We give expansions about the Gumbel distribution in inverse powers of n and log n for Mn, the maximum of a sample size n or n+1 when the j-th observation isμ( jn)+ej,μis any smooth trend function and the residuals{ej}are independent and identically distributed with ?∞P (e>r)≈exp(?δx)xd0 ck x?kβk=1 as x→∞. We illustrate practical value of the expansions using simulated data sets.