Estimators and bias

Two organisations are each asked to carry out a survey to find out the proportion, p, of the population that would vote for a particular political party.

The first organisation finds that out of m people, X would vote for this particular political party.

The second organisation finds that out of n people, Y would vote for this particular political party.

An unbiased estimator, Q, of p is proposed where

A second unbiased estimator, R, of p is proposed where

Show that a + b = 1

Given that m = 100 and n = 200 and that R is a better estimator of p than Q

calculate the range of possible values of a Show your working clearly.

A bag contains a large number of marbles of which an unknown proportion, p, is yellow.

Three random samples of size n are taken, and the number of yellow marbles in each sample, Y₁, Y₂ and Y₃, is recorded.

Two estimators ˆp₁ and ˆp₂ are proposed to estimate the value of p

Show that ˆp₁ and ˆp₂ are both unbiased estimators of p

Find the variance of ˆp₁

State, giving a reason, which is the better estimator.

Find the pair of values of a and b such that ˆp₃ is a better unbiased estimator of p

Korhan and Louise challenge each other to find an estimator for the mean, μ, of the continuous random variable X which has variance σ²

X₁, X₂, X₃, …, X_n are n independent observations taken from X Korhan’s estimator is given by

Louise’s estimator is given by

Show that K and L are both unbiased estimators of μ

Find Var (K)

Find Var (L)

The winner of the challenge is the person who finds the better estimator.

Determine the winner of the challenge for large values of n. Give reasons for your answer.

The continuous random variable X is uniformly distributed over the interval [0, 4β], where β is an unknown constant.

Three independent observations, X₁, X₂ and X₃, are taken of X and the following estimators for β are proposed

Calculate the bias of A, the bias of B and the bias of C

By calculating the variances, explain which of B or C is the better estimator for β

Find an unbiased estimator for β

A biased coin has a probability p of landing on heads, where 0 < p < 1 Simon spins the coin n times and the random variable X represents the number of heads. Taruni spins the coin m times, m ≠ n, and the random variable Y represents the number of heads.

Simon and Taruni want to combine their results to find unbiased estimators of p.

Show that both S and T are unbiased estimators of p.

Prove that, for all values of m and n, S is the better estimator.