Estimations for parameters in a bivariate beta distribution^†

Estimations for parameters in a bivariate beta distribution ^†

Changsoo Lee ¹

1 Department of Flight Operation, Kyungwoon University

Received 27 March 2021, revised 3 May 2021, accepted 12 May 2021

Abstract

We will consider the distribution and the moment of the ratio R = X/(X + Y ) in a bivariate beta distribution for (X, Y ). And we will observe the skewness of the ratio and numerically study trends for the skewness of the ratio based on a bivariate beta distribution. Also, we will derive approximate MLEs of three parameters and the reliability in the bivariate beta distribution.

Keywords: Approximate MLE, bivariate beta distribution, ratio, reliability, skewness.

1. Introduction

Many authors have studied estimations and characterizations in a bivariate beta distri- bution with three shape parameters α, δ and γ, whose distribution was widely used in engineering applications in Johnson et al. (1995).

An example of some beginning importance is the use of some distributions with parameters to apply life times of lights and machines. For two random variables X and Y , and a real number c, the probability P (X < c · Y ) is a distribution of the ratio R = X/(X + Y ) when c = t/(1 − t) for 0 < t < 1.

For given random variables X and Y , the distribution of the ratio R = X/(X + Y ) is of interest in biological and physical sciences, econometrics, engineering and selection. For example, ratios of normal variables appears as sampling distributions in single equation models in simultaneous equations models. Other area of applications include the mass to energy ratios in nuclear physics. Lee and Ahn (2019) have studied properties for the ratio of two independent weighted Lindley variables. Lee and Ahn (2020) have studied the reliability estimation and the ratio distribution in two independent Pareto-Pareto and power function The problem of estimating the probability that a random variable X is less than another random variable Y arises in many practical situations, like biometry, reliability study. The problem has been studied by many authors for different distributions of X and Y , see, for example Pal et al. (2005), Ali et al. (2010) and Raqab et al. (2007).

† This research was supported by a Research Grant of Kyungwoon University in 2021.

1

Associate professor, Department of Flight Operation, Kyungwoon University, Gumi 730-850, Korea.

E-mail : [email protected]

Balakrishnan and Cohen (1991) have proposed a method of finding approximate MLEs of parameters in several distributions. Son and Woo (2009) have studied approximate MLEs in a skewed double Weibull distribution. Kim (2012) have studied the moment of the ratio and approximate MLEs of parameters in a bivariate Pareto distribution

Here, we will derive the distribution and k-th moment of the ratio R = X/(X + Y ) in a bivariate beta distribution for (X, Y ). And we will observe coefficients of skewness of the ratio R = X/(X + Y ) with parameters (α, δ) and study numerically trends for the skewness of the ratio R = X/(X + Y ) based on a bivariate beta distribution. Also, we will derive approximate MLEs for three parameters α, δ, γ using determinant and will consider the reliability R = P (X < Y ) in the bivariate beta distribution for (X, Y ).

2. Moments of the ratio

A bivariate beta density function of (X, Y ) is given in Rohatgi (1976) as :

f (x, y) = Γ(α + δ + γ)

Γ(α)Γ(δ)Γ(γ) · x ^α−1 y ^δ−1 (1 − y − x) ^γ−1 , x + y < 1 for x > 0, y > 0, (2.1) where α > 0, δ > 0, γ > 0.

From the density function (2.1), following results are well-known in Rohatgi (1976):

Lemma 2.1 Let (X, Y ) be the bivariate beta density function (2.1). Then

(a) The marginal density function of X is a beta density function with parameters (α, δ+γ).

(b) The marginal density function of Y is a beta density function with parameters (α, δ+γ).

(c) The conditional density function X given Y = y is f (x|y) = Γ(α + γ)

Γ(α)Γ(γ) x ^α−1 (1 − x − y) ^γ−1 /(1 − y) ^α+γ−1 , 0 < x < 1 − y.

