Percentage Points of the Largest Among Student's T Random Variable

Abstract

Let us consider k(≥ 2) independent random variables U₁, . . . ,U_k where U_i is distributed as the Student's t random variable with a degree of freedom m_i, i=1, . . . ,k. Here, m₁, . . . ,m_k are arbitrary positive integers. We denote m=(m₁, . . . ,m_k) and U_k:k=max {U₁, . . . ,U_k}, the largest Student's t random variable. Having fixed 0<α <1, let a≡ a(k,α) and h_m ≡ h_m (k,α) be two positive numbers for which we can claim that (i) Φ^k(a)−Φ^k(−a)=1−α, and (ii) P{−h_m≤ U_k:k≤ h_m}=1−α. Then, we proceed to derive a Cornish–Fisher expansion (Theorem 3.1) of the percentage point h_m. This expansion involves “a” as well as expressions such as Σ_i=1 ^k m_i ⁻¹, Σ_i=1 ^km_i ⁻², and Σ_i=1 ^k m_i ⁻³. The corresponding approximation of h_m is shown to be remarkably accurate even when k or m₁, . . . ,m_k are not very large.