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

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< <1, let a a(k,) and h_m h_m (k,) be two positive numbers for which we can claim that (i) ^k(a)–^k(–a)=1–, and (ii) P{–h_m U_k:k h_m}=1–. 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 _i=1 ^k m_i ^–1, _i=1 ^km_i ^–2, and _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.