By Tang, Wan; He, Hua; Tu, Xin M
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3. If c = 0, Xn /Yn ∼a X/c. d. sample 2 Zi with mean µ = E (Zi ) and variance σ = V ar (Zi ). To show that s2n has an asymptotic distribution, first reexpress s2n as: s2n − σ 2 = 1 n n 2 (Zi − µ) − σ 2 + Z n − µ 2 . i=1 By CLT, √ Xn = n n n 2 2 (Zi − µ) − σ 2 ∼a N 0, V ar (Zi − µ) , i=1 √ n n Z n − µ ∼a N 0, σ 2 . By LLN, Z n − µ = n1 i=1 (Zi − µ) →p √ 2 0. Thus, by Slutsky’s theorem, n Z n − µ ∼a 0, and s2n − σ 2 ∼a and 2 N 0, n1 V ar (Zi − µ) 2 . 3 n i=1 Zi and µ4 = 1 n n i=1 4 (Zi − µ) . Maximum Likelihood Estimate One of the most popular inference approaches for parametric models is maximum likelihood.
Let g (θ) = (g1 (θ) , . . , gm (θ)) be a continuous vector-valued function from Rk to Rm . Introduction 19 Then, the function g θ n is a consistent estimate of g (θ). This result helps to find a consistent estimate of the variance of the asymptotic distribution of g θ n above. , θn →p θ d g (θ) = exp (θ). By the and θn ∼a N θ, n1 σ 2 . Let g (θ) = exp (θ). Then, dθ delta method, the estimate exp θn for exp (θn ) is also consistent, with the asymptotic distribution N exp (θ) , n1 σ 2 exp (2θ) .
9. 2 Let h (x1 , x2 ) = 21 (x1 − x2 ) . Since E [h(X1 , X2 )] = 21 E X12 − 2X1 X2 + X22 = V ar(X). 24) is an unbiased estimate of σ 2 . 11). Further, since 1 2 h (X1 ) = E [h (X1 , X2 ) | X1 ] − σ 2 = (X1 − µ) − σ 2 , 2 it follows that V ar h1 (X1 ) = 1 1 2 V ar (X1 − µ) = µ4 − σ 4 . 4 4 Thus, σn2 − σ 2 ∼a N 0, n1 µ4 − σ 4 . 5 using a different approach. 22) in closed form. 13) V ar [E [h (Z 1 , Z 2 ) | Z 1 ]] = E h (Z 1 , Z 2 ) h (Z 1 , Z 3 ) − θθ . 25) We can estimate θθ by θ θ . 25), we can construct another U-statistic.