Extremes of locally stationary chi-square processes with trend

Abstract

Chi-square processes with trend appear naturally as limiting processes in various statistical models. In this paper we are concerned with the exact tail asymptotics of the supremum taken over (0, 1) of a class of locally stationary chi-square processes with particular admissible trends. An important tool for establishing our results is a weak version of Slepian's lemma for chi-square processes. Some special cases including squared Brownian bridge and Bessel process are discussed.

Tail asymptotic behavior of the supremum of a class of chi-square processes

“…This section concerns a result derived in [19], which is crucial for the derivation of (6). Based on the discussions therein, we shall consider 1/2 0 (C(s)) 1/α ds = ∞ or 1 1/2 (C(s)) 1/α ds = ∞.…”

“…For the subsequent discussions we present a tailored version of Theorem A.1 of [19], focusing on |f (S)| = ∞. We define…”

“…The supremum of chi-square process appears naturally as limiting test statistic in various statistical models; see, e.g., [1,2,3,4]. It also plays an important role in reliability applications in the engineering sciences, see [5,6,7,8] and the references therein.…”

“…Organization of the rest of the paper: In Section 2 we present a preliminary result which is a tailored version of Theorem A.1 in [19]. The main results are given in Section 3, followed by examples.…”

“…This completes the proof. We shall first discuss the finiteness of sup t∈(0,1] Further, we have from the calculations in the proof of Corollary 2.7 in [19] that E B H (t), B H (s) < 1 for s = t, s, t ∈ (0, 1], and conditions B(0) and C(0) are satisfied by B H (t), t ∈ (0, 1]. Moreover, we have that J c,wρ,ε (0) ≤ 1 2 1/(2H) 1 (ln (e 2 /ǫ)) ρc 1/2 ε∧1/2 1 t dt + ε 0 1 t (ln (e 2 /t)) ρc dt < ∞ for any c > 1/ρ.…”