1.1 Parametric Inference, Recalled

Parametric and non-parametric are the two broad classifications of statistical procedure. A parametric procedure assumes the observations come from a distribution of known form, indexed by finitely many unknown parameters, and draws its conclusions about those parameters.

The normal distribution. The best known family is the normal, indexed by the mean \(\mu \) and variance \(\sigma ^{2}\). Given independent observations from it, the sample mean \(\overline {X}\) is a point estimate of \(\mu \); the \(t\)-test measures the strength of evidence against an a priori hypothesised value \(\mu _0\); and a confidence interval gives a range of plausible values for the true mean. All of this is parametric inference, and all of it leans on the assumed normality.

The normal distribution is strictly appropriate only to certain continuous measurements, though it often works well enough when the measurement scale is fine and the distribution roughly symmetric.

The binomial distribution. For counts, the binomial family has parameters \(n\), the number of observations, and \(P\), the probability that the event of interest occurs at any one of them. The number of occurrences \(R\), with \(0 \leq R \leq n\), satisfies \[P(R = r) = \binom {n}{r}P^{r}(1-P)^{\,n-r},\] and \(\widehat {P} = r/n\) is the point estimate of \(P\). Again one may test an a priori value \(P_0\) or form a confidence interval.

The binomial is relevant to counts in discrete outcome situations — the number of males in a family of size \(n\), for instance. The outcome being counted is conventionally called the “favourable” event, a name that sits awkwardly when the event is a positive diagnosis of an illness.

Other families. Uniform, multinomial, Poisson, exponential, double exponential, gamma and many others. Sometimes theory or experience makes it reasonable to assume a particular family; often it does not.

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