Chapter 5 Parameter Functions

In the previous chapters we have developed techniques for making inferences on a parameter vector $\overrightarrow{\theta }$. We now look at the situation where the quantity we are required to estimate is a one-dimensional function, $g(\overrightarrow{\theta })$, for example $g(\overrightarrow{\theta })={\theta }_1+{\theta }_2$, or $g(\overrightarrow{\theta })=\max ({\theta }_1,{\theta }_2)$.

Perhaps the simplest example is of the form $g(\overrightarrow{\theta })={\theta }_1$, say, and we have seen in the previous chapter that such inferences can be affected by the uncertainty in the other parameters. In this chapter we look at the more general situation.

Suppose that observations are available on random variables from a probability distribution with unknown parameters $\overrightarrow{\theta }$, and that $\varphi =g(\overrightarrow{\theta })$ is the only parameter of interest. The remaining parameters, termed nuisance parameters, can be written as $\overrightarrow{\lambda }=({\lambda }_1,\mathrm{\dots },{\lambda }_{d-1})$.

Though we are not really interested in estimates of $\overrightarrow{\lambda }$ or their precision, their uncertainty may affect the precision of estimation of $\widehat{\varphi }$.

Example 5.1:  Function of Normal Parameters, ctd.

Recall from Example 5.3 — we are interested in whether cables exceed a length $u$. We have $\overrightarrow{\theta }=(\mu ,\sigma )$.

In the first case

so $\overrightarrow{\lambda }=\sigma$ say. In the second case

and again $\overrightarrow{\lambda }=\sigma$.

Example 5.2:  Exponential Regression, ctd.

Referring back to Example 5.3, here we have $\overrightarrow{\theta }=(\alpha ,{\beta }_1,\mathrm{\dots },{\beta }_d)$. In this case,

Exercise: What are the nuisance parameters here?