(d) The conditional density function Y given X = x is f (y|x) = Γ(δ + γ)

Γ(δ)Γ(γ) x ^δ−1 (1 − x − y) ^γ−1 /(1 − x) ^δ+γ−1 , 0 < y < 1 − x.

(e) Let W = X/Y . The density function of W is given by f _W (w) = 1

B(α, δ) · w ^α−1

(1 + w) ^α+δ , w > 0, where B(a, b) is a beta function.

From the bivariate density function (2.1), we can obtain the following moment : E(X ^s Y ^t ) = Γ(α + s)Γ(t + δ)Γ(α + δ + γ)

Γ(α)Γ(δ)Γ(α + δ + γ + s + t) .

Especially, if s = t = 1, then

E(XY ) = αδ/((α + δ + γ + 1)(α + δ + γ)). (2.2) From the result (2.2) and Lemma 2.1 (a) and (b), we can obtain the correlation coefficient between X and Y :

ρ X,Y = − s

αδ

(α + γ)(δ + γ) (2.3)

is negative. From Lemma 2.1 (c) and (d), we can obtain the following conditional expecta- tions:

E(Y |X = x) = − δ

δ + γ x + δ

δ + γ , 0 < x < 1 and

E(X|Y = y) = − α

α + γ y + α

α + γ , 0 < y < 1 for y > 0, (2.4) which they are linear functions of X and Y , respectively.

From the results (2.3) and (2.4), we can obtain a well-known relationship between slopes of two linear functions for conditional expectations and correlation coefficient in (2.3).

Proposition 2.1 Let (X, Y ) be the bivariate beta distribution (2.1). Then ρ ² _X,Y = (slope of E(Y |X = x)) ∗ (slope of E(X|Y = y)) = αδ

(α + γ)(δ + γ) .

Define the ratio R = X/(X + Y ). Then R = W/(1 + W ) for W = X/Y and then from Lemma 2.1 (e), the density function of the ratio R = X/(X + Y ) follows a beta density function with parameters (α, δ) where the density function does not involve a parameter γ.

Therefore, we can obtain the k-th moments of the ratio R = X/(X + Y ) as follows :

E(R ^k ) = Γ(α + δ)Γ(α + k)

Γ(α)Γ(α + δ + k) , k = 1, 2, · · · , (2.5) which it is independent of the parameter γ.

From the result (2.5), we can obtain means, variances, and coefficients of skewness for the ratio R = X/(X + Y ) for α = 1/4, 1/2, 1, 2, 4 and δ = 1/4, 1/2, 1, 2, 4 as given in Table 2.1.

From Table 2.1, we can observe the following trends of the density function for the ratio

R = X/(X + Y ).

Table 2.1 Means, variances and coefficients of skewness for the ratio R α δ mean variance skewness

1/4 1/4 0.50000 0.16667 0.0000 1/2 0.33333 0.12698 0.68030

1 0.20000 0.07111 1.38462 2 0.11111 0.03039 2.09960 4 0.05882 0.01054 2.74955 1/2 1/4 0.66667 0.12698 -0.68030 1/2 0.50000 0.12500 0.00000 1 0.33333 0.08889 0.63888 2 0.20000 0.04547 1.24722 4 0.11111 0.01796 1.78587 1 1/4 0.80000 0.07111 -1.38462 1/2 0.66667 0.08889 -0.63888 1 0.50000 0.08333 0.00000 2 0.33333 0.05556 0.56569 4 0.20000 0.02667 1.04978 2 1/4 0.88889 0.03039 -2.09960 1/2 0.80000 0.04571 -1.24722 1 0.66667 0.05556 -0.56569 2 0.50000 0.05000 0.00000 4 0.33333 0.03175 0.46771 4 1/4 0.94118 0.01054 -2.74955 1/2 0.88889 0.01796 -1.78587 1 0.80000 0.02667 -1.04978 2 0.66667 0.03175 -0.46771 4 0.50000 0.02778 0.00000

Fact 2.1 Let (X, Y ) be the bivariate beta density function (2.1). Then

(a) the skewness of the density function of the ratio R = X/(X + Y ) is zero at R = 1/2 when .

