as m – the sum of the two largest digits in your student’s number

Mathematical Statistics 2019/2020, Homework 4

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).

4. Let X₁ be a random variable from a distribution with density

f_σ(x) =

√2

√πmσexp

 −x² 2σ²m²



1_[0,∞)(x), where σ > 0 is an unknown parameter.

Let ˆσ be the maximum likelihood estimator of σ (based on the single observation X₁).

a) Provide the value of the ˆσ estimator of σ, if we know that X₁ = n.

b) For ˆσ, calculate the bias of the estimator, assuming that the true value of parameter σ is equal to k;

c) For ˆσ, calculate the variance of the estimator, assuming that the true value of parameter σ is equal to k;

d) For ˆσ, calculate the MSE of the estimator, assuming that the true value of parameter σ is equal to k.

You may use the properties of the half-normal distribution, and in particular:

EX₁ = σm√ 2

π , VarX₁ = σ²m²

 1 − 2

π



ANSWER:

a) value of ˆσ: b) bias of ˆσ: c) variance of ˆσ: d) MSE of ˆσ:

Solution: