Expansions for the distribution and the maximum from distributions with an asymptotically gamma tail when a trend is present

We give expansions about the Gumbel distribution in inverse powers of n and log n for M _n , the maximum of a sample size n or n+1 when the j-th observation is \(\mu (\tfrac{j}{n}) + e_j \), µ is any smooth trend function and the residuals {e _j } are independent and identically distributed with

$$P(e > r) \approx \exp ( - \delta x)x^{d_0 } \sum\limits_{k = 1}^\infty {c_k x^{ - k\beta } } $$

as x→∞. We illustrate practical value of the expansions using simulated data sets.