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Algorithmic Inference of Two-Parameter Gamma Distribution Bruno Apolloni, Simone Bassis

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Algorithmic Inference of Two-Parameter Gamma Distribution Bruno Apolloni, Simone Bassis Algorithmic Inference of Two-Parameter Gamma Distribution Bruno Apolloni, Simone Bassis To cite this version: Bruno Apolloni, Simone Bassis. Algorithmic Inference of Two-Parameter Gamma Distribution. Com- munications in Statistics - Simulation and Computation, Taylor & Francis, 2009, 38 (09), pp.1950-1968. 10.1080/03610910903171821. hal-00520670 HAL Id: hal-00520670 https://hal.archives-ouvertes.fr/hal-00520670 Submitted on 24 Sep 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Communications in Statistics - Simulation and Computation For Peer Review Only Algorithmic Inference of Two-Parameter Gamma Distribution Journal: Communications in Statistics - Simulation and Computation Manuscript ID: LSSP-2009-0057.R1 Manuscript Type: Original Paper Date Submitted by the 01-Jul-2009 Author: Complete List of Authors: Apolloni, Bruno; University of Milano, Department of Computer Science Bassis, Simone; University of Milano, Department of Computer Science Two-parameters Gamma distribution, Algorithmic inference, Keywords: Population bootstrap, Gibbs sampling, Adaptive rejection sampling, Maximum likelihood estimator We provide an estimation procedure of the two-parameter Gamma distribution based on the Algorithmic Inference approach. As a key feature of this approach, we compute the joint probability distribution of these parameters without assuming any prior. To this end we Abstract: propose a numerical algorithm which is often beneficial of a highly efficient speed up based on an approximate analytical expression of the probability distribution. We contrast our interval and point estimates with those recently obtained in Son and Oh (2006). We realize that our estimates are both unbiased and more accurate, albeit more dispersed than Bayesian methods. Note: The following files were submitted by the author for peer review, but cannot be converted to PDF. You must view these files (e.g. movies) online. URL: http://mc.manuscriptcentral.com/lssp E-mail: [email protected] Page 1 of 49 Communications in Statistics - Simulation and Computation 1 2 3 4 infgamma.zip 5 6 7 8 9 10 11 12 13 14 For Peer Review Only 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 URL: http://mc.manuscriptcentral.com/lssp E-mail: [email protected] Communications in Statistics - Simulation and Computation Page 2 of 49 1 2 INFERENCE 3 4 ALGORITHMIC INFERENCE OF TWO-PARAMETER GAMMA DISTRIBUTION 5 6 7 8 B. Apolloni and S. Bassis 9 10 Dipartimento di Scienze dell’Informazione 11 12 University of Milano 13 Via Comelico 39/41, 20135 14 15 Milano, Italy 16 For Peer Review Only 17 {apolloni,bassis}@dsi.unimi.it 18 19 20 Key Words: Two-parameters Gamma distribution; Algorithmic inference; Population boot- 21 22 strap; Gibbs sampling; Adaptive rejection sampling; Maximum likelihood estimator. 23 24 ABSTRACT 25 26 We provide an estimation procedure of the two-parameter Gamma distribution based 27 28 on the Algorithmic Inference approach. As a key feature of this approach, we compute the 29 30 joint probability distribution of these parameters without assuming any prior. To this end we 31 32 propose a numerical algorithm which is often beneficial of a highly efficient speed up based on 33 an approximate analytical expression of the probability distribution. We contrast our interval 34 35 and point estimates with those recently obtained in Son and Oh (2006) for the same problem. 36 37 From this benchmark we realize that our estimates are both unbiased and more accurate, 38 39 albeit more dispersed, in some cases, than the competitor methods’, where the dispersion 40 41 drawback is notably mitigated w.r.t. Bayesian methods by a greater estimate decorrelation. 42 43 We also briefly discuss the theoretical novelty of the adopted inference paradigm which 44 45 actually represents a brush up on a Fisher perspective dating to almost a century, made 46 feasible today by the available computational tools. 