STAT 461/561- 2021 Assignments 1
1. [3+3+3] Assume X1, X2, . . . is a sequence of i.i.d. standard Poisson
random variables.
(a) max{X1, X2, . . . , Xn} = Op(log n).
(b) max{X1, X2, . . . , Xn} = op(n).
(c) max{X1, X2, . . . , Xn} 6= Op(
p
log n).
2. [5] Let an be a sequence of real values and Xn be a sequence of random
variables. Suppose an !1 and an(Xnµ) d! Y . If g(x) is a function
satisfying Lipschitz condition:
|g(x) g(y)|  C|x y|
for some constant C. Show that
an{g(Xn) g(µ)} = Op(1).
3. [8] Let Xn, Yn, be a pair of independent Poisson distributed random
variables with mean n1 and n2. Define
Tn = (Yn/Xn) (Xn > K)
for some K > 0. Show that
p
n(Tn 2/1) d! N(0, 2)
and work out the expression of 2.
4. [2+2+2] Suppose we have two i.i.d. samples fromN(0, 21) andN(0,
2
2).
Let the observations be denoted as x1, x2, . . . , xm and y1, y2, . . . , yn.
Design a test for H0 : 21 =
2
2 versus H1 :
2
1 6= 22.
The requirements of this problem include:
(1) Present the (same) test in all three ways as in Section 12.7;
(2) Does your test statistic have the two desirable properties?
(3) No actual data analysis is required.
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5. [2+2+2] Suppose we have two i.i.d. samples fromN(µ1, 2) andN(µ2, 2).
Let the observations by denoted as x1, x2, . . . , xm and y1, y2, . . . , yn.
Sample I:
-0.2212993 0.4495635 -0.9740405 -0.7886435 1.7338719
0.8720555 0.2192809 -1.8462444 0.3230189 1.0007282
Sample II:
-0.4507532 -2.1045739 -1.3628046 1.2556175
-0.5325500 0.3671854 -0.7482886 -1.9928208
(a) Carry out the classical two-sample t-test for H0 : µ2µ1  0 versus
H1 : µ2 µ1 > 0 based on the above data.
(b) Plot the type I error of this test as function of (µ2 µ1)/.
(c) Plot the type II error of this test as function of (µ2 µ1)/.
Hints: Review the material about non-central t-distribution. Choose
sensible regions in (b) and (c), show your codes.
6. [3+2+2] Let (Xi, Yi), i = 1, 2, . . . , n be a set of iid bivariate observa-
tions with their joint probably density function given by
f(x, y; ✓1, ✓2) =
xy
✓21✓
2
2
exp( x
✓1
y
✓2
).
Consider the test problem for H0 : ✓1 = ✓2 versus H1 : ✓1 > ✓2.
Let X¯n and Y¯n be sample means and define Tn = log{X¯n} log{Y¯n}.
(a) Illustrate that Tn has the desired properties for the purpose of
statistical significance test.
(b) Suppose the observed value of Tn = t0. What is the p-value of the
test based on Tn in a probability expression?
Remark: I look for an expression in the spirit of P (Tn 2 [1, 2]).
(c) Show that X¯n
Y¯n
has an F-distribution with some degrees of freedom.
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