hw9_stat210a

hw9_stat210a - UC Berkeley Department of Statistics STAT...

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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 9 Fall 2006 Issued: Thursday, November 2, 2006 Due: Thursday, November 9, 2006 Some useful notation: Let Φ denote the CDF of a standard normal variate, and let φ denote its PDF. Problem 9.1 Suppose that X 1 , . . . X n are i.i.d from the normal location model N ( θ, 1), and that we wish to estimate the critical or cuto± value g a ( θ ) = P [ X 1 a ], where a R is some ²xed number. (a) Let Φ denote the CDF of the standard normal distribution. Considering the estimator δ n ( X ) = Φ µr n n - 1 ( a - ¯ X n ) , prove that n ( δ n - g a ( θ )) d N (0 , φ 2 ( a - θ )), where φ is the PDF of the standard normal. (b) If we had doubts about the assumption of normality, we might prefer to use the non- parametric estimator δ 0 n ( X ) = 1 n # { X i a } , that simply counts the fraction of X i that are less than or equal to a . Prove that n ( δ 0 n - g a ( θ )) d
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hw9_stat210a - UC Berkeley Department of Statistics STAT...

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