By Tang, Wan; He, Hua; Tu, Xin M

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Sample of random vectors and let h be a vector-valued symmetric function m arguments. Then, θ= n m −1 h (X i1 , . . ,im )∈Cm Introduction 29 is an unbiased estimate of θ. 22) is an unbiased estimate of θ. , 2 n 1 σ 2 can be expressed as σ 2 = n−1 i=1 Xi − X n . 12 For random variables X and Y , show that E [E (X | Y )] = E (X) and V ar(X) = V ar(E(X | Y )) + E(V ar(X | Y )). 22). a) Show: V ar h1 (Z 1 ) = E h1 (Z 1 ) h1 (Z 1 ) − θθ . b) Use the iterated conditional expectation to show: E h1 (Z 1 ) h1 (Z 1 ) = E h (Z 1 , Z 2 ) h (Z 1 , Z 3 ) .

For example, if we measure an individual’s blood pressure twice, the two readings are correlated since they reflect the health condition of this particular individual; if he or she has high blood pressure, both readings tend to be higher than the normal range (positively correlated) despite the variations over 12 Applied Categorical and Count Data Analysis repeated assessments. The existence of such within-subject correlations invalidates the independent sampling assumption required for most classic models, precluding applications of statistical methods developed for cross-sectional data based on such an independence assumption.

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. Let g (Z 1 , Z 2 , Z 3 ) = h (Z 1 , Z 2 ) h (Z 1 , Z 3 ) , 1 g (Z 1 , Z 2 , Z 3 ) = (g (Z 1 , Z 2 , Z 3 ) + g (Z 2 , Z 1 , Z 3 ) + g (Z 3 , Z 2 , Z 1 )) . 3 Then, g (Z 1 , Z 2 , Z 3 ) is a symmetric with respect to the permutations of Z 1 , −1 Z 2 and Z 3 . The matrix Ψ = n3 (i,j,k)∈C3n g (Z i , Z j , Z k ) is a U-statistic and thus is a consistent estimate of Ψ.