Kullback-Leibler Information of the Equilibrium Distribution Function and its Application to Goodness of Fit Test Park, Sangun; Choi, Dongseok; Jung, Sangah;
Kullback-Leibler (KL) information is a measure of discrepancy between two probability density functions. However, several nonparametric density function estimators have been considered in estimating KL information because KL information is not well-defined on the empirical distribution function. In this paper, we consider the KL information of the equilibrium distribution function, which is well defined on the empirical distribution function (EDF), and propose an EDF-based goodness of fit test statistic. We evaluate the performance of the proposed test statistic for an exponential distribution with Monte Carlo simulation. We also extend the discussion to the censored case.
Cumulative residual KL information;exponential distribution;Fisher information;Goodness of fit test;
Abo-Eleneen, Z. A. (2011). The entropy of progressively censored samples, Entropy, 13, 437-449.
Andrews, F. C and Andrews, A. C. (1962). The form of the equilibrium distribution function, Trans-actions of the Kansas Academy of Science., 65, 247-256.
Baratpour, S. and Rad, A. H. (2012). Testing goodness-of-fit for exponential distribution based on cumulative residual entropy, Communications in Statistics-Theory and Methods, 41, 1387-1396.
Bowman, A. W. (1992). Density based tests for goodness-of-fit, Journal of Statistical Computation and Simulation, 40, 1-13.
Dudewicz, E. and van der Meulen, E. (1981). Entropy based tests of uniformity, Journal of the American Statistical Association, 76, 967-974.
Ebrahimi, N., Habibullah, M. and Soofi, E. S. (1992). Testing exponentiality based on Kullback-Leibler information, Journal of the Royal Statistical Society, 54, 739-748.
Jaynes, E. T. (1957). Information theory and statistical mechanics, Physical Revies, 106, 620-630.
Kullback, S. (1959). Information Theory and Statistics, Wiley, New York.
Lawless, J. F. (1982). Statistical Models and Methods for Lifetime Data, Wiley, New York.
Mosayeb, A. and Borzadaran, M. G. R. (2013). Kullback-Leibler information in view of an extended version of k-records, Communications for Statistical Applications and Methods, 20, 1-13.
Nakamura, T. K. (2009). Relativistic equilibrium distribution by relative entropy maximization, Europhysics letters, 88, 40009.
Park, S. (1995). The entropy of consecutive order statistics, IEEE Transactions on Information Theory, 41, 2003-2007.
Park, S. (2005). Testing exponentiality based on Kullback-Leibler information with the type II cnesored data, IEEE Transactions on Reliability, 54, 22-26.
Park, S. and Park, D. (2003). Correcting moments for goodness of fit tests based on two entropy estimates, Journal of Statistical Computation and Simulation, 73, 685-694.
Rao, M., Chen, Y., Vemuri, B. C. and Wang, F. (2004). Cumulative residual entropy: A new measure of information, IEEE Transactions on Information Theory, 50, 1220-1228.
Samanta, M. and Schwarz, C. J. (1988). The Shapiro-Wilk test for exponentiality based on censored data, Journal of the American Statistical Association, 83, 528-531.
Soofi, E. S. (2000). Principal information theoretic approaches, Journal of the American Statistical Association, 95, 1349-1353.
Stacy, E. W. (1962). A generalization of the Gamma distribution, Annals of Mathematical Statistics, 33, 1187-1192.
Theil, H. (1980). The entropy of the maximum entropy distribution, Economics Letters, 5, 145-148.
Wong, K. M. and Chen, S. (1990). The entropy of ordered sequences and order statistics, IEEE Transactions on Information Theory, 36, 276-284.