(b) it is right skewed when α = 1/4 for δ = 1/2 (1, 2, 4), α = 1/2 for δ = 1 (2, 4), α = 1 for δ = 2 (4), and α = 2 for δ = 4, elsewhere it is left skewed.

Next, we consider a probability P (X < c · Y ) in the bivariate beta density function (2.1) of (X, Y ). The probability P (X < c · Y ) is very interesting for us, because the distribution F _R (r) of a ratio R = X/(X + Y ) is a special case of P (X < c · Y ) only when c = r/(1 − r) for 0 < r < 1.

Proposition 2.2 Let (X, Y ) be the bivariate beta density function (2.1). Then P (X < c · Y ) = Γ(α + δ)

Γ(α + 1)Γ(δ) c ^α 2 F 1 (α + δ, α; α + 1; −c) for c > 0,

where ₂ F ₁ (a, b; c; x) is a hypergeometric function in Gradshteyn and Ryzhik (1965).

Especially, if c = 1 in Proposition 2.1, the reliability R = P (X < Y ) of two random variables X and Y is given by :

R(α, δ) ≡ R = Γ(α + δ)

Γ(α + 1)Γ(δ) ² F 1 (α + δ, α; α + 1; −1)

= Γ(α + δ)

Γ(α + 1)Γ(δ) 2 ^1−α−δ 2 F 1 (1 − δ, 1; α + 1; −1), which it is a function of parameters α and δ.

Because the approximate MLE usually performs better than the moment estimator in the sense of MSE in Balakrishnan and Cohen (1991) and Son and Woo (2009), an estimator of the reliability R = P (X < Y ) could be recommended as follows:

R = ˆ Γ( ˆ α + ˆ δ)

Γ( ˆ α + 1)Γ(ˆ δ) 2 ^{1− ˆ α−ˆ δ} 2 F 1 (1 − ˆ δ, 1; ˆ α + 1; −1), where ˆ α and ˆ δ are approximate MLEs of α and δ, respectively in Section 3.

Next, since c moves over (0, ∞), when c is considered as a random variable over (0, ∞), we obtain the following :

Proposition 2.3 Let (X, Y ) be the bivariate gamma density function (2.1). Then

(a) if c is an exponential random variable with density function f c (x) = e ^−x , x > 0, then E _c (P (X < cY )) = Γ(α + δ)

Γ(δ) Ψ(α + δ, δ + 1; 1),

where Ψ(a, b; x) is a degenerate hypergeometric function in Gradshteyn and Ryzhik (1965).

(b) if c is a gamma random variable having density function f c (x) = _Γ(γ) ^β

x ^γ−1 e ^−βx , x > 0, then

E c (P (X < cY )) = β ^−α

Γ(α)Γ(δ)Γ(γ) · E(α + δ, α, α + γ; α + 1; β), where E(a, b, c; d; x) is a MacRobert’s E-function in Gradshteyn and Ryzhik (1965).

Proof

(a) It comes from formula 7.522 (4) in Gradshteyn and Ryzhik and the result in Proposition 2.3.

(b) It comes from formula 7.522 (1) in Gradshteyn and Ryzhik and the result in Proposition

2.2.

3. Approximate MLEs of three parameters α, δ and γ

Assume (x 1 , y 1 ), (x 2 , y 2 ), · · · , (x n , y n ) be observed samples from the bivariate gamma den- sity function (2.1) of (X, Y ). From Lemma 2.1 (a), (b) and (2.2) in Section 2, we can obtain the following moment estimates α 0 , δ 0 and γ 0 of α, δ and γ, respectively.

α 0 =

n

X

i=1

x i ·

n

X

i=1

x i y i /(

n

X

i=1

x i ·

n

X

i=1

y i − n

n

X

i=1

x i y i ),

δ 0 =

n

X

i=1

y i ·

n

X

i=1

x i y i /(

n

X

i=1

x i ·

n

X

i=1

y i − n

n

X

i=1

x i y i ) and

γ 0 = (1/y − 1) · δ 0 − α 0 . (3.1)

Example 3.1 (Random numbers) The followings (x, y) are random generating numbers satisfying x + y < 1, which numbers are rounding up from the 2th decimal point below :

(x, y) : (0.53,0.22) (0.32,0.56) (0.04,0.70) (0.79,0.18) (0.66,0.21) (0.56,0.33) (0.48,0.38) (0.72,0.18) (0.34,0.53) (0.26,0.57)

Assume pairs (x _i , y _i )s come from a bivariate beta density function (2.1).

Solution From moment estimates (3.1) of α, δ and γ,

x = 0.47, y = 0.386,

n

X

i=1

x _i y _i = 1.4298, which values are substituted in equations (3.1).

And then we can obtain the following moment estimates α 0 , δ 0 and γ 0 of α, δ and γ:

α 0 = 1.74819, δ 0 = 1.43575 and γ 0 = 0.53562.

Now, we consider the likelihood function to derive MLEs of α, δ and γ in density function (2.1). The log-likelihood function l(α, δ, γ) of α, δ and γ in density function (2.1) is given by :

l(α, δ, γ) =n ln Γ(α + δ + γ) − n ln Γ(α) − n ln Γ(δ) − n ln Γ(γ) + (α − 1)

n

X

i=1

ln x i + (δ − 1)

n

X

i=1

y i + (γ − 1)

n

X

i=1

ln(1 − x i − y i ). (3.2)

As partial differentiating l(α, δ, γ) with respect α, δ and γ to derive MLEs ˆ α, ˆ δ and ˆ γ of

α, δ and γ, respectively :

∂l

∂α = −nψ(α + δ + γ) − nψ(α) +

n

X

i=1

ln x _i ≡ p(α, δ, γ)

∂l

∂δ = −nψ(δ + δ + γ) − nψ(δ) +

n

X

i=1

ln y i ≡ q(α, δ, γ)

∂l

∂γ = nψ(α + δ + γ) − nψ(γ) +

n

X

i=1

ln(1 − x i − y i ) ≡ r(α, δ, γ), (3.3)

where ψ(x) is the psi-function.

When if p( ˆ α, ˆ δ, ˆ γ) = q( ˆ α, ˆ δ, ˆ γ) = r( ˆ α, ˆ δ, ˆ γ) = 0, then ˆ α, ˆ δ and ˆ γ are MLEs of α, δ and γ respectively. And from the results (3.3), MLEs ˆ α, ˆ δ and ˆ γ of α, δ and γ can’t explicitly be represented by closed form.

Since the approximate MLE usually performs better than the moment estimator in the sense of MSE in Balakrishnan and Cohen (1991) and Son and Woo (2009), the approximate MLE could be useful in a parametric estimation only when MLE can’t be represented by closed form. From equations (3.3), since MLEs ˆ α, ˆ δ and ˆ γ can’t explicitly be represented by closed form. And then we consider approximate MLEs ˆ α, ˆ δ and ˆ γ of α, δ and γ, respectively.

Based on the method of finding approximate MLE of the parameter in a distribution in Balakrishnan and Cohen (1991), from equations (3.3), as taking first two terms of Taylor’s series for p(α, δ, γ), q(α, δ, γ) and r(α, δ, γ) about moment estimates (α 0 , δ 0 , γ 0 ), approximate MLE ˆ α, ˆ δ and ˆ γ of α, δ and γ are obtained by the followings :

Define p ₀ = p(α ₀ , δ ₀ , γ ₀ ), q ₀ = q(α ₀ , δ ₀ , γ ₀ ) and r ₀ = r(α ₀ , δ ₀ , γ ₀ ), p _α ≡ p _α (α ₀ , δ ₀ , γ ₀ ), p _δ ≡ p _δ (α ₀ , δ ₀ , γ ₀ ) and p _β ≡ p _β (α ₀ , δ ₀ , γ ₀ ) are partial derivatives of p(α, δ, γ) with respect to α, δ and γ, respectively.

