Journal of Modern Applied Statistical Methods Volume 3 | Issue 1 Article 10 5-1-2004 Beta-Normal Distribution: Bimodality Properties and Application Felix Famoye Central Michigan University, [email protected] Carl Lee Central Michigan University Nicholas Eugene Coppin State College Follow this and additional works at: http://digitalcommons.wayne.edu/jmasm Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Famoye, Felix; Lee, Carl; and Eugene, Nicholas (2004) "Beta-Normal Distribution: Bimodality Properties and Application," Journal of Modern Applied Statistical Methods: Vol. 3 : Iss. 1 , Article 10. DOI: 10.22237/jmasm/1083370200 Available at: http://digitalcommons.wayne.edu/jmasm/vol3/iss1/10 This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Journal of Modern Applied Statistical Methods Copyright © 2004 JMASM, Inc. May, 2004, Vol. 3, No. 1, 85- 103 1538 – 9472/04/$95.00 Beta-Normal Distribution: Bimodality Properties and Application Felix Famoye Carl Lee Nicholas Eugene Department of Mathematics Department of Mathematics Computer Science & Mathematics Central Michigan University Central Michigan University Coppin State College The beta-normal distribution is characterized by four parameters that jointly describe the location, the scale and the shape properties. The beta-normal distribution can be unimodal or bimodal. This paper studies the bimodality properties of the beta-normal distribution. The region of bimodality in the parameter space is obtained. The beta-normal distribution is applied to fit a numerical bimodal data set. The beta-normal fits are compared with the fits of mixture-normal distribution through simulation. Key words: Bimodal region, percentiles, curve estimation, egg size distribution Introduction Let Fx() be the cumulative distribution Bimodal distributions occur in many function (CDF) of a random variable X. The areas of science. Withington et al. (2000), in cumulative distribution function for a their study of cardiopulmonary bypass in infants generalized class of distributions for the random showed that plasma vecuronium and vecuronium variable X can be defined as the logit of the beta clearance requirements have bimodal random variable given by distributions. They concluded that their findings on bimodal distributions for plasma vecuronium Γ()αβ+ Fx() Gx()= ∫ tαβ−−11 (1− t ) dt , 0<<αβ , ∞ . and vecuronium clearance requirements ΓΓ()()αβ0 highlight the need for individual monitoring of (1.1) neuromuscular blockade. Espinoza et al. (2001) discussed the importance of bimodal distributions in the study of size distribution of Eugene et al. (2002) considered Fx() as the metals in aerosols. Bimodal distributions also CDF of the normal distribution with parameters occur in the study of genetic diversity (Freeland µ and σ. Thus, the random variable X has the et al., 2000), in the study of agricultural farm beta-normal distribution with probability density size distribution (Wolf & Sumner, 2001), in the function (pdf) study of atmospheric pressure (Zangvil et al., 2001), and in the study of anabolic steroids on Γ()αβ+ xxx−−−µµµαβ−−11 gx()= ⎡⎤⎡⎤Φ−Φ ( ) 1 ( )σ−1φ ( ) animals (Isaacson, 2000). ΓΓ()()αβ⎣⎦⎣⎦⎢⎥⎢⎥σσσ (1.2) Acknowledgement ⎛⎞x − µ where φ ⎜⎟ is the normal pdf and The first two authors gratefully acknowledge the ⎝⎠σ support received from the President Research ⎛⎞x − µ Investment Fund at Central Michigan University Φ⎜⎟ is the normal CDF. We denote the under the Grant #65139. Address ⎝⎠σ correspondence to the first author at Department beta-normal distribution with parameters α, β, µ, of Mathematics, Central Michigan University Mt. Pleasant, Michigan 48859-0001. Email: and σ as BN (α, β, µ, σ). [email protected]. Telephone: (989) 774- The distribution in (1.2) may be 5497. symmetric, skewed to the left, or skewed to the 85 BETA-NORMAL DISTRIBUTION 86 right. The distribution may be unimodal or Corollary 1: If α = β and one mode of bimodal. Eugene et al. (2002) discussed the BN(αβ , , µ, σ) is atx , then the other mode shape properties of the unimodal beta-normal 0 distribution. Furthermore, they considered the is at the point 2µ − x0 . estimation of its parameters by the method of maximum likelihood. Proof: If BN(αβ , , µ, σ) is unimodal, then In the analysis of bimodal data, a mixture of two normal densities is often used as the only mode occurs at the point x0 = µ. For a model (e.g., Cobb et al., 1983). The mixture of bimodal case, we need to show that if we replace normal distribution is used as a model to analyze x0 with 2µ − x0 , then equation (2.1) remains bimodal data because the mixture of normal the same. When α = β , equation (2.1) becomes densities can take on bimodal shapes depending on the parameters of the distribution. x − µ Eisenberger (1964) showed how the parameters σφ()(1)0 α − σ x0 − µ x0 = xx−−µµ{}12− Φ().