Abstract
We propose a general formula for the probability density function of transformations of continuous or discrete random variables. Approximations and estimations are derived. Particular cases are treated when transformations are sum or products of random variables. The formula has a simple form when probability density functions are expressed with respect to a reference measure which belongs to the class of natural exponential families with quadratic variance functions. Some numerical results are provided to illustrate the method.
Similar content being viewed by others
References
Abramowitz M, Stegun IA (1972) Orthogonal polynomials. Ch. 22 in handbook of mathematical functions. 9th printing. Dover, New York
Asmussen S, Rojas-Nandapaya L (2008) Asymptotics of sums of lognormal random variables with Gaussian copula. Statist Probab Lett 78:2709–2714
Asmussen S, Jensen JL, Rojas-Nandayapa L (2015) Exponential family techniques in the lognormal left tail. Scand J Stat 43:774–787
Barndorff-Nielsen O (1978) Information and exponential families: in statistical theory. Wiley Series in Probability and Mathematical Statistics Series J. Wiley
Belomestny D, Comte F, Genon-Catalot V (2016) Nonparametric Laguerre estimation in the multiplicative censoring model. Electron J Stat 10:3114–3152
Bodnar T, Mazur S, Okhrin Y (2013) On the exact and approximate distributions of the product of a Wishart matrix with a normal vector. J Multivariate Anal 122:70–81
De Vylder FE, Goovaerts MJ (2000) Homogeneous risks models with equalized claims amounts. Insurance Math Econom 26:223–238
Efromovich S (2010) Orthogonal series density estimation. Wiley Interdiscip Rev Comput Statist 2:467–476
Glickman T, Xu F (2008) The distribution of the product of two triangular random variables. Statist Probab Lett 78:2821–2826
Goffard PO, Loisel S, Pommeret D (2016) A polynomial expansion to approximate the ultimate ruin probability. In the compound Poisson ruin model. J Comput Appl Math 296:499–511
Goffard PO, Loisel S, Pommeret D (2017) Polynomial approximations for bivariate aggregate claims amount probability distributions. Methodol Comput Appl Probab 19:151–174
Joarder AH (2009) Moments of the product and ratio of two correlated chi-square variables. Stat Papers 50:581–592
Lefevre C, Picard P (2014a) Appell pseudopolynomials and Erlang type risk models. Stochastics 86:676–695
Lefevre C, Picard P (2014b) Ruin probabilities for risk models with ordered claim arrivals. Methodol Comput Appl Probab 16:885–905
Lefevre C, Picard P (2015) Risk models in insurance and epidemics: a bridge through randomized polynomials. Probab Engrg Inform Sci 29:399–420
Letac G (1992) Lectures on natural exponential families and their variance functions. Instituto de matemática pura e aplicada: Monografias de matemática, 50, Río de Janeiro, Brésil
Marques FJ, Loingeville F (2016) Improved near-exact distributions for the product of independent generalized gamma random variables. Comput Statist Data Anal 102:55–66
Morris CN (1982) Natural exponential families with quadratic variance functions. Ann Statist 10:65–82
Morris CN (1983) Natural exponential families with quadratic variance functions: statistical theory. Ann Statist 11:512–529
Nadarajah S (2012) Exact ddistribution of the product of N student’s t RVs. Methodol Comput Appl Probab 14:997–1009
Nadarajah S, Kotz S (2005) On the linear combination of laplace random variables. Probab Engrg Inform Sci 19:463–470
Nadarajah S, Kotz S (2006) On the product and ratio of gamma and Weibull random variables. Econ Theory 22:338–344
Shakil M, Golam Kibria BM, Chang KC (2008) Distributions of the product and ratio of Maxwell and Rayleigh random variables. Stat Papers (2008) 49:729–747
Withers C, Nadarajah S (2013) On the product of gamma random variables. Quality Quant: Int J Methodol 47:545–552
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pommeret, D., Reboul, L. Approximating the Probability Density Function of a Transformation of Random Variables. Methodol Comput Appl Probab 21, 633–645 (2019). https://doi.org/10.1007/s11009-018-9629-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-018-9629-0
Keywords
- Approximations
- Natural exponential families
- Orthogonal polynomials
- Probability density function
- Product of random variables
- Ratio
- Reference measure
- Sum of random variables