STAT 710: Discussion #17
April 1, 2008
1 UMP Tests
Example 1. Let (X1, ..., Xn) be a random sample from the uniform distri-
bution on (θ, θ + 1), θ∈ R. Suppose that n≥2.
(i) Show that a UMP test of size α∈(0,1) for testing H0:θ≤0 versus
H1:θ > 0 is of the form
T∗(X(1), X(n)) = 0X(1) <1−α1/n, X(n)<1
1 otherwise,
where X(j)is the jth order statistic.
(ii) Does the family of all densities of (X(1), X(n)) have monotone likelihood
ratio?
2 UMPU Tests
Example 2. Let Fand Gbe two known cumulative distribution functions
on Rand Xbe a single observation from the cumulative distribution function
θF (x) + (1 −θ)G(x), where θ∈[0,1] is unknown. Consider H0:θ∈[θ1, θ2]
versus H1:θ6∈ [θ1, θ2], where 0 < θ1≤θ2<1 are constants.
(i) Show that a UMP test does not exist.
(ii) Obtain a UMPU (uniformly most power unbiased) test of size α.
Example 3. Let X1and X2be independently distributed as the negative
binomial distributions with sizes n1and n2and probabilities p1and p2, re-
spectively, where ni’s are known and pi’s are unknown.
(i) Show that there exists a UMPU test of size αfor testing H0:p1≤p2
versus H1:p1> p2.
(ii) Let Y=X1and U=X1+X2. Determine the conditional distribution
of Ygiven Uwhen n1=n2= 1.
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