Theory of Statistical Prediction

Abstract

The author started the studies of problems of statistical prediction around 1965 and has written a series of papers on them, giving talks in academic meetings and seminars and also publishing papers. This chapter is a reorganization of the main results of those studies. The problems of ‘prediction’ for time-series data are not dealt within this chapter. We are mainly interested in simpler cases where the data $X_1,\dots ,X_n$ and the value Y to be predicted are jointly distributed real random variables, in most cases independently distributed or with rather simple structure. The purpose of our study is to construct an analogous theory of prediction corresponding to the theory of statistical inference on parameters. It has been established that in correspondence to the theory of point estimation and of interval estimation, a quite similar theory of point prediction and interval prediction can be constructed and corresponding to the theory of testing hypothesis and of multiple decisions, the theory of dual or multiple-choice prediction can be constructed.

This chapter is a reorganization of the main results of Takeuchi (1975) Theories of Statistical Prediction (Tōkei-teki Yosoku-ron) and Akahira and Takeuchi (1980) A note on prediction sufficiency (adequacy) and sufficiency. Austral. J. Statist. 22 (3), 332–335.