Point Estimation
1 What is a Point Estimator?
Definition 1 (Point Estimator) A single estimate of a parameter e.g. \(\mean{Y}\) estimates \(\mu\) and \(S^2\) estimates \(\sigma^2\)
Example 1 (Example 1) We have a bent coin and are interest in \(p\), the probability of heads coming up on a single toss. How can we estimate \(p\)?
Solution
First we need some data (observable random variable or variables).
For example, we toss the coin \(n\) times and observe the number of heads that comes up.
The data is then that number, and we may call it \(Y\).
We next need a model for data, e.g. \(Y \sim \Binomial(n, p)\). Here \(p\) may be called the target parameter or estimand. We now need to choose an estimator of \(p\), which may really be any statistic. For example, we may, conforming to our intuitions, use the statistic \(X \triangleq Y/n\) as the estimator.
Finally, we need to actually carry out the experiment and do the calculations.
For example, we toss the coin \(n = 10\) times and get 6 heads. Then, the realised value of the data \(Y\) is \(y = 6\), and the realised value of our estimator \(X\) is \(x = y/n = 6/10 / 0.6\). We call \(x\) an estimate of \(p\). Because \(x\) is a single number, we may also call it a point estimate of \(p\). Likewise, we may call \(X\) a point estimator of \(p\). (Note that this is random over repeated sampling).
A common practice is to denote both the estimator and estimate of a parameter \(\theta\) by \(\hat{\theta}\). Thus in our example:
- the estimator of \(p\) is \(\hat{p} = X = Y / n\) (a random variable)
- the estimate of \(p\) is \(\hat{p} = x = y/n = 0.6\) (a constant)
This may be a bit confusing because the same symbol is used for a random variable and a constant. But usually the symbol’s meaning is clear from the context.