Basic sequences and unbiased estimation in quasi power series distributions

Year 2018, Volume: 47 Issue: 4, 877 - 887, 01.08.2018

Faqir Muhammad M. Kazim Khan

Abstract

By using results from function space theory we give a characterization of when lacunary quasi power series sequences are basic in $C[0, 1]$. The paper discusses the links with unbiased estimable functions and the subspaces generated by the density of the lacunary quasi power series distributions. The paper also provides the rates of convergence of all the moments of the classic odds ratio estimator. This extends some known
results in Bleimann, Butzer and Hahn's approximation operator.

Keywords

UMVU estimators, unbiased estimators, lacunary sequences, Schauder basis

References

• Berg, C. and Vignat, C. Linearization coffiecients of Bessel polynomials and properties of Student t-distribution, Constructive Approximation, 27, 1532, 2008.
• Bianchi, G. and Sorrentino, R. Electronic Filter Simulation and Design, (McGraw Hill, NY. 2007).
• Bleimann, G.; Butzer, P. L. and Hahn, L. A Bernstein-type operator approximating continuous functions on the semi-axis, Nederl. Akad. Wetensch. Indag. Math., 42, 255262, 1980.
• Della Vecchia, B. Some properties of a rational operator of Bernstein-type, In: Progress in Approximation Theory, 177185, (Academic Press, Boston, 1991).
• Eno, Per. A counter example to the approximation problem in Banach spaces, Acta Math- ematica, 130 (1), 309317, 1973.
• Gurariy, V.I. and Matsaev, V.I. Lacunary power sequence in the spaces $C$ and $L_p$, Izv. Akad. Naud SSR Ser. Mat. 30, 314, 1966 (in Russian). Amer. Math. Soc. Trans. 72, 921, 1968 (in English).
• Gurariy, V.I. and Lusky, W. Geometry of Müntz Spaces and Related Questions, Lect. Notes Math. 1870, Springer-Verlag, Berlin, 2005.
• Joshi, S.W. and Park, C.J. Minimum variance unbiased estimation for truncated power series distribution, Sankhya: The Indian Journal of Statistics, Series A, 36(3), 305314, 1974.
• Khan, R. A. A note on a Bernstein-type operator of Bleimann, Butzer and Hahn, J. Approx. Theory, 53, 295303, 1988.
• Khan, R. A. Reverse martingales and approximation operators, J. Approx. Theory, 80, 367377, 1995.
• Khatri, C.G. On certain properties of power series distributions, Biometrika, 46, 486490, 1959.
• Krall, H. L., and Fink, O., A new class of orthogonal polynomials: The Bessel polynomials, Trans. Amer. Math. Soc., 65, 100115, 1948.
• Lehmann, E. L. and Casella, George. Theory of Point Estimation, 2nd ed. (Springer-Verlag, N.Y. 1998).
• Noack, A. A class of random variables with discrete distributions, Ann. Math. Stat. 21, 127132, 1950.
• Patil, G. P. Minimum Variance Unbiased Estimation and Certain Problems of Additive Number Theory, Ann. Math. Stat. 34, 10501056, 1963.
• Patil, G. P. and Joshi, S. W. Further Results on Minimum Variance Unbiased Estimation and Additive Number Theory, Ann. Math. Stat. 41(2), 567575, 1970.
• Patil, G. P. and Wani, J. K. On Certain Structural Properties of the Logarithmic Series Distribution and the First Type Stirling Distribution, Sankhya: The Indian Journal of Sta- tistics, Series A, 27(2/4), 271280, 1965.
• Plackett, R. L. The truncated Poisson distribution, Biometrics, 9, 485488, 1953.
• Tate, R. F. and Goen, R. L. Minimum Variance Unbiased Estimation for the Truncated Poisson Distribution, Ann. Math. Stat., 29, 755765, 1958.
• Wijsman, R. A. On the Attainment of the Cramér-Rao Lower Bound, Ann. Statist., 1(3), 538542, 1973.
• Zacks, S. Theory of Statistical Inference, (John Wiley & Sons, N.Y. 1971).