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Estimation for the Mean of Generalizeed Random Functions Alexander Ponomarenko Kyiv Taras Schevchenko University Department of Probability and Mathematical Statistics Volodimirska, 64 Kyiv, Ukraine [email protected] Denote by D(G) the space of test functions with compact supports on a locally compact DEHOLDQ JURXS * VHH 0DXULQ /HW ; ∈ D(G) be a real wide-sense stationary ndimensional generalized random function on G with zero mean, random spectral measure Z and spectral measure F, and let L2 )KHUH LVWKHFKDUDFWHUJURXSRI*EH+LOEHUWVSDFHRIVTXDUH integrable multivariate functions with respect to F (see Ponomarenko (1974)). &RQVLGHU D JHQHUDOL]HG UDQGRP IXQFWLRQ < $ ; VXSS ⊂ ZKHUH $ LV XQNQRZQUHDOQGLPHQVLRQDOGHWHUPLQDWHJHQHUDOL]HGIXQFWLRQRQ* LVDFRPSDFWVXEVHWRI*,Q this investigation we are interested in conditions for existence of and the form of the best linear XQELDVHGHVWLPDWRUV%/8( VIRUOLQHDUIXQFWLRQDOVRIWKHPHDQ$ RIWKHIXQFWLRQ< WKDWDUH FRQVWUXFWHGIURPWKHREVHUYDWLRQV^< VXSS ⊂ `XQGHUWKHDVVXPSWLRQWKDWSRVVLEOHYDOXHVRI $ ∈ D(G) form a closed subspace H of L2 )JHQHUDWHGE\WKHIXQFWLRQZKLFKFRPSRQHQWV DUHWKH)RXULHUWUDQVIRUPVRI VXSS ⊂ The integral characterization of BLUE lˆ for continuous linear functional l(A), A ∈ H by the mean of the measures Z and F is given, and the variance of lˆ (A) is obtained. This results are applied to solving of regression problem, when the space H has finite dimension with basis {Ak,k=1,…,m} and any element A ∈ H has a linear representation m $ ∑k =1 k Ak k=(A,A*k), where A*k , k=1,…,m is the system dual to Ak, k=1,…,m. In this case the characterization of BLUE Â and BLUE’s α̂ are given. REFERENCES Maurin Krzysztof (1968). General Eigenfunction Expansions and Unitary Representation of Topological Groups. Polish Scientific Publishers. Warszawa. Ponomarenko A.I. (1974). Harmonic analysis of generalized wide-sense homogeneous random fields on a locally compact commutative group, Theor. Probab. and Math. Statist., 3, 119-137. ESTIMATION DE LA MOYENNE DES FONCTIONS GENERALISES ALEATOIRES On etudie les estimateurs de la moyenne des fonctions generalisés aléatoires sur groupe localement compact abélien.
