MINIMUM VARIANCE UNBIASED ESTIMATION OF THE SCALE PARAMETER OF EXPONENTIAL DISTRIBUTIONS AND RELATED LOGARITHMIC INTEGRALS
Keywords:
unbiased estimator, minimum variance, exponential scale parameter, logarithmic integrals, density of primesAbstract
The paper tackles two(2) problems related to the exponential distribution. The first concerns a detailed derivation of the minimum variance unbiased estimator of the scale parameter. The second focuses on the relationship of the expected value of the reciprocal of an exponential random variable which is shown to be equivalent to evaluating the logarithmic integral and the density of primes as found in the Prime Number Theorem. In the first problem, we showed that the minimum variance unbiased estimator of the scale parameter has a variance larger than the Cramer-Rao lower bound. In the second problem, we demonstrated that the expected value of the reciprocal of an exponential random variable also obtains the density of primes less or equal to a given large number x. The minimum variance unbiased estimator found in the first problem can then be utilized to find such an approximation to the density of primes for the second problem. The second problem provides a new way of viewing the problem of finding the density of primes less or equal to x.
References
Granville, Andrew (1995). "Harold Cramér and the distribution of prime numbers" (PDF). Scandinavian Actuarial Journal 1: 12–28.
Hardy, G. H. & Littlewood, J. E. (1916). "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes". Acta Mathematica 41: 119–196.
Lehman, P. (1984) Theory of Point Estimation. (Wiley Series, New York, 151-153)
Padua, R., & Libao, M. (2017). ON STOCHASTIC APPROXIMATIONS TO THE DISTRIBUTION OF PRIMES AND PRIME METRICS. Journal Of Higher Education Research Disciplines, 1(1), 33-38. Retrieved from http://nmsc.edu.ph/ojs/index.php/jherd/article/view/8
Rao, S. (2010) . Advanced Statistical Inference. (Unpublished Lecture Notes, 7-10).
Regalado, D., & Azura, R. (2017). ON A MIXED REGRESSION ESTIMATOR FOR THE DENSITY OF PRIME GAPS. Journal Of Higher Education Research Disciplines, 1(2), 48-57. Retrieved from http://nmsc.edu.ph/ojs/index.php/jherd/article/view/12
Schoenfeld, Lowell (1976). "Sharper Bounds for the Chebyshev Functions ?(x) and ?(x). II". Mathematics of Computation 30 (134): 337–360.
Tao, T. (2013). “Structure and Randomness in the Distribution of Primes” (Video Lectures, University of California)
Temme, N. M. (2010), "Exponential, Logarithmic, Sine, and Cosine Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press
Von Koch, Helge (1901). "On the distribution of prime numbers”. Acta Mathematica (in French) 24 (1): 159–182
