Example 3.1: Simple Normal Data.
Let be IID random variables with common distribution .
In this case the
parameter vector represents the
unknown mean and standard deviation of the Normal population.
Example 3.2: Simple Linear Regression.
Let be independent random variables such that the distribution of is given by for .
In this case . This is a generalisation of the previous example in which each of the has a normal distribution with the same variance but a different mean.
In fact, this is exactly the simple
linear regression model discussed in MATH235. Though each of the
has a different distribution, likelihood techniques are still
applicable in such circumstances; we will find that this is also the
case in the multi-parameter situation.
Example 3.3: Function of Normal Parameters.
As in the simple normal case, let be IID random variables with common distribution .
It is often the case that inference is not required for the parameter vector itself, but for some function of that parameter vector.
To give a concrete example: suppose the ’s correspond to cable lengths which are unsuitable if they exceed a specified length .
There are two interesting possibilities:
If is fixed, then it will be the probability a cable is unsuitable that is required. This is given by
where , which is equal to, say,
Thus, the problem amounts to making inference on a function of .
Alternatively, we may be required to choose in such a way that the probability of a cable being unsuitable is equal to a specified value of . In this case, inference is required for , where
Example 3.4: ‘Exponential’ Regression.
Let be independent random variables such that
where
for , where the are covariates (explanatory variables).
In this case . Moreover, suppose that the data relate to a medical trial in which represents the age at death of a person with attributes corresponding, for example, to smoking status, sex, weight, etc. Thus,
and so on. Hence, gives a measure of the extent of attribute on lifetime.
In medical trials it will often be the case that the main interest is not in the complete vector , but perhaps just a single component, say, corresponding to the effect of smoking status on lifetime. The issue then is how best to make inference on the single parameter in the presence of the other nuisance parameters. We will deal with this issue later in the course.