Volume 5, Issue 2-1, March 2016, Page: 40-48
A Novel Approach to Finding Sampling Distributions for Truncated Laws Via Unbiasedness Equivalence Principle
Nicholas A. Nechval, Department of Mathematics, Baltic International Academy, Riga, Latvia
Sergey Prisyazhnyuk, Department of Geoinformation Systems, National Research University of Information Technologies, Mechanics and Optics, St-Petersburg, Russia
Vladimir F. Strelchonok, Department of Mathematics, Baltic International Academy, Riga, Latvia
Received: Jan. 26, 2016;       Accepted: Jan. 28, 2016;       Published: Feb. 23, 2016
DOI: 10.11648/j.ajtas.s.2016050201.16      View  3709      Downloads  59
Abstract
Truncated distributions arise naturally in many practical situations. In this paper, the problem of finding sampling distributions for truncated laws is considered. This problem concerns the very important area of information processing in Industrial Engineering. It remains today perhaps the most difficult and important of all the problems of mathematical statistics that require considerable efforts and great skill for investigation. In a given problem, most would prefer to find a sampling distribution for truncated law by the simplest method available. For many situations encountered in textbooks and in the literature, the approach discussed here is simple and straightforward. It is based on use of the unbiasedness equivalence principle (UEP) that represents a new idea which often allows one to provide a neat method for finding sampling distributions for truncated laws. It avoids explicit integration over the sample space and the attendant Jacobian but at the expense of verifying completeness of the recognized family of densities. Fortunately, general results on completeness obviate the need for this verification in many problems involving exponential families. The proposed approach allows one to obtain results for truncated laws via the results obtained for non-truncated laws. It is much simpler than the known approaches. In many situations this approach allows one to find the results for truncated laws with known truncation points and to estimate system reliability in a simple way. The approach can also be used to find the sampling distribution for truncated law when some or all of its truncation parameters are left unspecified. The illustrative examples are given.
Keywords
Truncated Law, Unbiasedness Equivalence Principle, Sampling Distribution, Reliability Estimation
To cite this article
Nicholas A. Nechval, Sergey Prisyazhnyuk, Vladimir F. Strelchonok, A Novel Approach to Finding Sampling Distributions for Truncated Laws Via Unbiasedness Equivalence Principle, American Journal of Theoretical and Applied Statistics. Special Issue: Novel Ideas for Efficient Optimization of Statistical Decisions and Predictive Inferences under Parametric Uncertainty of Underlying Models with Applications. Vol. 5, No. 2-1, 2016, pp. 40-48. doi: 10.11648/j.ajtas.s.2016050201.16
Reference
[1]
B. R. Cho and M. S. Govindaluri, “Optimal screening limits in multi-stage assemblies,” International Journal Production Research, vol. vol. 40, pp. 1993–2009, 2002.
[2]
A. Jeang, “An approach of tolerance design for quality improvement and cost reduction,” International Journal Production Research, vol. 35, pp. 1193–1211, 1997.
[3]
K. C. Kapur and B. R. Cho, “Economic design and development of specification,” Quality Engineering, vol. 6, pp. 401–417, 1994.
[4]
K. C. Kapur and B. R. Cho, “Economic Design of the Specification Region for Multiple Quality Characteristics,” IIE Transactions, vol. 28, pp. 237–248, 1996.
[5]
M. D. Phillips and B. R. Cho, “Quality improvement for processes with circular and spherical specification region,” Quality Engineering, vol. 11, pp. 235–243, 1998.
[6]
M. D. Phillips and B. R. Cho, “Modeling of Optimum Specification Regions,” Applied Mathematical Modelling, vol. 24, pp. 327–341, 2000.
[7]
M. T. Khasawneh, S. R. Bowling, S. Kaewkuekool, and B. R. Cho BR (2004). “Tables of a truncated standard normal distribution: a singly truncated case,” Quality Engineering, vol. 17, pp. 33–50, 2004.
[8]
M. T. Khasawneh, S. R. Bowling, S. Kaewkuekool, and B. R. Cho, “Tables of a truncated standard normal distribution: a doubly truncated case,” Quality Engineering, vol. 18, pp. 227–241, 2005.
[9]
K. Bhowmick, A. Mukhopadhyay, and G. B. Mitra, “Edgeworth series expansion of the truncated cauchy function and its effectiveness in the study of atomic heterogeneity,” Zeitschrift fur Kristallographie, vol. 215, pp. 718–726, 2000.
[10]
U. Shmueli, “Symmetry and composition dependent cumulative distribution of the normalized structure amplitude for use in intensity statistics,” Acta Crystallography, vol. A35, pp. 282–286, 1979.
[11]
U. Shmueli, A. J. Wilson, “Effects of space group symmetry and atomic heterogeneity on intensity statistics,” Acta Crystallography, vol. A37, pp. 342–353, 1981.
[12]
D. G. Chapman, “Estimating the parameters of a truncated gamma distribution,” Ann. Math. Statist. vol. 27, pp. 498506, 1956.
[13]
K. W. Kenyon, V. B. Scheffer, and D. G. Chapman, “A Population Study of the Alaska Fur Seal Herd,” U. S. Wildlife, vol. 12, pp. 177, 1954.
[14]
S. Nadarajah, “Some Truncated Distributions,” Acta Appl. Math., vol. 106, pp. 105-123, 2009.
[15]
L. J. Bain and D. L. Weeks, “A note on the truncated exponential distribution,” Ann. Math. Statist., vol. 35, pp. 13661367, 1964.
[16]
C. A. Charalambides, “Minimum variance unbiased estimation for a class of left-truncated discrete distributions,” Sankhyā, vol. 36, pp. 397418, 1974.
[17]
T. Cacoullos, “A combinatorial derivation of the distribution of the truncated poisson sufficient statistic,” Ann. Math. Statist., vol. 32, pp. 904905, 1961.
[18]
N. A. Nechval, K. N. Nechval, G. Berzins, and M. Purgailis, “Unbiasedness equivalence principle and its applications to finding sampling distributions for truncated laws,” in Proceedings of the Second International Conference on Mathematics: Trends and Developments, Vol. I. Cairo: The Egyptian Mathematical Society, 2007, pp. 165180.
[19]
J. W. Tukey, “Sufficiency, truncation and selection,” Ann. Math. Statist., vol. 20, pp. 309311, 1949.
[20]
C. Jordan, Calculus of Finite Differences. New York: Chelsea, 1950.
[21]
R. F. Tate and R. L. Goen, “Minimum Variance Unbiased Estimation for the Truncated Poisson Distribution,” Ann. Math. Statist., vol. 29, pp. 755765, 1958.
[22]
P. R. Halmos, Measure Theory. New York: Van Nostrand, Inc., 1950.
[23]
S. Zacks, The Theory of Statistical Inference. New York: John Wiley & Sons, Inc., 1971.
[24]
A. P. Basu, “Estimation of reliability for some distributions useful in life testing,” Technometrics, vol. 6, pp. 215219, 1964.
Browse journals by subject