### Discrete Power Distributions and Inference Using Likelihood

#### Abstract

Discrete power distributions are proposed and studied, by considering the positive jumps on the discontinuities of an original discrete distribution function. Inequalities in moments and distribution functions are studied, allowing the definition of discrete intermediate distributions that lie between an original distribution and a power distribution. Original uniform, binomial, Poisson, negative binomial, and hypergeometric distributions are considered, to propose new power and intermediate distributions. Stochastic orders and unimodality are discussed. Estimation problems using likelihood are investigated. Simulation experiments are performed, to evaluate the bias and the mean square error of the maximum likelihood estimates, that are numerically calculated, with classic tools for numerical optimization.

#### Keywords

#### Full Text:

PDF (English)#### References

N. BALAKRISHNAN, V. B. NEVZOROV (2003). A Primer on Statistical Distributions. John Wiley & Sons, Hoboken, New Jersey.

F. BELZUNCE, C. MARTINEZ-RIQUELME, J. MULERO (2016). An Introduction to Stochastic Orders. Academic Press, San Diego, California.

R. P. BRENT (1973). Algorithms for Minimization without Derivatives. Prentice-Hall, Englewood Cliffs, New Jersey.

S. CHAKRABORTY, R. D.GUPTA (2015). Exponentiated geometric distribution: Another generalization of geometric distribution. Communications in Statistics - Theory and Methods, 44, pp. 1143–1157.

F. DALY, R. E. GAUNT (2016). The Conway-Maxwell-Poisson distribution: Distributional theory and approximation. Latin American Journal of Probability and Mathematical Statistics, 13, pp. 635–658.

S. DHARMADHIKARI, K. JOAG-DEV (1988). Unimodality, Convexity, and Applications. Academic Press, San Diego, California.

S. R. DURRANS (1992). Distributions of fractional order statistics in hydrology. Water Resources Research, 28, pp. 1649–1655.

Y. M. GÓMEZ, H. BOLFARINE (2015). Likelihood-based inference for the power halfnormal distribution. Journal of Statistical Theory and Applications, 14, pp. 383–398.

R. D. GUPTA, R. C. GUPTA (2008). Analyzing skewed data by power normal model. Test, 17, pp. 197–210.

G. HARDY, J. E. LITTLEWOOD, G. PÓLYA (1951). Inequalities. Second Edition, Cambridge University Press, Cambridge.

J. L. W. V. JENSEN (1906). Sur les fonctions convexes et les inégalités entre les valeurs moyenne. Acta Mathematica, 30, pp. 175–193.

N. L. JOHNSON, A. W. KEMP, S. KOTZ (2005). Univariate Discrete Distributions. Third Edition, JohnWiley & Sons, Hoboken, New Jersey.

M. C. JONES (2004). Families of distributions arising from distributions of order statistics (with discussion). Test, 13, pp. 1–43.

J. B. KADANE (2016). Sums of possibly associated Bernoulli variables: The Conway-Maxwell-binomial distribution. Bayesian Analysis, 1, pp. 403–420.

E. L. LEHMANN (1953). The power of rank tests. The Annals of Mathematical Statistics, 24, pp. 23–43.

E. L. LEHMANN, G. CASELLA (1998). Theory of Point Estimation. Second Edition, Springer, New York.

R. MIURA, H. TSUKAHARA (1993). One-sample estimation for generalized Lehmann's alternative models. Statistica Sinica, 3, pp. 83–101.

A. MÜLLER, D. STOYAN (2002). Comparison Methods for Stochastic Models and Risks. John Wiley & Sons, Chichester, England.

S. NADARAJAH, S. A. A. BAKAR (2016). An exponentiated geometric distribution. Applied Mathematical Modelling, 40, pp. 6775–6784.

S. NADARAJAH, S. KOTZ (2006). The exponentiated type distributions. Acta Applicandae Mathematicae, 92, pp. 97–111.

J. A. NELDER, R. MEAD (1965). A simplex method for function minimization. The Computer Journal, 7, pp. 308–313. Correction: 8, p. 27.

Y. PAWITAN (2001). In All Likelihood: Statistical Modelling and Inference Using Likelihood. Clarendon Press, Oxford.

A. PEWSEY, H.W.GÓMEZ, H. BOLFARINE (2012). Likelihood-based inference for power distributions. Test, 21, pp. 775–789.

N. S. PISKUNOV (1979). Calcolo Differenziale e Integrale. Volumi 1 e 2, Seconda Edizione, Editori Riuniti, Edizioni MIR, Roma, Mosca.

R CORE TEAM (2017). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/.

S. M. ROSS (2013). Simulation. Fifth Edition, Academic Press, San Diego, California.

G. SHMUELI, T. P. MINKA, J. B. KADANE, S. BORLE, P. BOATWRIGHT (2005). A useful distribution for fitting discrete data: Revival of the Conway-Maxwell-Poisson distribution. Journal of the Royal Statistical Society. Series C, 54, pp. 127–142.

G. R. SHORACK (2000). Probability for Statisticians. Springer-Verlag, New York.

M. SPIVAK (1994). Calculus. Third Edition, Cambridge University Press, Cambridge.

A. WALD (1949). Note on the consistency of the maximum likelihood estimates. The Annals of Mathematical Statistics, 20, pp. 595–601.

DOI: 10.6092/issn.1973-2201/8598