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 · Standard +0.0

Sort by: Default | Easiest first | Hardest first

Explain why $\frac { X _ { 1 } - X _ { 2 } } { \sigma }$ is not a statistic. $$S = \frac { 3 } { 5 } X _ { 1 } + \frac { 5 } { 7 } X _ { 2 }$$
Show that $S$ is a biased estimator of $\mu$
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,
find an equation, in terms of $a$ and $b$, that must be satisfied.
Using your answer to part (d), show that $\operatorname { Var } ( Y ) = \left( 2 a ^ { 2 } - 2 a + 1 \right) \sigma ^ { 2 }$

an unbiased estimator,
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
the best estimator,
the worst estimator.

Explain what is meant by the sampling distribution of an estimator $T$ of the population parameter $\theta$.
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.
Calculate the bias of each of the following estimators of $\mu$. $$\begin{aligned} & \hat { \mu } _ { 1 } = \frac { X _ { 3 } + X _ { 5 } + X _ { 7 } } { 3 } \\ & \hat { \mu } _ { 2 } = \frac { 5 X _ { 1 } + 2 X _ { 2 } + X _ { 9 } } { 6 } \\ & \hat { \mu } _ { 3 } = \frac { 3 X _ { 10 } - X _ { 1 } } { 3 } \end{aligned}$$
Find the variance of each of these three estimators.
State, giving a reason, which of these three estimators for $\mu$ is
1. the best estimator,
2. the worst estimator.

Determine the bias, if any, of each of the following estimators of $\mu$. $$\theta_1 = \frac{X_1 + X_4 + X_5}{3}$$ $$\theta_2 = \frac{X_{10} - X_1}{3}$$ $$\theta_3 = \frac{3X_1 + 2X_5 + X_{10}}{6}$$ [4]
Find the variance of each of these estimators. [5]
State, giving reasons, which of these three estimators for $\mu$ is
1. the best estimator,
2. the worst estimator.
[4]