Bayesian one sided testing for the scale parameter of Lindley distribution ^†

Bayesian one sided testing for the scale parameter of Lindley distribution ^†

Jeonghwan Ko ¹

1 Department of Information Statistics, Andong National University

Received 12 October 2020, revised 12 November 2020, accepted 16 November 2020

Abstract

In this paper, we consider the problem of one sided testing on the scale parameter of Lindley distribution. We propose default Bayesian testing procedures for the scale parameters under Jeffreys’ priors. Jeffreys’ prior was usually improper which yielded a calibration problem that made the Bayes factor to be defined up to a multiplicative constant. Therefore we propose the default Bayesian testing procedures based on the fractional Bayes factor and the intrinsic Bayes factors under Jeffreys’ priors. We com- pared the proposed procedure using a Simulation study and an example are provided.

Keywords: Fractional Bayes factor, Intrinsic Bayes factor, Jeffreys’ prior, Lindley dis- tribution.

1. Introduction

The Lindley distribution was introduced by Lindley (1958) in the context of Bayesian statistics, as a counter example of fiducial statistics. The Lindley distribution is given by

f (x|θ) = θ ²

1 + θ (1 + x)e ^−θx , x > 0, (1.1) where θ > 0 is scale parameter. We denote this distribution as Lindley(θ).

Recently, many authors have paid great attention to the Lindley distribution as a lifetime model. Ghitany et al. (2008) showed that Lindley distribution was a better lifetime model than exponential distribution. They studied different properties and the necessary infer- ential procedure for the Lindley distribution, and showed that the density of the Lindley distribution was unimodal when 0 < θ < 1, and was decreasing when θ ≥ 1. Al-Mutairi et al. (2013) developed the inferential procedure of the stress-strength parameter when both stress and strength variables followed Lindley distribution. Gomoz-Deniz and Calderin-Ojeda (2011) developed a discrete Lindley model with its applications in collective risk modeling.

Mazucheli and Achcar (2011) studied a competing risk model when the causes of failures fol- lowed Lindley distribution. Krishna and Kumar (2011) estimated the parameter of Lindley

† This work was supported by a Research Grant of Andong National University.

1

Professor, Department of Information Statistics, Andong National University, Kyungpook 36729, Ko-

rea. E-mail: [email protected]

distribution with progressive type-II censoring scheme. Singh and Gupta (2012) studied a k-component load-sharing parallel system model in which each component’s lifetime followed Lindley distribution.

The present paper focuses on one sided testing the scale parameters in Lindley distribu- tion. In Bayesian model selection or testing problem, the Bayes factor under proper priors or informative priors has been very successful. However, limited information and time con- straints often required the use of noninformative priors. Since noninformative priors such as Jeffreys’ prior or reference prior (Berger and Bernardo, 1989, 1992) were typically improper, such priors were only defined up to arbitrary constants which affected the values of Bayes factors.

Berger and Pericchi (1996) introduced the intrinsic Bayes factor using a data-splitting idea, which would eliminate the arbitrariness of improper prior. O’Hagan (1995) proposed the fractional Bayes factor. He used a portion of the likelihood with a so-called the fraction b in order to remove the arbitrariness. These approaches have shown to be quite useful in many statistical areas (Lee et al., 2017; Kang, 2020). Berger and Pericchi(2001) have developed an excellent exposition of the objective Bayesian method for model selection.

In this paper, we propose the objective Bayesian one sided hypothesis testing procedures for the scale parameters in Lindley distribution based on the Bayes factors. Simulation study and an example are provided.

2. Bayesian hypothesis testing procedures

Let X i , i = 1, · · · , n, denote observations from Lindley(θ) with the scale parameter θ.

Then likelihood function of θ is given by

f (x|θ) = θ ²ⁿ (1 + θ) ⁿ

n

Y

i=1

(1 + x i ) exp (

−θ

n

X

i=1

x i

)

, (2.1)

where x = (x 1 , · · · , x n ). We are interested in testing the hypotheses H 0 : 0 < θ < 1 versus H 1 : θ ≥ 1 based on the fractional Bayes factor and the intrinsic Bayes factors.

2.1. The fractional Bayes factor methodology

From (2.1) the likelihood function under the hypothesis H 0 : 0 < θ < 1 is

L 0 (θ|x) = θ ²ⁿ (1 + θ) ⁿ

n

Y

i=1

(1 + x i ) exp (

−θ

n

X

i=1

x i

)

, (2.2)

where 0 < θ < 1. And under the hypothesis H 0 , Jeffreys’ prior for θ is

π ₀ ^N (θ) ∝ θ ⁻¹ (1 + θ) ⁻¹ (θ ² + 4θ + 2)

, (2.3) where 0 < θ < 1. Now the fractional Bayes factor (FBF) of O’Hagan (1995) under hypothesis H 1 versus hypothesis H 0 is given by

B ₁₀ ^F = B ₁₀ ^N · R L ^b (x|θ ₀ )π ^N ₀ (θ ₀ )dθ ₀

R L ^b (x|θ ₁ )π ^N ₁ (θ ₁ )dθ ₁ = B ^N ₁₀ · m ^b ₀ (x)

m ^b ₁ (x) , (2.4)

where π ^N ₀ and π ^N ₁ are the noninformative prors, b is a fraction and

B ₁₀ ^N = m ^N ₁ (x) m ^N ₀ (x) .

Then from the likelihood (2.2) and Jeffreys’ prior (2.3), the element m ^b ₀ (x) in the FBF under H ₀ is given by

m ^b ₀ (x) = Z 1

0

L ^b ₀ (θ|x)π ₀ ^N (θ)dθ

=

n

Y

i=1

(1 + x _i ) ^b Z 1

0

θ ^2bn−1 (θ ² + 4θ + 2)

(1 + θ) ^bn+1 exp

(

−bθ

n

X

i=1

x _i )

dθ. (2.5)

For the hypothesis H 1 , Jeffreys’ prior for θ is

π ^N (θ) ∝ θ ⁻¹ (1 + θ) ⁻¹ (θ ² + 4θ + 2)

, (2.6) where θ ≥ 1. The likelihood function under the hypothesis H 1 is

L 1 (θ|x) = θ ²ⁿ (1 + θ) ⁿ

n

Y

i=1

(1 + x i ) exp (

−θ

n

X

i=1

x i

)

, (2.7)

where θ ≥ 1. Thus from the likelihood (2.7) and Jeffreys’ prior (2.6), the element m ^b ₁ (x) of FBF under H ₁ is given as follows.

m ^b ₁ (x) = Z ∞

1

L ^b ₂ (θ|x)π ₂ ^N (θ)dθ

=

n

Y

i=1

(1 + x i ) ^b Z ∞

1

θ ^2bn−1 (θ ² + 4θ + 2)

(1 + θ) ^bn+1 exp

(

−bθ

n

X

i=1

x i

)

dθ. (2.8)

Therefore the element B ₁₀ ^N of FBF is given by

B ^N ₁₀ = S 1 (x)

S ₀ (x) , (2.9)

where

S ₀ (x) = Z 1

0

θ ²ⁿ⁻¹ (θ ² + 4θ + 2)

(1 + θ) ⁿ⁺¹ exp

(

−θ

n

X

i=1

x _i )

dθ

and

S ₁ (x) = Z ∞

1

θ ²ⁿ⁻¹ (θ ² + 4θ + 2)

(1 + θ) ⁿ⁺¹ exp

(

−θ

n

X

i=1

x _i )

dθ.

And the ratio of marginal pdf’s with fraction b is m ^b ₀ (x)

m ^b ₁ (x) = S 0 (x; b)

S ₁ (x; b) , (2.10)

where

S ₀ (x; b) = Z 1

0

θ ^2bn−1 (θ ² + 4θ + 2)

(1 + θ) ^bn+1 exp

(

−bθ

n

X

i=1

x _i )

dθ

and

S 1 (x; b) = Z ∞

1

θ ^2bn−1 (θ ² + 4θ + 2)

(1 + θ) ^bn+1 exp

(

−bθ

n

X

i=1

x i

) dθ.

Thus the FBF of H ₁ versus H ₀ is given by

B ₁₀ ^F = S 0 (x; b)S 1 (x)

S 0 (x)S 1 (x; b) . (2.11)

Note that the calculations of the FBF of H 1 versus H 0 require one dimensional integration.