Define p α ≡ p α (α 0 , δ 0 , γ 0 ), p δ ≡ p δ (α 0 , δ 0 , γ 0 ) and p β ≡ p β (α 0 , δ 0 , γ 0 ) are partial deriva- tives of q(α, δ, γ) with respect to α, δ and γ, respectively,

And define p α ≡ p α (α 0 , δ 0 , γ 0 ), p δ ≡ p δ (α 0 , δ 0 , γ 0 ) and p β ≡ p β (α 0 , δ 0 , γ 0 ) are partial derivatives of r(α, δ, γ) with respect to α, δ and γ, respectively.

Then from equations (3.3), as taking first two terms of Taylor’s series for p(α, δ, γ), q(α, δ, γ) and r(α, δ, γ) about (α 0 , δ 0 , γ 0 ), we can obtain the following asymptotic equations :

p(α, δ, γ) ≈ p ₀ + p _α · (α − α 0 ) + p _δ · (δ − δ 0 ) + p _β · (γ − γ 0 ), q(α, δ, γ) ≈ q 0 + q α · (α − α 0 ) + q δ · (δ − δ 0 ) + q β · (γ − γ 0 ),

r(α, δ, γ) ≈ r ₀ + r _α · (α − α ₀ ) + r _δ · (δ − δ ₀ ) + r _β · (γ − γ ₀ ). (3.4)

From equations (3.3), since

p _α = nψ ⁰ (α ₀ + δ ₀ + γ ₀ ) − nψ ⁰ (α ₀ ), p δ = p γ = nψ ⁰ (α 0 + δ 0 + γ 0 ), q _δ = nψ ⁰ (α ₀ + δ ₀ + γ ₀ ) − nψ ⁰ (δ ₀ ), q α = q γ = nψ ⁰ (α 0 + δ 0 + γ 0 ), r γ = nψ ⁰ (α 0 + δ 0 + γ 0 ) − nψ ⁰ (γ 0 ), r _δ = r _α = nψ ⁰ (α ₀ + δ ₀ + γ ₀ ),

and then, as they are substituted in (3.4), MLEs ˆ α, ˆ δ and ˆ γ satisfy the following asymptotic linear equations :

−p 0 /n ≈(ψ ⁰ (α 0 + δ 0 + γ 0 ) − ψ ⁰ (α 0 )) · ( ˆ α − α 0 ) + ψ ⁰ (α 0 + δ 0 + γ 0 ) · (ˆ δ − δ 0 ) + ψ ⁰ (α 0 + δ 0 + γ 0 ) · (ˆ γ − γ 0 ),

−q ₀ /n ≈ψ ⁰ (α ₀ + δ ₀ + γ ₀ ) · ( ˆ α − α ₀ ) + (ψ ⁰ (α ₀ + δ ₀ + γ ₀ ) − ψ ⁰ (δ ₀ )) · (ˆ δ − δ ₀ ), + ψ ⁰ (α 0 + δ 0 + γ 0 ) · (ˆ γ − γ 0 )

−r ₀ /n ≈ψ ⁰ (α ₀ + δ ₀ + γ ₀ )( ˆ α − α ₀ ) + ψ ⁰ (α ₀ + δ ₀ + γ ₀ )(ˆ δ − δ ₀ )

+ (ψ ⁰ (α 0 + δ 0 + γ 0 ) − ψ ⁰ (γ 0 ))(ˆ γ − γ 0 ), (3.5) where ψ ⁰ (x) = dψ(x)/dx.

