BIOS760 Final Exam, Fall 2007
1. Suppose that we observe nindependent pairs (Yi, Xi), where
Xi=1
2log i+Ui, Yi=β+eXi²i
with Ui, i = 1, ..., n i.i.d from Uniform(0,1), ²i, i = 1, ..., n from N(0,1), and Uiindependent of ²i.
By noting
Yie−Xi=βe−Xi+²i,
we can obtain an alternative least-square estimator for βgiven as
ˆ
β=Pn
i=1 Yie−2Xi
Pn
i=1 e−2Xi.
(a) (5 points) Write down the observed likelihood function for β.
(b) (5 points) What is the Cramer-Rao lower bound for β?
We want to derive the asymptotic distribution for ˆ
βafter a proper normalization. Particulary,
we take the following steps.
(c) (5 points) Derive the asymptotic distribution of Pn
i=1 i−1/2e−Ui²i,after a proper normaliza-
tion;
(d) (5 points) Show
Pn
i=1 i−1e−2Ui
(1 −e−2)/2Pn
i=1 i−1→p1;
(e) (5 points) Obtain the asymptotic distribution for ˆ
βafter a proper normalization.
2. Suppose X1, ..., Xnare i.i.d from density
f(x) = I(x≥θ) exp{−(x−θ)}.
(a) (5 points) Find a complete and sufficient statistic for θ.
(b) (5 points) Derive the UMVUE for θand denote it as ˆ
θ1.
(c) (5 points) Find the maximum likelihood estimate for θand denote it as ˆ
θ2.
(d) (5 points) Calculate the variance of ˆ
θ2and show that ˆ
θ2is consistent using the Chebyshev’s
inequality.
(e) (5 points) Find some constant ansuch that an(ˆ
θ2−θ0) converges in distribution to a non-
degenerate distribution. Identify the limiting distribution. Here, θ0is the true value for
θ.
(f) (5 points) Find some constant bnsuch that bn(ˆ
θ1−θ0) converges in distribution to a non-
degenerate distribution. Identify the limiting distribution.
3. Let Z1, ..., Znbe the measurements of body mass index (BMI) and X1,..., Xnbe the daily fat
intake from nindependent adults. Assume
Zi=β0+β1Xi+²i,
where ²ifollows distribution N(0, σ2) and is independent of Xi. Assume Xihas a known density
g(x). In a real study, we observe X1, ..., Xnbut instead of Zi, only the categories of BMI are
observed: such categories are defined as
Yi=(normal, Zi<25,
overweight,25 ≤Zi.
Therefore, the observed data are (Y1, X1), ..., (Yn, Xn). We may code Yi= 0 for “normal” category
and Yi= 1 for “overweight” category.
