1 n 2 Var ( n∑i=1Var [( X i − µ ′ ) r ]i,1)(Xi − µ ′ ) ri,1(21)(22)Var( Cn r )= 1 n Var[ ( X − µ ′ 1 ) r ] (23)where X denotes any one of the random variables (because they are all identical). In the case where r =1,we obtainVar ( Cn1 ) 1 =n Var[ X − µ′ 1 ]= 1 n Var[ X − µ ]= 1 (24)n σ2 − 2 Cov[ X, µ]+Var[ µ ]= 1 n σ24. SAMPLE ABOUT THE AVERAGE4.1. Definitions. Assume there is a sequence of random variables, X 1 ,X 2 ,...X n . Define the rth samplemoment about the average asm r nMnr = 1 n∑ (Xi −n¯X) rn , r =1, 2, 3,..., (25)i=1This is clearly a statistic of which we can compute a numerical value. We denote the numerical value by,, and define it asmIn the special case where r = 1 we haver n = 1 nn∑( x i − ¯x n ) r (26)i =1Mn 1 = 1 n= 1 nn∑ (Xi − ¯X)ni =1n∑X i − ¯X ni =1= ¯X n − ¯X n = 0(27)4.2. Properties of Sample Moments about the Average when r = 2.
SAMPLE MOMENTS 54.2.1. Alternative ways to write M r n. We can write M 2 n in an alternative useful way by expanding the squaredterm and then simplifying as followsM r n = 1 nn∑ ( ) rXi − ¯X ni =1⇒ M 2 n= 1 n= 1 n= 1 nn∑ (Xi − ¯X) 2ni=1( n∑i =1n∑i =1[X2iX 2i− 2 X i ¯Xn + ¯X] )n2− 2 ¯X nn∑ ni =1 X i + 1 nn∑i =1¯X 2 n(28)n∑= 1 ni =1( n∑X 2 i − 2 ¯X 2 n + ¯X 2 n)= 1 ni =1X 2i− ¯X 2 n4.2.2. Expected value of Mn 2. The expected value of M n r is then given byE ( [) nMn2 1 =n E ∑i =1n∑= 1 n= 1 ni=1n∑i =1X 2i]− E [ ] ¯X2 nE [ Xi2 ] ( [ ]) 2− E ¯Xn − Var( ¯Xn )µ ′ i,2 −(1nn∑i =1µ ′ i,1) 2− Var( ¯X n )(29)The second line follows from the alternative definition of varianceVar( X )=E ( X 2 ) − [ E ( X )] 2⇒ E ( X 2 ) =[ E ( X )] 2 + Var( X )(30)⇒ E ( ¯X 2n) =[ E( ¯X n)] 2+ Var( ¯X n )and the third line follows from equation 12. If the X i are independent and identically distributed, then
Page 1: SAMPLE MOMENTS1. POPULATION MOMENTS
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 10 and 11: 10 SAMPLE MOMENTSNow consider the t
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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