Student’s number

Mathematical Statistics 2019/2020, Homework 5

Name and Surname ... Student’s number ...

In the problems below, please use the following: as k – the sum of digits in your student’s number; as m – the sum of the two largest digits in your student’s number;

and as n – the smallest digit in your student’s number plus 1. For example, if an index number is 609999: k = 42, m = 18, n = 1.

Please write down the solutions (transformations, substitutions etc.), and additio- nally provide the final answer in the space specified (the answer should be a number in decimal notation, rounded to four digits).

5. Let X₁, X₂, . . . , X_k be a sample from a gamma distribution with parameters n + 1 and θ, i.e. with a density function equal to

f_θ(x) =

(_(θ)n+1

n! xⁿe^−θx x > 0

0 otherwise,

where θ > 0 is an unknown parameter.

Let

T_m = X₁+ . . . + X_m m(n + 1) be an estimator of the function ¹_θ.

Calculate the Fisher information I_k(θ) connected with the k-element sample, the variance of estimator T_m and the efficiency of estimator T_m and provide these values assuming that the true value of parameter θ amounts to m.

You may use the properties of the gamma distribution (in particular, the formula for the expected value and the variance). Please note that the sample size is k but the estimator T_m is based on the first m observations.

ANSWER: I_k(θ) = var T_m = eff T_m =

Solution: