Application of laplace approximation to record values

Let Y1, Y2…,Yn,. be a sequence of independent and identically distributed random variables with a continuous distribution function F(y) and corresponding probability density function f(y). If {K(n), n > 1} is defined by K(1) = 1, K(n) = min{j|j > K(n - 1),Yj > YK(n- 1)} for n > 2, then (Y(n) = F/K(n),n > 1} is said to be a sequence of record values, We develop accurate approximations based on the Laplace approximation method as developed by Tierney and Kadane (1986) to the predictive distribution of a future observation based on (1) the availability of just a sequence of record values, and (2) the full likelihood. It is seen that this approximation is remarkable in small sample sizes leading to tractable results for any distribution of the observations.