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Problem Sheet 2
Problem 1. (Duality between con?dence intervals and hypothesis tests) Prove the theorem stated in Lecture 3, which relates con?dence intervals and hypothesis tests.
Problem 2. (Duality between con?dence intervals and hypothesis tests) Let X1 , . . . Xn be i.i.d. copies of a random variable X with absolutely continuous cumulative distribution function F. We wish to test the null hypothesis H0 that the median of X equals m, i.e.
F-1 (1/2) = m.
The sign test uses the test statistic Tn,m =Σ 1[Xi>m] and is de?ned by
?(x) = 1, if |Tn,m ? 2/n| > c(n, α),
where x = (x1, . . . , xn ) denotes the sample and Q the con?dence level of the test. Given that under the null hypothesis Tn,m is binomially distributed with success probability 0.5, use a suitable theorem from Lecture 3 to construct a con?dence interval with con?dence level Q.
Problem 3. (Maximum likelihood estimate) You wish to estimate the number N of ?shina pond. You catch ?ve ?sh, mark them in a clear manner and return them to the pond. Assume that after some time the marked ?sh have intermingled with the unmarked ones. In a second round you catch eleven ?sh out of which three are marked and eight are unmarked. Construct the maximum likelihood estimate of N.
Problem 4. (Maximum likelihood estimate) Let X1 , . . . Xn be i.i.d. copies of a random variable X with density
where θ ∈ (1, ∞) is the unknown parameter. Compute the maximum likelihood estimate Tn for θ.