Confidence Interval - Statistical Theory - Definition

Definition

Let X be a random sample from a probability distribution with statistical parameters θ, which is a quantity to be estimated, and φ, representing quantities that are not of immediate interest. A confidence interval for the parameter θ, with confidence level or confidence coefficient γ, is an interval with random endpoints (u(X), v(X)), determined by the pair of random variables u(X) and v(X), with the property:

The quantities φ in which there is no immediate interest are called nuisance parameters, as statistical theory still needs to find some way to deal with them. The number γ, with typical values close to but not greater than 1, is sometimes given in the form 1 − α (or as a percentage 100%·(1 − α)), where α is a small non-negative number, close to 0.

Here Pr_θ,φ indicates the probability distribution of X characterised by (θ, φ). An important part of this specification is that the random interval (u(X), v(X)) covers the unknown value θ with a high probability no matter what the true value of θ actually is.

Note that here Pr_θ,φ need not refer to an explicitly given parameterised family of distributions, although it often does. Just as the random variable X notionally corresponds to other possible realizations of x from the same population or from the same version of reality, the parameters (θ, φ) indicate that we need to consider other versions of reality in which the distribution of X might have different characteristics.

In a specific situation, when x is the outcome of the sample X, the interval (u(x), v(x)) is also referred to as a confidence interval for θ. Note that it is no longer possible to say that the (observed) interval (u(x), v(x)) has probability γ to contain the parameter θ. This observed interval is just one realization of all possible intervals for which the probability statement holds.