+ µ of a mixture of normal distributions determine Φ−Φ()[1()]00 σ its shape. When a mixture assumption is not σσ required or justified the beta-normal distribution (2.2) can serve as a model to analyze data since only one distribution has to be used and one less If x0 in (2.2) is replaced with 2µ − x0 , we parameter to estimate. obtain In the rest of the paper, we provide some bimodality properties of the beta-normal µ −x σφ()(1)0 α − distribution. We obtain the region of bimodality σ µ −x0 2()µµ−Φx0 = µµ−−xx{}12− + in the parameter space. We also illustrate the Φ−Φ()[1()]00σ application of beta-normal distribution to a σσ numerical data set that exhibits two modes and (2.3) compare the fit with mixture-normal distribution. A simulation study is conducted to ⎛⎞⎛⎞x00−−µµx By using φφ⎜⎟⎜⎟= and compare the performance between beta-normal ⎝⎠⎝⎠σσ and mixture-normal distributions in fitting bimodal data. ⎛⎞x00−−µµ ⎛⎞x Φ=−Φ⎜⎟1 ⎜⎟ in (2.3) and on ⎝⎠σσ ⎝⎠ Bimodality Properties simplification, we get the result in (2.2). In this section, some results on the bimodality properties of beta-normal distribution Corollary 2: If BN(αβ , , µ, σ) has a mode at are obtained. x , then BN(,βα , µ, σ) has a mode at 0 Fact: A mode of the BN(αβ , , µ, σ) is any 2µ − x0 . point xx00= (,αβ ) that satisfies Proof: We need to show that if we replace α xx−−µµ σφ()00(1)α− φ () σ with β, and 2µ − x withx , equation (2.1) σσ 0 0 x0 = xxx−−−µµµ{}2.−−αβ++µ 1(−Φ000 ) Φ ( )[1()] −Φ remains the same. Equation (2.1) can be written σσσas (2.1) x −µ σφ()0 σ x0 −µ Proof: Differentiating BN(αβ , , µ, σ) in x0 =+xx−−µµ{}(2−−Φαβ ) ( )+ ( α− 1)µ . Φ−Φ()[1()]00 σ (1.2) with respect to x, setting it equal to zero, σσ and solving for x gives the result in (2.1). (2.4) 87 FAMOYE, LEE, & EUGENE If x0 is replaced with 2µ − x0 and α is replaced Hence x0(,αβ ) is an increasing function of α ⎛⎞⎛⎞x00−−µµx and a decreasing function of β. with β in (2.4), using φφ⎜⎟⎜⎟= ⎝⎠⎝⎠σσ Eugene et al. (2002) showed that the ⎛⎞x00−−µµ ⎛⎞x and Φ=−Φ⎜⎟1 ⎜⎟, and on beta-normal distribution is symmetric about µ ⎝⎠σσ ⎝⎠ when α = β. From this result and corollary 3, the simplification, we obtain the result in (2.4). modal value is greater than µ if α > β. Also, the modal value is less than µ if α < β. The beta- Corollary 3: The modal point x0(,αβ ) is an normal distribution has a very distinct property increasing function of α and a decreasing in that it can be used to describe both bimodal and unimodal data. function of β. Region of Bimodality Proof: Differentiating the result in (2.1) with The beta-normal distribution becomes respect to α and β gives bimodal for certain values of the parameters α and β, and the analytical solution of α and β , x − µ where the distribution becomes bimodal, cannot σφ()0 ∂x0(,αβ ) σ be solved algebraically. A numerical solution is =>x − µ 0 ∂α Φ()0 obtained, however, by solving the number of σ roots of the derivative of BN(α, β, µ, σ ). Table 1 shows a grid of values where the distribution and is bimodal. The “2” in Table 1 indicates that the x − µ −σφ()0 beta-normal distribution has two turning points ∂x0(,αβ ) σ =<x − µ 0 . which implies that the distribution is bimodal ∂β 1(−Φ 0 ) σ and the “1” indicates that the beta-normal distribution has one turning point which implies that the distribution is unimodal. BETA-NORMAL DISTRIBUTION 88 Table 1. Number of turning points of BN(,αβ , 0, 1) for various values of α and β Beta .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .02 .21 .22 Alpha .01 2 2 2 2 2 2222222222211 1 1 1 1 .02 2 2 2 2 2 2222222222211 1 1 1 1 .03 2 2 2 2 2 2222222222211 1 1 1 1 .04 2 2 2 2 2 2222222222211 1 1 1 1 .05 2 2 2 2 2 2222222222211 1 1 1 1 .06 2 2 2 2 2 2222222222211 1 1 1 1 .07 2 2 2 2 2 2222222222211 1 1 1 1 .08 2 2 2 2 2 2222222222221 1 1 1 1 .09 2 2 2 2 2 2222222222221 1 1 1 1 .1 2 2 2 2 2 2222222222221 1 1 1 1 .11 2 2 2 2 2 2222222222221 1 1 1 1 .12 2 2 2 2 2 2222222222221 1 1 1 1 .13 2 2 2 2 2 2222222222222 1 1 1 1 .14 2 2 2 2 2 2222222222222 1 1 1 1 .15 2 2 2 2 2 2222222222222 1 1 1 1 .16 2 2 2 2 2 2222222222222 1 1 1 1 .17 1 1 1 1 1 1122222222222 2 1 1 1 .18 1 1 1 1 1 1111111222222 2 1 1 1 .19 1 1 1 1 1 1111111111122 2 2 1 1 .2 1 1 1 1 1 1111111111111 2 2 1 1 .21 1 1 1 1 1 1111111111111 1 1 2 1 .22 1 1 1 1 1 1111111111111 1 1 1 1 .23 1 1 1 1 1 1111111111111 1 1 1 1 .24 1 1 1 1 1 1111111111111 1 1 1 1 .25 1 1 1 1 1 1111111111111 1 1 1 1 .26 1 1 1 1 1 1111111111111 1 1 1 1 Note: “2” indicates where bimodality occurs and “1” indicates where unimodality occurs Numerically, the largest value of α or β that gives bimodal property is approximately 0.214.