47 48 49 1. INTRODUCTION 50 51 The Gamma distribution is a formidable touchstone for any inference framework. On 52 53 54 55 56 1 57 58 59 60 URL: http://mc.manuscriptcentral.com/lssp E-mail: [email protected] Page 3 of 49 Communications in Statistics - Simulation and Computation 1 2 the one hand it is widely employed in many application fields, ranging from reliability (Ju- 3 ran and Godfrey, 1999; Corp., 2009) to telecommunications (Titus and Wheatley, 1996), 4 5 insurance (Ammeter, 1970) and so on. On the other, the estimation of its parameters still 6 7 poses a challenging problem. Adopting an analogous notation as in Son and Oh (2006), we 8 9 introduce the Gamma distribution through its probability density function (PDF) applied 10 1 11 to the random variable X as follows : 12 13 xα−1e−x/β f (x)= , x> 0, α,β> 0, (1) 14 X Γ(α)βα 15 16 where α and Forβ are respectively Peer a shape Review and scale parameter. Only Of the conventional point 17 18 estimators, the one based on the method of moments (MM) is very simple but inefficient 19 20 as well – with some exceptions in asymptotic conditions (Wiens et al., 2003). Maximum 21 22 likelihood (ML) estimators may rely on a pair of joint sufficient statistics, but they are 23 24 penalized by the absence of a closed analytical expression. To overcome this drawback, we 25 26 find in the literature either approximate expressions (Thom, 1958; Greenwood and D., 1960) 27 28 or recursive procedures to get a numerical solution (Bowman and Shenton, 1988). With the 29 latter we work with a sequence of parameter values converging with some dynamics to the 30 31 planned estimators. With a dual perspective, other researchers adopt the Bayes approach to 32 33 collect a population of parameter estimators – possibly generated through a Gibbs sampler 34 35 (Ripley, 1988) or more sophisticated techniques (Gilks and P., 1992) – and deduce point 36 37 estimators as central values of this population. 38 39 Our work goes in the same direction, but without requiring to set any a priori distri- 40 41 bution of the parameters. We have implemented this kind of procedure in many inference 42 problems, mainly within the machine learning framework (Apolloni et al., 2008a) and in 43 44 specific operational fields such as survival analysis (Apolloni et al., 2007b), human mobility 45 46 modeling (Apolloni et al., 2007a) and statistical tests of hypotheses (Apolloni and Bassis, 47 48 2007). The goal of this paper is twofold: on the one hand we want to efficiently solve the 49 1 50 By default, capital letters (such as U, X) will denote random variables and small letters (u, x) their 51 corresponding realizations; vectorial versions of the above variables will be denoted through bold symbols 52 53 (X, U, x, u). 54 55 56 2 57 58 59 60 URL: http://mc.manuscriptcentral.com/lssp E-mail: [email protected] Communications in Statistics - Simulation and Computation Page 4 of 49 1 2 questioned estimation problem as a distinguished inference task. To this aim, we show our 3 procedures to quickly compute accurate and unbiased estimators based on independent ap- 4 5 proximations of the unknown parameters. On the other hand, we would use this solution as 6 7 a business card to introduce our inference framework, which we call Algorithmic Inference, 8 9 to a more probability methodologically oriented audience and invite the reader to read more 10 11 widely about it in a couple of books we have published on the subject (Apolloni et al., 2008c, 12 13 2006). Therefore the paper is organized as follows. In Section 2 we introduce our method 14 15 from a strictly operational perspective, through definitions and a few logical links, having 16 the questionedFor inference Peer problem as Review lead example. In SectionOnly3 we apply the proposed 17 18 procedure to the same benchmark as Son and Oh (2006), contrasting our results with the 19 20 competitor methods considered therein. We devote Section 4 to a discussion of the benefits 21 22 of our approach and its connections to some inference strategies animating the debate since 23 24 the early 20th century, to conclude the paper. 25 26 2. COMPATIBLE PARAMETERS 27 28 Given a distribution law, think of its unknown parameters as random parameters. But, 29 30 in place of a specific prior distribution, they inherit probabilities from a standard source 31 32 of random seeds that are also at the basis of the random variables they are deputed to 33 34 characterize. Like with a barrel with two interlocking taps, through the former you get, 35 36 from the source, samples of the random variable for given parameters; while from the latter 37 38 you get samples of the parameters for given observed variables. You measure what happens 39 with the former and compute what would happen with the latter. The basic steps of this 40 41 twisting task are the following: 42 43 M 44 1. Sampling mechanism X . It consists of a pair hZ,gθi, where the seed Z is a random 45 θ 46 variable without unknown parameters, while the explaining function g is a function 47 mapping from samples {z ,...,z } of Z to samples {x ,...,x } of the random variable 48 1 m 1 m 49 X we are interested in.
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