From asymptotic linear equations (3.5) and setting ψ ₀ ⁰ ≡ ψ ⁰ (α 0 + δ 0 + γ 0 ), we obtain the following :

Proposition 3.1 Let (x 1 , y 1 ), (x 2 , y 2 ), · · · , (x n , y n ) be a random sample from a bivariate beta density function (2.1). Then approximate MLEs ˆ α, ˆ δ and ˆ γ of α, δ and γ are given by :

ˆ

α ≈ α ₀ + det(D ₁ )/ det(D), δ ≈ δ ˆ ₀ + det(D ₂ )/ det(D), and

ˆ

γ ≈ γ 0 + det(D 3 )/ det(D), where

D =





ψ ⁰ ₀ − ψ ⁰ (α 0 ) ψ ⁰ ₀ ψ ⁰ ₀ ψ ₀ ⁰ ψ ⁰ ₀ − ψ ⁰ (delta 0 ) ψ ⁰ ₀ ψ ₀ ⁰ ψ ⁰ ₀ ψ ₀ ⁰ − ψ ⁰ (γ ₀ )



 ,

D ₁ =





−p 0 /n ψ ₀ ⁰ ψ ⁰ ₀

−q ₀ /n ψ ₀ ⁰ − ψ ⁰ (δ ₀ ) ψ ⁰ ₀

−r ₀ /n ψ ₀ ⁰ ψ ⁰ ₀ − ψ ⁰ (γ ₀ )



 ,

D ₂ =





ψ ₀ ⁰ − ψ ⁰ (α ₀ ) −p ₀ /n ψ ⁰ ₀ ψ ⁰ ₀ −q 0 /n ψ ⁰ ₀ ψ ⁰ ₀ −r 0 /n ψ ₀ ⁰ − ψ ⁰ (γ 0 )



 ,

and

D 3 =





ψ ₀ ⁰ − ψ ⁰ (α 0 ) ψ ₀ ⁰ −p 0 /n ψ ⁰ ₀ ψ ₀ ⁰ − ψ ⁰ (δ 0 ) −q 0 /n ψ ⁰ ₀ ψ ₀ ⁰ −r 0 /n



 .

and p 0 = p(α 0 , δ 0 , γ 0 ), q 0 = q(α 0 , δ 0 , γ 0 ) and r 0 = r(α 0 , δ 0 , γ 0 ) for moment estimates α 0 , δ 0

and γ 0 of α, δ and γ.

To assure the result in Proposition 3.1, the following example provides only numerical values of approximate MLEs of parameters after doing a fit of goodness for the bivariate density function (2.1).

Example 3.2 In Example 3.1, we can derive approximates MLEs as the followings:

First, we can obtain the following numerical values:

From Example 3.1, α ₀ = 1.74819, δ ₀ = 1.43575 and γ ₀ = 0.53562.

From data in Example 3.1,

n

X

i=1

ln x i = −9.7126,

n

X

i=1

ln y i = −10.7141,

n

X

i=1

ln(1 − x i − y i ) = −20.6886, and then from equations (3.3), p 0 = −0.44149, q 0 = 1.27136, and r 0 = 9.02063.

From formulas 6.3.5 and 6.4.6 in Abramowitz and Stegun (1970), ψ ⁰ ₀ − ψ ⁰ (α 0 ) = −0.45694, ψ ⁰ ₀ − ψ ⁰ (δ 0 ) = −0.68294,

ψ ₀ ⁰ − ψ ⁰ (γ 0 ) = −4.08364, ψ ⁰ ₀ = 0.30318.

And then from Proposition 3.1, approximate MLEs ˆ α, ˆ δ and ˆ γ of α, δ and γ are given by:

ˆ

α ≈ 2.184, ˆ δ ≈ 1.943 and ˆ γ ≈ 0.826.

which approximate MLE are more overestimated than moment estimates.

4. Conclusions

In this paper, we have derived the distribution and k-th moment of the ratio R = X/(X + Y ) in a bivariate beta distribution for (X, Y ). And we have observed coefficients of skewness of the ratio R = X/(X + Y ) with parameters (α, δ) and studied numerically trends for the skewness of the ratio R = X/(X + Y ) based on a bivariate beta distribution. Also, we have derived approximate MLEs for three parameters α, δ, γ using determinant and the reliability R = P (X < Y ) in the bivariate beta distribution for (X, Y ).

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