Calculating bias of estimator

Questions asking to find the bias (E(T) - θ) of a given estimator, where the estimator is known or expected to be biased.

4 questions

Edexcel S3 2023 June Q3
  1. A random sample of 2 observations, \(X _ { 1 }\) and \(X _ { 2 }\), is taken from a population with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\)
    1. Explain why \(\frac { X _ { 1 } - X _ { 2 } } { \sigma }\) is not a statistic.
    $$S = \frac { 3 } { 5 } X _ { 1 } + \frac { 5 } { 7 } X _ { 2 }$$
  2. Show that \(S\) is a biased estimator of \(\mu\)
  3. Hence find the bias, in terms of \(\mu\), when \(S\) is used as an estimator of \(\mu\) Given that \(Y = a X _ { 1 } + b X _ { 2 }\) is an unbiased estimator of \(\mu\), where \(a\) and \(b\) are constants,
  4. find an equation, in terms of \(a\) and \(b\), that must be satisfied.
  5. Using your answer to part (d), show that \(\operatorname { Var } ( Y ) = \left( 2 a ^ { 2 } - 2 a + 1 \right) \sigma ^ { 2 }\)
Edexcel S3 Q3
3. The discrete random variable \(X\) has the probability distribution given below.
\(x\)247\(k\)
\(\mathrm { P } ( X = x )\)0.050.150.30.5
  1. Find the mean of \(X\) in terms of \(k\).
  2. Find the bias in using ( \(2 \bar { X } - 5\) ) as an estimator of \(k\). Fifty observations of \(X\) were made giving a sample mean of 8.34 correct to 3 significant figures.
  3. Calculate an unbiased estimate of \(k\).
    (2 marks)
Edexcel S4 2004 June Q5
5. (a) Explain briefly what you understand by
  1. an unbiased estimator,
  2. a consistent estimator.
    of an unknown population parameter \(\theta\). From a binomial population, in which the proportion of successes is \(p , 3\) samples of size \(n\) are taken. The number of successes \(X _ { 1 } , X _ { 2 }\), and \(X _ { 3 }\) are recorded and used to estimate \(p\).
    (b) Determine the bias, if any, of each of the following estimators of \(p\). $$\begin{aligned} & \hat { p } _ { 1 } = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } } { 3 n }
    & \hat { p } _ { 2 } = \frac { X _ { 1 } + 3 X _ { 2 } + X _ { 3 } } { 6 n }
    & \hat { p } _ { 3 } = \frac { 2 X _ { 1 } + 3 X _ { 2 } + X _ { 3 } } { 6 n } \end{aligned}$$ (c) Find the variance of each of these estimators.
    (d) State, giving a reason, which of the three estimators for \(p\) is
  3. the best estimator,
  4. the worst estimator.
Edexcel S4 2014 June Q6
  1. (a) Explain what is meant by the sampling distribution of an estimator \(T\) of the population parameter \(\theta\).
    (b) Explain what you understand by the statement that \(T\) is a biased estimator of \(\theta\).
A population has mean \(\mu\) and variance \(\sigma ^ { 2 }\)
A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 10 }\) is taken from this population.
(c) Calculate the bias of each of the following estimators of \(\mu\). $$\begin{aligned} & \hat { \mu } _ { 1 } = \frac { X _ { 3 } + X _ { 5 } + X _ { 7 } } { 3 }
& \hat { \mu } _ { 2 } = \frac { 5 X _ { 1 } + 2 X _ { 2 } + X _ { 9 } } { 6 }
& \hat { \mu } _ { 3 } = \frac { 3 X _ { 10 } - X _ { 1 } } { 3 } \end{aligned}$$ (d) Find the variance of each of these three estimators.
(e) State, giving a reason, which of these three estimators for \(\mu\) is
  1. the best estimator,
  2. the worst estimator.