Interval Estimates

The notion of an interval estimate of a parameter $\theta$ with a confidence coefficient is assumed to be familiar. A point estimate, on its own, doesn't convey any indication of reliability, but a point estimate together with its standard error would do so. This idea is incorporated into a confidence interval, which is a range of values within which we are ``fairly confident'' that the true (unknown) value of the parameter $\theta$ lies. The length and location of the interval are random variables and we cannot be certain that $\theta$ will actually fall within the limits evaluated from a single sample. So the object is to generate narrow intervals which include $\theta$ with a high probability.

Examples such as

(i)

A CI for $\mu$ in a normal distribution where $\sigma$ is either known or unknown;

(ii)

A CI for $p$ where $p$ is the probability of success in a binomial distribution;

(iii)

A CI for $\sigma^2$ in a normal distribution where the mean is either known or unknown; etc.

will not be repeated here. However, we will mention the general method of construction of a confidence interval using a pivotal quantity. Further, we will find a confidence interval for a population quantile.

Suppose $\hat{\theta}_L$ and $\hat{\theta}_U$ (both functions of $X_1, \ldots, X_n$ and hence random variables) are the lower and upper confidence limits respectively, for a parameter $\theta$. Then if

$$P(\hat{\theta}_L < \theta < \hat{\theta}_U)=\gamma,$$ (8.15)

the probability $\gamma$ is called the confidence coefficient. The interval ( $\hat{\theta}_L,\,\hat{\theta}_U$) is referred to as a two-sided confidence interval, both endpoints being random variables.

It is possible to construct 1-sided intervals such that

$$P(\hat{\theta}_L < \theta)=\gamma$$

or

$$P(\theta < \hat{\theta}_U)=\gamma,$$

in which case only one end-point is random. The confidence intervals are respectively, $(\hat{\theta}_L, \infty)$, $(-\infty, \, \hat{\theta}_U)$.