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hw2 - PubH 8452 Fall 2011 Homework#2 Due 1 Consider the...

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PubH 8452 Fall 2011 Homework #2 Due October 21, 2011 1. Consider the following model that separate within-subject from between-subject varia- tion: Y i j = u i + ² i j , i = 1,..., m ; j = 1,..., n i where E( u i ) = μ , Var( u i ) = σ 2 b = between-subject variance, Var( ² i j ) = σ 2 w = within-subject variance and E[ ² i j ] = 0, Var( Y i j ) = σ 2 t = σ 2 b + σ 2 w = total variance. (a) Define N = m i = 1 n i , ¯ Y i · = n i j = 1 Y i j / n i and ¯ Y ·· = m i = 1 ¯ Y i · / m . Show that ˆ σ 2 w = m i = 1 n i j = 1 ( Y i j - ¯ Y i · ) 2 N - m is an unbiased estimator of σ 2 w . (b) Assume for the rest of the problem that n i = n for all i . Derive var ( ¯ Y i · ) and compute the expected value of m i = 1 ( ¯ Y i · - ¯ Y ·· ) 2 . Thereby derive an unbiased estimator of σ 2 b . Call this estimator ˜ σ 2 b . (c) Following (a) and (b), derive an unbiased estimator of σ 2 t . Call this estimator ˜ σ 2 t . (d) Are ˆ σ 2 w , ˜ σ 2 b and ˜ σ 2 t consistent as m → ∞ , holding n fixed? Are they consistent as n , holding m fixed? (Hint 1: If Y 1 ,..., Y n N ( μ , σ 2 ), then σ - 2 n i = 1 ( Y i - ¯ Y ) χ 2 ( n - 1). Hint 2: When p → ∞ , χ 2 ( p )/ p 1.) (e) Now consider the estimator ˆ σ 2 t = m X i = 1 n X j = 1 ( Y i j - ¯ Y ) 2 /( N - 1), where ¯ Y = m i = 1 n j = 1 Y i j / N . By rewriting this estimator as a linear combination of m i = 1 n j = 1 ( Y i j - ¯
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