2.2. The intrinsic Bayes factor methodology

The element B ₁₀ ^N of the intrinsic Bayes factor is computed in the fractional Bayes factor. So under minimal training sample, we only calculate the marginal densities for the hypotheses H 0 and H 1 , respectively. The marginal densities of X j are finite for all 1 ≤ j ≤ n under each hypothesis. Thus we conclude that any training sample of size 1 is a minimal training sample.

The marginal densities m ^N ₂ (x _j ) under H ₂ (= H ₀ ∪ H 1 ) : θ > 0 are given by

m ^N ₂ (x j ) = Z ∞

0

f (x j |θ)π ₁ ^N (θ)dθ

= (1 + x j ) Z ∞

0

θ(θ ² + 4θ + 2)

(1 + θ) ² exp {−θx j } dθ.

The marginal pdf m ^N ₀ (x _j ) under H ₀ is given by

m ^N ₀ (x j ) = Z 1

0

f (x j |θ)π ₀ ^N (θ)dθ

= (1 + x j ) Z 1

0

θ(θ ² + 4θ + 2)

(1 + θ) ² exp {−θx j } dθ.

And the marginal pdf m ^N ₁ (x _j ) under H ₁ is given by

m ^N ₁ (x j ) = Z ∞

1

f (x j |θ)π ₁ ^N (θ)dθ

= (1 + x j ) Z ∞

1

θ(θ ² + 4θ + 2)

(1 + θ) ² exp {−θx j } dθ.

Now the encompassing arithmetic intrinsic Bayes factor (EIBF) of Berger and Pericchi (1996) is then defined as

B ₁₀ ^EI = B ₁₀ ^N × P L

l=1 B ₀₂ ^N (x(l)) P L

l=1 B ₁₂ ^N (x(l)) , (2.12)

where B ^N _i2 (x(l)) = m ^N _i (x(l))/m ^N ₂ (x(l)), i = 0, 1 and x(l) is the minimal training sample.

Note the sum over all the possible minimal training samples contained in the sample is computed. Therefore the EIBF of H 1 versus H 0 is given by

B ₁₀ ^EI = S ₁ (x) S 0 (x)

" P n

j=1 T 0 (x j )/T 2 (x j ) P n

j=1 T 1 (x j )/T 2 (x j )

#

, (2.13)

where

T 2 (x j ) = Z ∞

0

θ(θ ² + 4θ + 2)

(1 + θ) ² exp {−θx j } dθ, T ₀ (x _j ) =

Z 1 0

θ(θ ² + 4θ + 2)

(1 + θ) ² exp {−θx _j } dθ and

T ₁ (x _j ) = Z ∞

1

θ(θ ² + 4θ + 2)

(1 + θ) ² exp {−θx _j } dθ.

Also the median intrinsic Bayes factor (MIBF) by Berger and Pericchi (1998) of H 1 to H 0

is

B ₁₀ ^{M I} = B ^N ₁₀ × M E[B ₀₁ ^N (x(l))], (2.14) where M E indicates the median for all the training sample Bayes factors. Then the MIBF of H 1 versus H 0 is given by

B ₁₀ ^{M I} = S ₁ (x)

S 0 (x) M E  T ₀ (x _j ) T 1 (x j )



. (2.15)

Note that the calculations of the EIBF and the MIBF of H ₁ versus H ₀ require one dimen- sional integration.

3. Numerical studies

In order to assess the Bayesian hypothesis testing procedures, we investigate the posterior probability for several configurations of θ and n. In particular, for fixed θ, we take 1,000 independent random samples of X _i with sample size n from Lindley distribution with θ. We want to test the hypotheses H 0 : 0 < θ < 1 versus H 1 : θ ≥ 1. The posterior probabilities of H 0 being true are computed assuming equal prior probabilities.

Table 3.1 shows the results of the averages and the standard deviations in parentheses of posterior probabilities. In Table 3.1, P ^F (·), P ^EI (·) and P ^{M I} (·) are the posterior probabilities of the hypothesis H 0 being true based on FBF, EIBF and MIBF, respectively. From the results of Table 3.1, the EIBF and the MIBF give a similar behavior for all sample sizes.

