soxumis saxelmwifo universitetis S r o m e b i VII

ˆsup F nxa,bxFx 0in probability (a.s.) for any fixed interval a, b0,1n such that a, bn, n n0.since there may exist0. The conditions of Theorem 2 are ful-Assume thatfilled:andh n , 0 n1121n h n if 0 ,2p22 pp n hn if 0 , p 2 .2 p4. Estimation of momentsIn considering the problem, there naturally arises a question of estimationof the integral functionals of F x, for example, moments m,m 1:1t m tm 1 1F dt .As estimates for we consider the statisticsmm0 t tn 1hm 1 m1j 1ˆ nm 1j t KF2ntdt.n j1h hhTheorem 3. Let xK x0 outside the interval 1,1 K satisfy condition 1 0 and, in addition to this, . If nh as n , then ˆnkisan asymptotically unbiased, consistent estimate for mand moreoverthatn ˆ E ˆnmProof. SincenmdNK xhas 1,1 12 2 2m20,1 , m t Ft1 Ft0dt . as a support, we establish from (5)20