Estimator versus estimate?

The estimator is the random variable (e.g. \bar X); the estimate is its value for one sample (e.g. \bar x=12.4).

When is an estimator unbiased?

When E(T)=\theta, so its bias is zero.

Unbiased estimate of \mu?

The sample mean \bar X, since E(\bar X)=\mu.

Unbiased estimate of \sigma^2?

S^2=\frac{1}{n-1}\sum(Xi-\bar X)^2

Why divide by n-1 rather than n?

Dividing by n gives E(V)=\frac{n-1}{n}\sigma^2, which is biased low; n-1 removes the bias.

Calculator-friendly form of s^2?

s^2=\frac{1}{n-1}\left(\sum x^2-\frac{(\sum x)^2}{n}\right)

Define standard error.

The standard deviation of the sampling distribution of an estimator; for \bar X it is \frac{\sigma}{\sqrt n}.

How do you choose between two unbiased estimators?

Choose the one with the smaller variance.

Variance of \frac{X1+2X2}{3}?

\frac{1+4}{9}\sigma^2=\frac{5\sigma^2}{9}

Estimators, standard error and biasEdexcel A-Level Further Maths: Flashcards

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What is an estimator?

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What is an estimator?: A statistic calculated from a random sample, used to estimate a population parameter.
Estimator versus estimate?: The estimator is the random variable (e.g. $\bar X$ ); the estimate is its value for one sample (e.g. $\bar x=12.4$ ).
Define the bias of an estimator $T$ of $\theta$ .: $\mathrm{E}(T)-\theta$
When is an estimator unbiased?: When $\mathrm{E}(T)=\theta$ , so its bias is zero.
Unbiased estimate of $\mu$ ?: The sample mean $\bar X$ , since $\mathrm{E}(\bar X)=\mu$ .
Unbiased estimate of $\sigma^2$ ?: $S^2=\frac{1}{n-1}\sum(X_i-\bar X)^2$
Why divide by $n-1$ rather than $n$ ?: Dividing by $n$ gives $\mathrm{E}(V)=\frac{n-1}{n}\sigma^2$ , which is biased low; $n-1$ removes the bias.
Calculator-friendly form of $s^2$ ?: $s^2=\frac{1}{n-1}\left(\sum x^2-\frac{(\sum x)^2}{n}\right)$
What is $\mathrm{Var}(\bar X)$ ?: $\frac{\sigma^2}{n}$
Define standard error.: The standard deviation of the sampling distribution of an estimator; for $\bar X$ it is $\frac{\sigma}{\sqrt n}$ .
Estimated standard error of $\bar X$ when $\sigma$ is unknown?: $\frac{s}{\sqrt n}$
What happens to the standard error of $\bar X$ if $n$ is multiplied by 4?: It is halved.
How do you choose between two unbiased estimators?: Choose the one with the smaller variance.
Variance of $\frac{X_1+2X_2}{3}$ ?: $\frac{1+4}{9}\sigma^2=\frac{5\sigma^2}{9}$

Exam questions on Estimators, standard error and bias

$X_1$ , $X_2$ , $X_3$ is a random sample from a population with mean $\mu$ and variance $\sigma^2$ . The statistic $U=\frac{X_1+2X_2+3X_3}{6}$ is proposed as an estimator of $\mu$ .
The sample mean $\bar X=\frac{X_1+X_2+X_3}{3}$ is also an estimator of $\mu$ . Compare $U$ and $\bar X$ and state, with a reason, which you would use.2 marks
A random sample of $10$ observations of a quantity $x$ from a population with unknown mean and variance gives $\sum x=124.0$ and $\sum x^2=1582.6$ .
A larger sample of $40$ observations has the same sample variance as in part (a). Calculate the estimated standard error of its mean and state what this shows about $\bar X$ as an estimator of the population mean.2 marks
$X_1,X_2,\ldots,X_n$ is a random sample from a population with mean $\mu$ and variance $\sigma^2$ . Two estimators of $\sigma^2$ are $S^2=\frac{1}{n-1}\sum(X_i-\bar X)^2$ , which is known to be unbiased, and $V=\frac{1}{n}\sum(X_i-\bar X)^2$ .
Show that $V$ is a biased estimator of $\sigma^2$ , and state the bias.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).