StatisticsConfidence Intervals

It is often of limited use to know the value of an estimator given an observed collection of observations, since the single value does not indicate how close we should expect to be to . For example, if a poll estimates that a randomly selected voter has 46% probability of being a supporter of candidate A and a 42% probability of being a supporter of candidate B, then knowing more information about the distributions of the estimators is essential if we want to know the probability of winning for each candidatewhich candidate is more likely to win. Thus we introduce the idea of a confidence interval.

Solution. We expect to be a little larger than the largest observation, so we look for a confidence interval of the form . We'd like to make the interval short so that it's more informativemore likely to trap , but we can't make it too short or else it won't be likely to trap it won't be informative.

For example, the probability that all 10 observations will be less than 90% of is 34.9%. So with probability about 65.1%, we will trap the value of in the interval .

We can replace 90% with a variable and solve the equation to find that is the shortest 90% confidence interval.

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This first example was exceptionally amenable to analysis because we can solve exactly for the relevant probabilities. Estimators based on sums of observations are more typical, and in those cases we usually use the normal approximation:

Solution. A normal random variable is within standard deviations of its mean with probability . Since the mean of is , this implies that includes with probability approximately .

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Solution. The standard deviation of a Bernoulli random variable with parameter is . Therefore, the average of 1000 independent observations from such a distribution is within units of (on the number line) with probability about 95%.

Although we don't know the value of in this expression, we don't lose too much by approximating it with . Making this substitution, we get a confidence interval of . The standard deviation for B works out to the same value to the nearest tenth, so we get as a 95% confidence interval for .

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Warning. In the standard confidence interval framework (as described above), the value of the statistical functional is not random. Furthermore, the values of our estimators are random, even though they will realize concrete real-number values once the data are collected. This is opposite to the way probability questions are usually framed (asking for a given random variable how much of its probability mass lies in a particular, fixed interval).

One way to avoid the pitfall of thinking of the parameter as random is to speak of the random confidence interval trapping the value of the statistical functional, rather than speaking of the unknown parameter as falling into the given interval.

Confidence bands

If we are estimating a function-valued feature of rather than a single number (for example, a regression function), then we might want to provide a confidence band which traps the whole graph of the function with specified probability (we'll see an example, the DKW theorem, in the next section).