However the FBF supports the hypothesis H 1 better than the EIBF and the MIBF. We can see that the FBF prefers the hypothesis H 1 , especially when the sample size is small and the values of θ is around 1.

Example 1. This example was taken from Zakerzadeh and Mahmoudi (2013). We consider

vinyl chloride data obtained from clean upgradient monitoring wells in mg/L, and this

Table 3.1 The Averages and the standard deviations in parentheses of posterior probabilities θ n P

(H

|x) P

(H

|x) P

(H

|x)

0.6 5 0.734(0.225) 0.803 (0.212) 0.780 (0.213) 10 0.888(0.160) 0.925 (0.130) 0.911 (0.138) 15 0.942(0.116) 0.962 (0.094) 0.961 (0.095) 0.8 5 0.577(0.221) 0.659 (0.236) 0.644 (0.229) 10 0.640(0.242) 0.719 (0.234) 0.703 (0.229) 15 0.705(0.238) 0.775 (0.221) 0.778 (0.214) 0.9 5 0.489(0.220) 0.567 (0.245) 0.554 (0.237) 10 0.536(0.248) 0.617 (0.253) 0.608 (0.247) 15 0.554(0.262) 0.632 (0.263) 0.642 (0.256) 1.0 5 0.428(0.201) 0.503 (0.235) 0.501 (0.230) 10 0.423(0.244) 0.497 (0.266) 0.494 (0.254) 15 0.423(0.249) 0.500 (0.267) 0.518 (0.262) 1.2 5 0.319(0.178) 0.375 (0.215) 0.390 (0.217) 10 0.260(0.190) 0.316 (0.226) 0.324 (0.219) 15 0.238(0.189) 0.294 (0.224) 0.318 (0.230) 1.5 5 0.217(0.145) 0.253 (0.177) 0.285 (0.192) 10 0.126(0.124) 0.155 (0.154) 0.170 (0.161) 15 0.077(0.101) 0.097 (0.126) 0.111 (0.136) 1.8 5 0.144(0.113) 0.164 (0.138) 0.199 (0.167) 10 0.054(0.069) 0.065 (0.085) 0.076 (0.095) 15 0.023(0.043) 0.028 (0.056) 0.035 (0.065)

Table 3.2 Bayes factor and posterior probabilities of H

: 0 < θ < 1 B

P

(H

|x) B

P

(H

|x) B

P

(H

|x)

0.135 0.881 0.063 0.941 0.056 0.947

data set was used for Bhaumik et al. (2009) in fitting the gamma distribution for small samples. The data are: 5.1, 1.2, 1.3, 0.6, 0.5, 2.4, 0.5, 1.1, 8.0, 0.8, 0.4, 0.6, 0.9, 0.4, 2.0, 0.5, 5.3, 3.2, 2.7, 2.9, 2.5, 2.3, 1.0, 0.2, 0.1, 0.1, 1.8, 0.9, 2.0, 4.0, 6.8, 1.2, 0.4, 0.2. For this data set, Zakerzadeh and Mahmoudi (2013) showed that Lindley distribution provides good fit based on Kolmogorov-Smirnov, Akaike Information Criterion, Bayesian Information Criterion statistics, etc. The maximum estimate of θ is 0.8238.

We want to test the hypotheses H 0 : 0 < θ < 1 versus H 1 : θ ≥ 1. The values of the Bayes factors and the posterior probabilities of H 0 are given in Table 3.2. From the results of Table 3.2, the posterior probabilities based on various Bayes factors give the same answer, and the FBF, the EIBF and the MIBF select the hypothesis H 1 . But the FBF supports the hypothesis H 1 better than the EIBF and the MIBF.

4. Concluding remarks

In this paper, we developed the objective Bayesian one sided hypothesis testing procedures

based on the fractional Bayes factor and the intrinsic Bayes factors for the scale parameter

of Lindley distribution under Jeffreys’ priors. From our numerical results, the developed

hypothesis testing procedures gave fairly reasonable answers for all parameter configurations.

However the FBF supports the hypothesis H ₁ better than the EIBF and the MIBF. From our simulation and example, we recommended the use of the FBF than the EIBF and MIBF for practical application in view of its simplicity and ease of implementation.

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