THE NUMBER OF PEAKS IN A STATIONARY SAMPLE AND ORTHANT PROBABILITIES

Abstract

Abstract. The essence of this paper is the exact evaluation of var Sn, where Sn is the number of peaks in a segment of n readings from a stationary Gaussian autoregressive moving‐average (ARMA) process, and of the asymptotic normality of (Sn–ESn)/(var Sn)1/2. The emphasis is on the AR(1) and MA(1) cases, motivated by Stigler (Estimating serial correlation by visual inspection of diagnostic plots. Am. Statistician 40 (1986), 111–16). The evaluation of var Sn is based on a discussion of closed‐form calculation of orthant probabilities for a zero mean quadrivariate normal with correlation structure new to the literature. Related issues are the power of the peaks test against stationary alternatives and the good fit of the normal even for small n. validated by simulation.