BETA BESSEL DISTRIBUTIONS

Three new distributions on the unit interval [0,1] are introduced which generalize the standard beta distribution. These distributions involve the Bessel function. Expression is derived for their shapes, particular cases, and the nth moments. Estimation by the method of maximum likelihood and Bayes estimation are discussed. Finally, an application to consumer price indices is illustrated to show that the proposed distributions are better models to economic data than one based on the standard beta distribution.


Introduction
Beta distributions are very versatile and a variety of uncertainties can be usefully modeled by them.Many of the finite range distributions encountered in practice can be easily transformed into the standard distribution.In reliability and life testing experiments, many times the data are modeled by finite range distributions, see, for example, [2].
A random variable X is said to have the standard beta distribution with parameters ν and μ if its probability density function (pdf) is for 0 < x < 1, ν > 0, and μ > 0, where denotes the beta function.Many generalizations of (1.1) involving algebraic, exponential, and hypergeometric functions have been proposed in the literature.Some of these are (see [6, Chapter 25] and [5] for comprehensive accounts) (i) the four-parameter generalization given by for c ≤ x ≤ d, a > 0, and b > 0 (see [7,Section 3.2] for a reparameterization of this); (ii) the McDonald and Richards [9,13] beta distribution given by f (x) = px ap−1 1 − (x/q) p b−1 q ap B(a,b) (1.4) for 0 ≤ x ≤ q, a > 0, b > 0, p > 0, and q > 0; (iii) the Libby and Novick [8] beta distribution given by for 0 ≤ x ≤ 1, a > 0, b > 0, and λ > 0; (iv) the McDonald and Xu [10] beta distribution given by , where a > 0, b > 0, 0 ≤ c ≤ 1, p > 0, and q > 0; (v) the Gauss hypergeometric distribution given by denotes the Gauss hypergeometric function, where ( f ) denotes the ascending factorial; (vi) confluent hypergeometric distribution given by is the confluent hypergeometric function.In this paper, we introduce the first generalizations of (1.1) involving the Bessel function.We refer to them as the beta Bessel (BB) distributions.We propose three BB distributions in all.
For each of the three BB distributions, we derive various particular cases, an expression for the nth moment as well as estimation procedures by the method of maximum likelihood and Bayes method (Sections 2 to 4).We also present an application of the proposed models to consumer price indices (Section 5).The calculations involve several special functions, including the modified Bessel function of the first kind defined by and the 2 F 3 hypergeometric function defined by where ( f denotes the ascending factorial.The properties of the above special functions can be found in [4,11,12].

BB distribution I
The first generalization of (1.1) is given by the pdf (2. 2) The standard beta pdf (1.1) arises as the particular case of (2.1) for c = 0 and ν = 0. Several other particular cases of (2.1) can be obtained using special properties of I ν (•).Note that x 2 + 3 sinh(x) − 3x cosh(x) x 5/2 , More generally, if ν − 1/2 ≥ 1 is an integer, then (2.4) Thus, several particular forms of (2.1) can be obtained for half-integer values of ν.For example, if ν = 3/2, then (2.1) reduces to The modes of (2.1) are the solutions of There could be more than one mode (see Figures 2.1 and 2.2).The nth moment of (2.1) can be written as (2.9) For a random sample w 1 ,...,w n , the maximum-likelihood estimators (MLEs) of the four parameters in (2.1) are the solutions of In (2.10) Assuming (2.1) as the prior, the Bayes estimate of the binomial parameter, say p, is where n is the number of trials and x is the number of successes.

BB distribution II
The second generalization of (1.1) is given by the pdf The standard beta pdf (1.1) arises as the particular case of (3.1) for c = 0 and ν = 0. Further particular cases of (3.1) can be obtained using (2.4).The modes of (3.1) are the solutions of There could be more than one mode (see Figures 2.1 and 2.2).The nth moment of (3.1) can be written as and an application of [12, equation (2.15.4.1)] shows that the above reduces to For a random sample w 1 ,...,w n , the MLEs of the four parameters in (3.1) are the solutions of n i=1 In (3.6) Assuming (3.1) as the prior, the Bayes estimate of the binomial parameter, say p, is where n is the number of trials and x is the number of successes.

BB distribution III
The third and final generalization of (1.1) is given by the pdf The standard beta pdf (1.1) arises as the particular case of (4.1) for c = 0 and ν = 0. Further particular cases of (4.1) can be obtained using (2.4).The modes of (4.1) are the solutions of There could be more than one mode (see Figures 2.1 and 2.2).The nth moment of (4.1) can be written as and an application of [12, equation (2.15.4.1)] shows that the above reduces to For a random sample w 1 ,...,w n , the MLEs of the four parameters in (4.1) are the solutions of In Assuming (4.1) as the prior, the Bayes estimate of the binomial parameter, say p, is where n is the number of trials and x is the number of successes.

Application
We now illustrate an application of the proposed beta distributions to consumer price index data.We collected the data on this index for the six countries: United States, United Kingdom, Japan, Canada, Germany, and Australia.The data were extracted from the website http://www.globalfindata.com/(go to "Sample Data" under "Database" and then look under "Consumer Price Indices" for the closing value of the index) and the range of data for each country is shown in Table 5.1.Taking the ratio W = X/(X + Y ), we attempted to model the relative economic performance of each country against another over the range of overlapping years.This yields 15 data sets for the variable W. As expected, some of the data for W appeared to concentrate to a subinterval of [0,1] and so suitable location-scale transformations were applied to make the data span from 0 to 1.For each data set, we fitted the standard beta distribution given by (1.1) and the BB III distribution given by (4.1) with ν fixed as ν = 1.The In where Ψ(x) = d In Γ(x)/dx is the digamma function.The MLEs of (α,β,c) in (4.1) with ν fixed as ν = 1 are obtained by solving (4.6).
The results of the fits were remarkable.In each fit, the maximized log-likelihood for (4.1) turned up significantly higher than that for the standard beta model.Here, we give details for just two of the 15 data sets.
(i) For the (United States, United Kingdom) data set shown in Table A.1 of the appendix the fitted estimates were α = 1.392, β = 1.230 with logL = 5.145 for the standard beta model (1.1); and α = 0.820, β = −3.180,c = 1.571 with logL = 7.647 for the BB III model (4.1).The corresponding fitted densities superimposed with the empirical density are shown in Figure 2.1 (the empirical density computed using the hist command in the R software package).(ii) For the (United States, Germany) data set shown in Table A.2 of the appendix the fitted estimates were α = 0.914, β = 1.130 with logL = 1.494 for the standard beta model (1.1); and α = 1.405, β = 2.370, c = 7.828 × 10 −6 with logL = 5.198 for the BB III model (4.1).The corresponding fitted densities superimposed with the empirical density are shown in Figure 2.2 (the empirical density computed using the hist command in the R software package).So, we can conclude at least in this situation that the beta Bessel models are better than the one based on the standard beta distribution.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Figure 2 .
Figure 2.1.The empirical and fitted densities for the consumer price indices of the United States and the United Kingdom (X = consumer price index of the United States and Y = consumer price index of the United Kingdom).

Figure 2 . 2 .
Figure 2.2.The empirical and fitted densities for the consumer price indices of the United States and Germany (X = consumer price index of the United States and Y = consumer price index of Germany).

First
Round of Reviews March 1, 2009

Table 5 .
1. Countries and years of data.

Table A .
1. Consumer price index data for the United States and the United Kingdom for the years 1820-2003.

Table A .
2. Consumer price index data for the United States and Germany for the years 1923-2003.