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Department of Statistics
STAT 210A: Introduction to Mathematical Statistics
Problem Set 10 Fall 2014

Problem 10.1
Read Chapter 14 of Keener.
Problem 10.2
Consider the general linear model with normality:
Y ∼ N(Xβ,σ2I), β ∈Rp, σ2 > 0.
If the rank r of X equals p, show that (β,Sˆ 2) is a complete sufficient statistic.
Problem 10.3
Inverse linear regression. Consider the model for simple linear regression:

studied in Section 14.5.
(a) Derive a level-α test of H0 : β2 = 0 versus H1 : β2 6= 0.
(b) Let y0 denote a “target” value for the mean of Y . The regression line β1 + β2(x − x¯) achieves this value when the independent variable x equals
.
(c) Use duality to find a confidence region, first discovered by Fieller (1954), for θ. Show that this region is an interval if the test in part (a) rejects β2 = 0.
Problem 10.4
Consider a regression version of the two-sample problem in which:
;
,…,n1 + n2 = n,
with i.i.d. from N(0,σ2). Derive a 1−α confidence interval for β4−β2, the difference between the two regression slopes.
Problem 10.5
A variable Y has a log-normal distribution with parameters µ and σ2 if logY ∼ N(µ,σ2).
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(a) Find the mean and density for the log-normal distribution.
(b) If Y1,…,Yn are i.i.d. from the log-normal distribution with unknown parameters µ and σ2, find the UMVU for µ.
(c) If Y1,…,Yn are i.i.d. from the log-normal distribution with parameters µ and σ2, with σ2 a known constant, find the UMVU for the common mean η = EYi.
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