Page 557 - Probability and Statistical Inference

534 11. Likehood Ratio and Other Tests

implementable form. {Hint: Proceed along Section 11.3.2. With 0 < c < d <
∞, a LR test rejects H if and only if < c or > d.}
0
11.3.13 Two neighboring towns wanted to compare the variations in the
time (minutes) to finish a 5k-run among the first place winners during each
town’s festivities such as the heritage day, peach festival, memorial day, and
other town-wide events. The following data was collected recently by these
two towns:

Town A (x ): 18 20 17 22 19 18 20 18 17
A
Town B (x ): 20 17 25 24 18 23
B
Assume that the performances of the first place winners are independent and
that the first place winning times are normally distributed within each town.
Then, test at 1% level whether the two town’s officials may assume σ = σ .
A
B
11.3.14 Let the random variables X , ..., X be iid N(µ , ∞ ), n ≥ 2, i = 1,
2
i
i
i1
ini
2, 3 and that the X ’s, X ’s and X ’s be all independent. Here we assume that
2j
1j
2j
µ , µ , µ , ∞ are all unknown and (µ , µ , µ ) ∈ ℜ , ∞ ∈ ℜ . With preassigned
3
+
3
3
2
1
1
2
α ∈ (0, 1), show that a level α LR test for H : µ + µ = 2µ versus H : µ +
1
1
2
1
0
3
µ ≠ 2µ would reject H if and only if
2 3 0
where
is understood to be the corresponding pooled sample variance based on n + n 2
1
+ n – 3 degrees of freedom. {Hint: Repeat the techniques from Section
3
11.3.1. Under H , while writing down the likelihood function, keep µ , µ but
2
1
0
replace µ by ½(µ + µ ) and maximize the likelihood function with respect to
2
3
1
µ , µ only.}
1 2
11.3.15 (Behrens-Fisher problem) Let X , ..., X be iid N(µ , ) random
i
ini
i1
variables, n ≥ 2, i = 1, 2, and that the X ’s be independent of the X ’s. Here we
i
1j
2j
assume that µ , µ , σ , σ are all unknown and (µ , µ ) ∈ ℜ , (σ , σ ) ∈ ℜ , σ 1
+2
2
2
1
1
1
1
2
2
2
≠ σ . Let respectively be the sample mean and variance, i = 1, 2. With
2
preassigned α ∈ (0, 1), we wish to have a level a test for H : µ = µ versus H 1
2
0
1
: µ ≠ µ in the implementable form. It may be natural to use the test statistic
1 2
U where Under H , the statistic U
calc 0 calc
has approximately a Student’s distribution with
Obtain a two-sided approximate t test based on
U calc . {Hint: This is referred to as the Behrens-Fisher problem. Its develop-
ment, originated from Behrens (1929) and Fisher (1935, 1939), is histori-
cally rich. Satterthwaite (1946) obtained the approximate distribution of U calc
under H , by matching “moments.” There is a related confidence inter-
0
val estimation problem for the ratio µ /µ which is referred to as the
1 2