On a Modified Yule Distribution
Keywords:Generalized likelihood ratio test, Maximum likelihood estimation, Model Selection, Probability generating function, Simulation
A modified version of Yule distribution is introduced here and discuss some of its properties by deriving expressions for its probability generating function, raw moments, factorial moments etc. Certain recursion formulae for its probabilities, raw moments and factorial moments are also developed. Various methods of estimation are employed for estimating the parameters of the distribution and certain test procedures are suggested for testing the significance of the additional parameters of the distribution. The distribution has been fitted to certain real-life data sets for illustrating its usefulness, compared with certain existing models available in the literature. Further, a simulation study is conducted for assessing the performance of the maximum likelihood estimators.
F. A.HAIGHT (1966). Some statistical problems in connection with word association data. Journal of Mathematical Psychology, 3, no. 1, pp. 217–233.
M. HEASMAN, D. REID (1961). Theory and observation in family epidemics of the common cold. British journal of preventive & social medicine, 15, no. 1, p. 12.
M. G. KENDALL (1961). Natural law in the social sciences : Presidential address, delivered to the Royal Statistical Society on Wednesday,
November 16th, 1960. Journal of the Royal Statistical Society, 124, no. 1, pp. 1–16.
C. S. KUMAR, B. U. NAIR (2014). A three parameter hyper-Poisson distribution and some of its properties. Statistica, 74, no. 2, pp. 183–198.
C. S. KUMAR, A. RIYAZ (2013). On zero-inflated logarithmic series distribution and its modification. Statistica, 73, no. 4, pp. 477–492.
A. M. MATHAI, H. J. HAUBOLD (2008). Special Functions for Applied Scientists. Springer-Verlag, New York.
A. MISHRA (2009). On a generalized Yule distribution. Assam Statistical Review, 23, pp. 140–150.
C. R. RAO (1947). Minimum variance and the estimation of several parameters. In Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press, Cambridge, vol. 43, no. 2, pp. 280–283.
J. RIORDAN (1968). Combinatorial identities. Wiley, New York.
H. A. SIMON (1955). On a class of skew distribution functions. Biometrika, 42, no. 3/4, pp. 425–440.
H. A. SIMON (1960). Some further notes on a class of skew distribution functions. Information and Control, 3, no. 1, pp. 80–88.
L. J. SLATER (1966). Generalized Hypergeometric Functions. Cambridge University Press, Cambridge.
C. WILLIAMS (1943). The numbers of publications written by biologists. Annals of Eugenics, 12, no. 1, pp. 143–146.
E. XEKALAKI (1983). A property of the Yule distribution and its applications. Communications in Statistics-Theory and Methods, 12, no. 10, pp. 1181–1189.
G. U. YULE (1925). A mathematical theory of evolution, based on the conclusions of Dr. JC Willis, FRS. Philosophical Transactions of the Royal Society of London, Series B, 213, pp. 21–87.