A New Approach to Distribution Fitting: Decision on Beliefs

Document Type: Research Paper

Authors

Department of Industrial Engineering, Sharif University of Technology, Tehran, IRAN

Abstract

We introduce a new approach to distribution fitting, called Decision on Beliefs (DOB). The objective is to identify the probability distribution function (PDF) of a random variable X with the greatest possible confidence. It is known that f X is a member of = { , , }. 1 m S f L f To reach this goal and select X f from this set, we utilize stochastic dynamic programming and formulate this problem as a special case of Optimal Stopping Problem. The decision is made on the basis of the outcome of a limited number of experiments. A real number, namely, belief is assigned to each candidate by considering the outcome of observations. At each stage and after a random observation, beliefs are updated by applying Bayesian formula and then either one element of S is selected as the desired PDF or another observation is made. At each stage, a PDF from S with the greatest belief is accepted as the desired PDF provided the belief is higher than a least acceptable designated level. We assume the total number of possible observations can not exceed N and a cost is incurred for each observation. Dynamic and nonlinear programming are applied to calculate the least acceptable belief value for each stage. To reduce the search of the optimal solution, the concept of entropy is utilized.

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[1] Ahn J.H., Kim J.J. (1998), Action-Timing Problem with Sequential Bayesian Belief Revision Process;
European Journal of Operational Research 105; 118-129.
[2] Bernardo J.M., Smith A.F.M. (2001), Bayesian Theory, 1st Edition; Wiley.
[3] Conver W.J. (2001), Practical Nonparametric Statistics, 3rd Edition; Wiley.
[4] Eshragh A. (2001), A New Approach to Distribution Fitting: Decision on Beliefs, Unpublished B.S.
Project, Industrial Eng. Dept., Sharif University of Technology, Tehran, IRAN.
[5] Eshragh A., Akhavan Niaki S.T. (2003), Application of Decision on Beliefs in Response Surface
Methodology; Proceeding of the 54th Session of International Statistical Institute; Berlin, Germany.
[6] Fallahnezhad M.S., Akhavan Niaki S.T., Eshragh A. (2006), Application of Decision on Beliefs in
Univariate Quality Control Environments; Proceeding of Operations Research Conference; Karlsruhe,
Germany.
[7] MacKay D.J.C. (2005), Information theory, Inference, and Learning Algorithm; Cambridge University
Press; 7th Edition.
[8] Ross S.M. (1983), Introduction to Stochastic Dynamic Programming; Academic Press; New York.
[9] Saniee Monfared M.A., Ranaeifar F. (2007), Further Analysis and Developments of DOB Algorithm
on Distribution Fitting; Scientia Iranica 14(5); 425-434.