hw2 - PubH 8452 Fall 2011 Homework #2 Due October 21, 2011...

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Unformatted text preview: 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 + i j , i = 1,..., m ; j = 1,..., n i where E( u i ) = , Var( u i ) = 2 b = between-subject variance, Var( i j ) = 2 w = within-subject variance and E[ i j ] = 0, Var( Y i j ) = 2 t = 2 b + 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 2 w = m i = 1 n i j = 1 ( Y i j- Y i ) 2 N- m is an unbiased estimator of 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 2 b . Call this estimator 2 b . (c) Following (a) and (b), derive an unbiased estimator of 2 t . Call this estimator 2 t . (d) Are 2 w , 2 b and 2 t consistent as m , holding n fixed? Are they consistent as n , holding m fixed? (Hint 1: If Y 1 ,..., Y n N ( , 2 ), then - 2 n i = 1 ( Y i- Y ) 2 ( n- 1). Hint 2: When p , 2 ( p )/ p 1.) (e) Now consider the estimator 2 t = m X i = 1 n X j = 1 ( Y i j- Y ) 2 /( N- 1), where...
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hw2 - PubH 8452 Fall 2011 Homework #2 Due October 21, 2011...

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