Year 2018, Volume 47, Issue 4, Pages 877 - 887 2018-08-01

Basic sequences and unbiased estimation in quasi power series distributions

Faqir Muhammad [1] , M. Kazim Khan [2]

23 142

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.
UMVU estimators, unbiased estimators, lacunary sequences, Schauder basis
  • 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).
Primary Language en
Subjects Mathematics
Journal Section Mathematics
Authors

Author: Faqir Muhammad

Author: M. Kazim Khan (Primary Author)

Dates

Publication Date: August 1, 2018

Bibtex @research article { hujms453112, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {Hacettepe University}, year = {2018}, volume = {47}, pages = {877 - 887}, doi = {}, title = {Basic sequences and unbiased estimation in quasi power series distributions}, key = {cite}, author = {Muhammad, Faqir and Khan, M. Kazim} }
APA Muhammad, F , Khan, M . (2018). Basic sequences and unbiased estimation in quasi power series distributions. Hacettepe Journal of Mathematics and Statistics, 47 (4), 877-887. Retrieved from http://dergipark.org.tr/hujms/issue/38872/453112
MLA Muhammad, F , Khan, M . "Basic sequences and unbiased estimation in quasi power series distributions". Hacettepe Journal of Mathematics and Statistics 47 (2018): 877-887 <http://dergipark.org.tr/hujms/issue/38872/453112>
Chicago Muhammad, F , Khan, M . "Basic sequences and unbiased estimation in quasi power series distributions". Hacettepe Journal of Mathematics and Statistics 47 (2018): 877-887
RIS TY - JOUR T1 - Basic sequences and unbiased estimation in quasi power series distributions AU - Faqir Muhammad , M. Kazim Khan Y1 - 2018 PY - 2018 N1 - DO - T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 877 EP - 887 VL - 47 IS - 4 SN - 2651-477X-2651-477X M3 - UR - Y2 - 2017 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics Basic sequences and unbiased estimation in quasi power series distributions %A Faqir Muhammad , M. Kazim Khan %T Basic sequences and unbiased estimation in quasi power series distributions %D 2018 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 47 %N 4 %R %U
ISNAD Muhammad, Faqir , Khan, M. Kazim . "Basic sequences and unbiased estimation in quasi power series distributions". Hacettepe Journal of Mathematics and Statistics 47 / 4 (August 2018): 877-887.
AMA Muhammad F , Khan M . Basic sequences and unbiased estimation in quasi power series distributions. Hacettepe Journal of Mathematics and Statistics. 2018; 47(4): 877-887.
Vancouver Muhammad F , Khan M . Basic sequences and unbiased estimation in quasi power series distributions. Hacettepe Journal of Mathematics and Statistics. 2018; 47(4): 887-877.