Statistical inference can be divided into two parts:
Hypothesis testing can help us answer questions such as whether the population mean is equal to 10% or not? The whole reading in the CFA curriculum is devoted to hypothesis testing.
As far as estimation is concerned, it helps us answer the following question: what is the value of a population parameter with a given probability? In other words, thanks to estimation we can for example state that with 95% probability the population mean will be within a given interval.
To fully understand estimation, we have to introduce the concepts of estimators and confidence intervals.
An estimator is a formula used to estimate the value of a parameter of a distribution. For example, if we want to estimate the average return on a population of bonds, we can draw a sample of bonds and compute the sample mean. The sample mean will be an estimator that we will use to estimate the value of the population mean. Note that the particular value that we calculate from a sample using an estimator formula is called an estimate.
To allow us to estimate the parameters of a population correctly, estimators should be:
Let's discuss the properties of good estimators that we've just enumerated one by one:
There are two types of estimation that we can perform:
We use a point estimate when we need a single value as an estimate of a population parameter. For example, the point estimate of the population mean is the sample mean. When we use a point estimate, we must simply find the correct estimator of the parameter.
In point estimation, we seek to arrive at a concrete number, that is the value of the parameter, while in the case of interval estimation we need to find a range of values that we expect to include the parameter with a particular degree of confidence. This range of values is called the confidence interval.
Confidence interval is a range of values that we expect to include the parameter with a particular degree of confidence.
For the confidence interval:
A confidence interval is a range for which one can assert with a given probability \(1-\alpha\), called the degree of confidence, that it will contain the parameter it is intended to estimate. This interval is often referred to as the \(100\times(1-\alpha)\%\) confidence interval for the parameter. Alpha is the probability of an error, namely that the parameter is not within the confidence interval.
