代写MATH0099 Statistical Methods and Data Analytics Problem Sheet 2代写Java编程

MSc Financial Mathematics

Statistical Methods and Data Analytics

MATH0099

Problem Sheet 2

Problem  1. (Duality  between confidence intervals and hypothesis tests) Prove the theorem stated in Lecture 3, which relates confidence intervals and hypothesis tests.

Problem  2. (Duality between confidence 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 defined by

ϕ(x) = 1, if |Tn,m − 2/n| > c(n, α),

where x = (x1, . . . , xn ) denotes the sample and Q the confidence 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 confidence interval with confidence level Q.

Problem 3. (Maximum likelihood estimate) You wish to estimate the number N of fishina pond. You catch five fish, mark them in a clear manner and return them to the pond.  Assume that after some time the marked fish have intermingled with the unmarked ones.  In a second round you catch eleven fish 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.