The problem of estimation of unknown mean value for some correlation models of homogeneous and isotropic random fields
Authors:
M. I. Yadrenko and N. Semenovs’ka
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 72 (2006), 193-201
MSC (2000):
Primary 60J60, 60J12, 60J25
DOI:
https://doi.org/10.1090/S0094-9000-06-00676-4
Published electronically:
September 6, 2006
MathSciNet review:
2168148
Full-text PDF Free Access
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Additional Information
Abstract: We obtain explicit formulas for the variance of a linear unbiased estimator of unknown mean value for some correlation models of homogeneous and isotropic random fields.
References
- M. Ĭ. Yadrenko, Spectral theory of random fields, Translation Series in Mathematics and Engineering, Optimization Software, Inc., Publications Division, New York, 1983. Translated from the Russian. MR 697386
- N. V. Semenovs′ka and M. Ĭ. Yadrenko, Explicit extrapolation formulas for some correlation models of homogeneous and isotropic random fields, Teor. Ĭmovīr. Mat. Stat. 69 (2003), 162–171 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 69 (2004), 175–185 (2005). MR 2110915, DOI https://doi.org/10.1090/S0094-9000-05-00624-1
- A. M. Yaglom, Certain types of random fields in $n$-dimensional space similar to stationary stochastic processes, Teor. Veroyatnost. i Primenen 2 (1957), 292–338 (Russian, with English summary). MR 0094844
- C. E. Buell, Correlation functions for wind and geopotential on isobaric surfaces, J. Appl. Meteorology 11 (1972), 51–59.
- I. S. Gradšteĭn and I. M. Ryžik, Tablitsy integralov, summ, ryadov i proizvedeniĭ, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR 0161996
- M. Ĭ. Yadrenko, Some statistical problems for the Whittle random field, Teor. Ĭmovīr. Mat. Stat. 63 (2000), 163–167 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 63 (2001), 179–183 (2002). MR 1870787
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
- Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
- Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
- M. Ĭ. Jadrenko, Isotropic random fields of Markov type in Euclidean and Hilbert spaces, Proc. All-Union Conf. Theory Prob. and Math. Statist. (Erevan, 1958) Izdat. Akad. Nauk Armjan. SSR, Erevan, 1960, pp. 263–279 (Russian). MR 0214123
References
- M. I. Yadrenko, Spectral Theory of Random Fields, “Vyshcha shkola”, Kiev, 1980; English transl., Optimization Software, Inc., Publications Division, New York, 1983. MR 0697386 (84f:60003)
- N. Semenovs’ka and M. I. Yadrenko, Explicit extrapolation formulas for some correlation models of homogeneous and isotropic random fields, Teor. Imovirnost. Matem. Statist. 69 (2003), 162–171; English transl. in Theory Probab. Math. Statist. 69 (2004), 175–185. MR 2110915 (2005k:60160)
- A. M. Yaglom, Certain types of random fields in $n$-dimensional space similar to stationary stochastic processes, Teor. Veroyatnost. i Primenen. 2 (1957), no. 3, 292–338. (Russian) MR 0094844 (20:1353)
- C. E. Buell, Correlation functions for wind and geopotential on isobaric surfaces, J. Appl. Meteorology 11 (1972), 51–59.
- I. S. Gradstein and I. M. Ryzhik, Tables of Integrals, Series, and Products, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963; English transl., Academic Press, Boston, 1994. MR 0161996 (28:5198)
- M. I. Yadrenko, Some statistical problems for the Whittle random processes, Teor. Imovirnost. Matem. Statist. 63 (2000), 163–167; English transl. in Theory Probab. Math. Statist. 63 (2001), 179–183. MR 1870787 (2002m:60097
- G. N. Watson, A Treatise on the Theory of Bessel Functions, vol. 1, Cambridge University Press, Cambridge–New York, 1944. MR 0010746 (6:64a)
- A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, vol. 1, McGraw–Hill Book Company, New York–Toronto–London, 1953. MR 0058756 (15:419i)
- ---, Higher Transcendental Functions, vol. 2, McGraw–Hill Book Company, New York–Toronto–London, 1953. MR 0058756 (15:419i)
- M. I. Yadrenko, Isotropic random fields of Markov type in Euclidean and Hilbert spaces, Proc. All-Union Conf. Theory Prob. and Math. Statist. (Erevan, 1958), Izdat. Akad. Nauk Armjan. SSR, Erevan, 1960, 263–285. (Russian) MR 0214123 (35:4974)
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Additional Information
M. I. Yadrenko
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
N. Semenovs’ka
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
semenovsky@hotmail.ru
Keywords:
Random field,
Whittle–Mattern model,
linear unbiased model
Received by editor(s):
June 5, 2004
Published electronically:
September 6, 2006
Article copyright:
© Copyright 2006
American Mathematical Society
