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The aim of this paper is to estimate probability distribution functions with maximum entropy and known quantiles. The paper formulates the problem as a nonlinear optimization problem, and converts it into a system of nonlinear equations by Lagrange multipliers method. Finally, an efficient method is proposed to obtain a solution of the nonlinear system. The method needs to solve a linear programming problem in each iteration. Since linear programming problems can be solved in a reasonable time, our proposed method is faster than generic methods of solving nonlinear programming problems. Several computational experiment are provided to demonstrate the performance and validation of our proposed method.
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- Aldrich, J. et al. (1997). Ra fisher and the making of maximum likelihood 1912-1922. Statistical science,12(3):162–176.
- Arandjelovi´ c, O., Pham, D.-S., and Venkatesh, S. (2014). Two maximum entropy-based algorithms for running quantile estimation in nonstationary data streams. IEEE Transactions on circuits and systems for video technology, 25(9):1469–1479.
- Bajgiran, A. H., Mardikoraem, M., and Soofi, E. S. (2020). Maximum entropy distributions with quantile information. European journal of operational research.
- Barzdajn, B. (2014). Maximum entropy distribution under moments and quantiles constraints. Measurement,
- Basset, N.(2015). A maximal entropy stochastic process for a timed automaton. Information and Computation, 243:50–74.
- Chliamovitch, G., Dupuis, A., and Chopard, B. (2015). Maximum entropy rate reconstruction of markov dynamics. Entropy, 17(6):3738–3751.
- Cover, T. and Thomas, J. (2006). Elements of Information Theory, (2nd edn, 2006).
- Dai, H., Zhang, H., and Wang, W. (2016). A new maximum entropy-based importance sampling for reliability analysis. Structural Safety, 63:71–80.
- Deuflhard, P. (2011). Newton methods for nonlinear problems: affine invariance and adaptive algorithms, volume 35. Springer Science & Business Media.
- Hosking, J. R. (1990). L-moments: Analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society: Series B (Methodological), 52(1):105–124.
- Jaynes, E. T. (1957). Information theory and statistical mechanics. ii. Physical review, 108(2):171.
- Krvavych, Y. and Mergel, V. (2000). Large loss distributions: probabilistic properties, evt tools, maximum entropy characterization. In Proceedings of the 31st ASTIN Colloquium, Sardinia, Italy.
- Landsman, Z. and Makov, U. E. (1999). Credibility evaluation for the exponential dispersion family. Insurance: Mathematics and Economics, 24(1-2):23–29.
- Najafabadi, A. T. P., Hatami, H., and Najafabadi, M. O. (2012). A maximum-entropy approach to the linear credibility formula. Insurance: Mathematics and Economics, 51(1):216–221.
- Oosterbaan, R. (2019). Software for generalized and composite probability distributions. International Journal of Mathematical and Computational Methods, 4.
- Sachlas, A. and Papaioannou, T. (2014). Residual and past entropy in actuarial science and survival models. Methodology and Computing in Applied Probability, 16(1):79–99.
- Shannon, C. E. (1948). A mathematical theory of communication. The Bell system technical journal, 27(3):379–423.
- Templeman, A. B. and Xingsi, L. (1987). A maximum entropy approach to constrained non-linear programming. Engineering Optimization+ A35, 12(3):191–205.
- Terlaky, T. (2013). Interior point methods of mathematical programming, volume 5. Springer Science & Business Media.
- Van der Straeten, E. (2009). Maximum entropy estimation of transition probabilities of reversible markov chains. Entropy, 11(4):867–887.
- Zhao, Z. and Zhang, Y. (2011). Design of ensemble neural network using entropy theory. Advances in Engineering Software, 42(10):838–845.
- Zografos, K. (2008). On some entropy and divergence type measures of variability and dependence for mixed continuous and discrete variables. Journal of statistical planning and inference, 138(12):3899–3914.