We study a new family of random variables that each arise as the
distribution of the maximum or minimum of a random number N of i.i.d. random
variables X1, X2,…, XN, each distributed as a variable X with support on [0,
1]. The general scheme is first outlined, and several special cases are studied
in detail. Wherever appropriate, we find estimates of the parameter θ in the
one-parameter family in question.
References
[1]
Durrett, R. (1991) Probability: Theory and Examples. Wadsworth and Brooks/Cole, Pacific Grove. http://dx.doi.org/10.4236/am.2012.34054
[2]
Louzada, F., Roman, M. and Cancho, V. (2011) The Complementary Exponential Geometric Distribution: Model, Properties, and a Comparison with Its Counterpart. Computational Statistics and Data Analysis, 55, 2516-2524. http://dx.doi.org/10.1016/j.csda.2010.09.030
[3]
Louzada, F., Bereta, E. and Franco, M. (2012) On the Distribution of the Minimum or Maximum of a Random Number of i.i.d. Lifetime Random Variables. Applied Mathematics, 3, 350-353.
[4]
Morais, A. and Barreto-Souza, W. (2011) A Compound Class of Weibull and Power Series Distributions. Computational Statistics and Data Analysis, 55, 1410-1425.
[5]
Johnson, N., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions. Vol. 2, Wiley, New York.
[6]
Kotz, S. and van Dorp, J. (2004) Beyond Beta: Other Continuous Families of Distributions with Bounded Support and Applications. World Scientific Publishing Co., Singapore. http://dx.doi.org/10.1142/5720
[7]
Leadbetter, M., Lindgren, G. and Rootzén, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer Verlag, New York. http://dx.doi.org/10.1007/978-1-4612-5449-2
[8]
Earthquake Data Set. http://www.data.scec.org/eq-catalogs/date_mag_loc.php
[9]
Chung, F., Lu, L. and Vu, V. (2003) Eigenvalues of Random Power Law Graphs. Annals of Combinatorics, 7, 21-33. http://dx.doi.org/10.1007/s000260300002
[10]
Aiello, B., Chung, F. and Lu, L. (2001) A Random Graph Model for Power Law Graphs. Experimental Mathematics, 10, 53-66. http://dx.doi.org/10.1080/10586458.2001.10504428
[11]
Marshall, A. and Olkin, I. (1997) A New Method of Adding a Parameter to a Family of Distributions with Applications to the Exponential and Weibull Families. Biometrika, 84, 641-652. http://dx.doi.org/10.1093/biomet/84.3.641
[12]
Kumaraswamy, P. (1980) A Generalized Probability Density Function for Double-Bounded Random Processes. Journal of Hydrology, 46, 79-88. http://dx.doi.org/10.1016/0022-1694(80)90036-0
[13]
Urzúa, C.M. (2000) A Simple and Efficient Test for Zipf’s Law. Economics Letters, 66, 257-260. http://dx.doi.org/10.1016/S0165-1765(99)00215-3
[14]
Aban, I.B., Meerschaert, M.M. and Panorska, A.K. (2006) Parameter Estimation for the Truncated Pareto Distribution. Journal of the American Statistical Association, 101, 270-277. http://dx.doi.org/10.1198/016214505000000411
[15]
Topp, C.W. and Leone, F.C. (1955) A Family of J-Shaped Frequency Functions. Journal of the American Statistical Association, 50, 209-219. http://dx.doi.org/10.1080/01621459.1955.10501259
