📂Mathematical Statistics

Easy Definition of Confidence Intervals

Definition ¹

Let the probability density function $f (x; \theta)$ of the random variable $X$ and the samples $X_{1} , \cdots , X_{n}$ with a Confidence Coefficient $\alpha \in (0,1)$ be given. $$ L := L \left( X_{1} , \cdots , X_{n} \right) \\ U := U \left( X_{1} , \cdots , X_{n} \right) $$ It is said that the statistic $L < U$ is defined as above, then the interval $(L,U) \subset \mathbb{R}$ satisfying the following is called the $( 1 - \alpha)100 \%$ confidence interval for the parameter $\theta$. $$ 1-\alpha = P \left[ \theta \in \left( L,U \right) \right] $$

Explanation

In fact, the confidence interval can be generalized into a broader space as a confidence region, and from the standpoint of learning basic statistics, defining it mathematically like above could be unnecessarily complicated. However, now that the definitions of random variables, samples, and statistics have become mathematically solid, we can continue a more rigorous discussion.

What’s important in this definition of confidence intervals is that $L$ and $U$ are statistics. Although the confidence interval is presented as an interval, which allows us to perceive it spatially, in fact, the confidence intervals that were calculated in statistics before mathematical statistics would have been found by calculating its upper $U$ and lower $L$ limits. For example, if $X$ follows the standard normal distribution $N(0,1)$, the $1-\alpha$ confidence interval for $\mu$ would have been calculated as follows. $$ L := \overline{X} - {{ S } \over { \sqrt{n} }} z_{\alpha/2} \\ U := \overline{X} + {{ S } \over { \sqrt{n} }} z_{\alpha/2} $$ Let’s pay attention to the fact that what actually “moves” in this interval estimation for $\mu$ to find the confidence interval are the ends of the interval $L,U$. At a glance, it might seem like we are interested in the probability of $\mu$ randomly changing and falling within the confidence interval, but calling $L,U$ a “statistic” is precisely to prevent such misinterpretations.

See Also

• Complex definitions of confidence intervals