PREDICTION OF RECORD-BREAKINGS

Abstract

For analyzing record statistics from a sequence of non-i.i.d. random vari- ables, a model with one or two parameters, controlling the occurrences of record- breakings, is proposed. Under the model, the number of record-breakings within n steps has a probability function including the Stirling-Carlitz polynomial of the first kind. Its expected number is still O(logn) and saturates in the long run. Waiting time to the sth occurrence also has a probability function of similar form. Under this model, methods for predicting future record-breakings are proposed, and applied to some practical data sets of weather and sports. 1. Introduction and Preliminaries 1.1. Upper new records Let (Zk) ∞=1 be a sequence of random variables (r.v.), in which Zk is an upper new record (u.n.r.), or a record breaking, if Zj <Z k , 1 ≤ j ≤ k − 1 . Following the usual convention Z1 is a u.n.r. Let Xn be the number of u.n.r.'s in (Zk) n=1 , and let Ws be the time, or sequential index, when the sth u.n.r. occurs. The theory of these 'record statistics' for a sequence of independent and identically distributed (i.i.d.) or exchangeable r.v.'s is well established, and was surveyed by Glick (1978) and Galambos (1978). In the usual applications like environment, sports and economy, the i.i.d. or exchageable assumption is too restrictive, and some generalizations have been proposed. See, for example, Nevzorov (1987) and Nagaraja (1988) for extensive surveys up to the date. In this paper, a simple approach, which is distribution-free but parametric, is proposed for the prediction of future u.n.r.'s. 1.2. Nevzorov's model Nevzorov (1984, 1987) considered the following model. Let (Zk) ∞=1 be a sequence of independent but nonidentically distributed r.v.'s, and let Zk itself be the maximum of a random sample of varying size αk from a common continuous distribution function (d.f.) H(·), that is, Zk has the d.f. H αk (·). Let Yk ,k = 2, 3 ,... be binary r.v.'s such that if Zk is a u.n.r. Yk =1 , elseYk =0 . Then (Yk) ∞=2 are independent, and