Large Quantile And Endpoint Estimation Of A Distribution And The Convergence Of Density Function Of The Double Extreme Of The Random Sequence With Two Subscript

In this paper,two questions on exetreme value theory are investigated as follows: I. The asymptotic properties of the large quantile and endpoint of a distribution:Let {Xi} be an i.i.d random sequence with common distribution F(x) which is unknown. Denote Mn (k) as the k - th maxima of X1 … Xn. If there exist contants an > 0, bn, such thatfor some non-degenerated distribution function H(x), then H(x) must be the following type:H(x) = Hr(x) = exp(-(l +rx) 1 + rx > 0 7 ∈ RHr(x) is said as exetreme value distribution,and 7 is exetreme value index. F(x) is said in the domain of atrraction of Hr(x), denoted as F ∈ D(H}.In this paper, the large quantile of F(x) is estimated by controling the convergence rate of regularly varing function; As 7 < 0, the endpoint of F(x) is also estimated under second regularly varing condition. Moreover, the results which have been proved enable us to construct a asymptotic confidence interval for the large quantile and endpoint of F(x). H. The convergence of the density function of the double exetreme value.Let {Xm,n;m,n > l}be an i.i.d random sequence with two subscript, having common absolutely continuous distribution function F(x). Let M(k)(m,n) be the k - th maxima of Xm,1,…, Xm,n, and Y(l)(m,n;k) be the l-th maxima of M(k)(1, n),… , M(k)(m,n). Y(l)m,n;k) is called the l-th double extreme value of the k - th maxima.The necessary and sufficient conditions on the convergence of the normalized density function of Y(l)(m, n;k) is given in two cases: m and n both approach to infinityjm approaches to infinity before n does. We also give the sufficient conditions on the convergence of another normalized density function of Y(l) (m, n; k) as n approaches to infinity before m does.