Nonparametric Predictive Inference

Excerpt

Nonparametric predictive inference (NPI) is a statistical method based on Hill’s assumption \({A}_{(n)}\) Hill (1968), which gives a direct conditional probability for a future observable random quantity, conditional on observed values of related random quantities (Augustin and Coolen 2004; Coolen 2006). Suppose that \({X}_{1},\ldots ,{X}_{n},{X}_{n+1}\) are continuous and exchangeable random quantities. Let the ordered observed values of \({X}_{1},\ldots ,{X}_{n}\) be denoted by \({x}_{(1)} < {x}_{(2)} < \ldots < {x}_{(n)} < \infty \), and let \({x}_{(0)} = -\infty \) and \({x}_{(n+1)} = \infty \) for ease of notation. For a future observation \({X}_{n+1}\), based on \(n\) observations, \({A}_{(n)}\) (Hill 1968) is \({A}_{(n)}\) does not assume anything else, and is a post-data assumption related to exchangeability. Hill discusses \({A}_{(n)}\) in detail. Inferences based on \({A}_{(n)}\) are predictive and nonparametric, and can be considered suitable if there is hardly any knowledge about the random quantity of interest, other than the \(n\) observations, or if one does not want to use such information, e.g., to study effects of additional assumptions underlying other statistical methods. \({A}_{(n)}\) is not sufficient to derive precise probabilities for many events of interest, but it provides optimal bounds for probabilities for all events of interest involving \({X}_{n+1}\). These bounds are lower and upper probabilities in the theories of imprecise probability (Walley 1991) and interval probability (Weichselberger 2001), and as such they have strong consistency properties (Augustin and Coolen 2004). NPI is a framework of statistical theory and methods that use these \({A}_{(n)}\)-based lower and upper probabilities, and also considers several variations of \({A}_{(n)}\) which are suitable for different inferences. For example, NPI has been presented for Bernoulli data, multinomial data and lifetime data with right-censored observations. NPI enables inferences for \(m \geq 1\) future observations, with their interdependence explicitly taken into account, and based on sequential assumptions \({A}_{(n)},\ldots ,{A}_{(n+m-1)}\). NPI provides a solution to some explicit goals formulated for objective (Bayesian) inference, which cannot be obtained when using precise probabilities (Coolen 2006). NPI is also exactly calibrated (Lawless and Fredette 2005), which is a strong consistency property, and it never leads to results that are in conflict with inferences based on empirical probabilities. …