Confidence Intervals

Suppose that we have an independent and identically distributed (iid) sample of data x1, . . . , xn where xi iid ∼ F from some distribution F . Furthermore, suppose that the distribution F depends on some parameter θ = t(F ), and we have formed an estimate of the parameter, denoted by θ̂ = s(x), where x = (x1, . . . , xn) > is the sample of data. As a reminder, the notation θ = t(F ) denotes that the parameter is a function of the distribution, and the notation θ̂ = s(x) denotes that the estimate is a function of the sample. Given that θ̂ is a function of a random sample of data, the estimate θ̂ is a random variable that has some distribution, which we will denote by Fθ̂. Note that the distribution of the estimate, i.e., Fθ̂, will depend on (i) the form of the estimator, i.e., the function s(·) that is used to compute the estimate, (ii) the form of the data generating distribution F , and (iii) the sample size n. Given that the estimate θ̂ is a random variable, there is an inherent amount of uncertainty in the estimate. In the previous chapter, we discussed some ways to explore the quality of an estimator: bias, variance, and mean squared error. As a reminder, the bias is concerned with the location of the estimator (i.e., the difference between the estimator’s expected value and the true unknown parameter) and the variance is concerned with the spread of the estimator (i.e., the expected squared difference between the estimate and its expected value). If the sampling distribution of the estimator is normally distributed, i.e., if θ̂ ∼ N(μθ̂, σ θ̂) where μθ̂ = E(θ̂) is the expected value and σ 2 θ̂ = Var(θ̂) is the variance, then we only need the parameters μθ̂ and σ 2 θ̂ to understand how confident we can be in our estimate θ̂. However, if the sampling distribution Fθ̂ is some generic distribution, we need to know the distributional form and parameters to assess the confidence we can have in our estimate.