# On Statistical Properties of a New Bivariate Modified Lindley Distribution with an Application to Financial Data

Authors: Ahmed Elhassanein
Journal: Complexity
 (2022)

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## Abstract

There is an increasing interest in expanding the one-parameter Lindley distribution to two-parameter, three-parameter, and five-parameter. The univariate one-parameter Lindley distribution is still one of the most applicable distributions in data analysis especially in lifetime data. Modeling dependent random quantities required bivariate parametric probability distributions. This study presents a new bivariate three-parameter probability distribution called bivariate modified Lindley distribution. The one-parameter modified Lindley distribution is used as a base line to construct the new model. Its statistical properties including cumulative function, density function, marginals, moments, conditional distributions, and copula are discussed. Simulation is constructed to declare theoretical properties, to show the flexibility of the new model and to investigate the goodness of fit. Two sets of real data, financial data and UEFA Champion’s League data, are used to show the applicability of the proposed model for different types of data.


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## Body

## 1. Introduction

Parametric probability distributions are very powerful tools for data analysis. The improvement of such distributions had a great attention [1–6]. Lindley [7] introduced the one-parameter Lindley distribution as a developed version of the exponential distribution. He considered the probability density function and cumulative function in the form(1)fx;θ=θ21+θ1+xe−θx,θ,x>0,(2)Fx;θ=1−1+θx1+θe−θx.Recently, Lindley distribution and its modified forms have been applied in different fields such as engineering, physics, medicine, and finance. Ghitany et al. [8] studied statistical properties of the Lindley distribution. They proved the applicability of the distribution for the waiting times before service of the bank customers and its superiority in comparison with the exponential distribution, see also [9–21]. Shankar and Mishra [22] developed a two-parameter version of the one-parameter Lindley distribution. The probability density function and cumulative function were formulated from (1) and (2) in the form(3)fx;α;θ=θ21+αθα+xe−θx,α,θ,x>0,(4)Fx;α;θ=1−1+α+xθ1+αθe−θx.Forα=1, the one-parameter Lindley distribution is obtained. Shanker et al. [23] constructed a new three-parameter Lindley distribution with probability density function and cumulative function:(5)fx;α;θ;β=θ2β+αθα+βxe−θx,αθ+β,θ,β,x>0,(6)Fx;α;θ;β=1−1+α+θβxβ+αθe−θx.The model was used to analyze lifetime data. Asgharzadeh et al. [24] pursued the method by Gupta and Kundu [25], to introduce a weighted Lindley (NWL) distribution. Wongrin and Bodhisuwan [26] modeled a count data using a new modified Lindley distribution called Poisson-generalised Lindley distribution. Maurya et al. [27] analyzed survival times’ data via a new three-parameter Lindley distribution formulated following the work of Cordeiro and De Castro [28]. Mota et al. [29] discussed mathematical properties of a reparameterized version of the weighted Lindley that can be used for data with bathtub-shaped or increasing hazard rate function distribution. Rather and Özel [30] propose a new weighted power Lindley distribution for modeling lifetime data. Elbatal et al. [31] extended the power Lindley distribution to a new five-parameter lifetime model named the exponentiated Kumaraswamy power-Lindley distribution. Chesneau et al. [32] derived a new inverted modified Lindley distribution. Diandarma et al. [33] formulated a discrete version of the Lindley distribution and conducted simulation to show that it has unimodal, right skew, high fluidity, and overdispersion.Dependent random data appear in many fields as reliability, survival analysis, queueing analysis, insurance risk analysis, life insurance, and finance. For example, in lifetime data analysis, the lifetimes of organisms or items are most often dependent. Carriere [34] reported a high positive correlation between the times of deaths of coupled lives. Jagger and Sutton [35] discussed the significantly increase of the risk of mortality after the marital bereavement. Bivariate parametric probability distributions play a fundamental role in analyzing dependent data. There is an increasing interest in generating new bivariate models. Common ideas for generating bivariate distribution introduced by Balakrishnan and Lai [36] and Kundu and Gupta [37, 38] are employed by many researchers to introduce new distribution based on different marginals [39–43]. Although this method suits lifetime data, it is not applicable for a lot of other types of data. Maximum likelihood estimators cannot be obtained in the closed form. The joint pdf can take different shapes and it has a singular part. The bivariate modeling is still a challenge issue. Vaidyanathan and Varghese [44] proposed a new bivariate Lindley distribution using Morgenstern approach which can be used for analyzing bivariate lifetime data. Thomas and Jose [45] considered a marginal Rayleigh distribution and impound it with the pdf of an extended form of Lindley distribution to generate a new bivariate distribution.In this study, we employed the idea discussed in [46, 47], to construct a new simple bivariate modified Lindley distribution based on the one-parameter modified Lindley distribution. The new distribution is flexible; its statistical quantities are available in explicit forms. The joint probability density function is absolutely continuous. The maximum likelihood function can be obtained. It is applicable for lifetime data, financial data, and other types of data including heavy-tailed data and approximately symmetric data. Its hazard function gives various shapes according to the parameters.The reset of the study is organized as follows. The new distribution is formulated in Section2. Statistical properties are discussed in Section 3. Maximum likelihood estimators are derived in Section 4. Simulation is presented in Section 5. Real financial data are used to apply the new model in Section 6. Conclusion and remarks are given in Section 7.


## 2. Model Description

Recently, Chesneau et al. [32] introduced a new distribution named modified Lindley (ML) distribution that depends on only one parameter, and its cumulative distribution function is given by(7)Fx=1−1+θx1+θe−θxe−θx,x>0.The cdf (7) will be used as a base line distribution to generate a bivariate modified Lindley distribution following Alzaatreh et al. [48] and Ganji et al. [49]. Let X,Y be a random vector with joint probability density function hX,Yx,y, where x∈α1,β1,y∈α2,β2, −∞<α1<β1<∞, and −∞<α2<β2<∞. Let Ψ,Ω be a random vectors with marginals HΨψ and HΩω, respectively, and let FΨHΨψ and FΩHΩω be functions such that FΨHΨψ and FΩHΩω∈α2,β2 and FΨHΨψ and FΩHΩω are differentiable and monotonically nondecreasing functions and FΨHΨψ⟶α1 as ψ⟶−∞, FΨHΨψ⟶β1 as ψ⟶∞,FΩHΩω⟶α2 as ω⟶−∞, and FΩHΩω⟶β2 as ω⟶∞. Then, the cdf of random variable Ψ,Ω is given by FΨ,ΩFΨHΨψ,FΩHΩω=∫α1F1HΨψ∫α2FΩHΩωhX,Yx,ydxdy.Suppose the transformation functionhX,Yx,y=1+ξ11−x+ξ21−y+2ξ31−x−y, where x,y∈0,1,ξ1,ξ2,ξ3∈−1,1,−1≤ξ1+ξ3≤1, and −1≤ξ2+ξ3≤1. Then, the cumulative function of the modified Lindley random vector Ψ,Ω is given by(8)GΨ,Ωψ,ω=1−1+θψ1+θe−θψe−θψ1−1+θω1+θe−θωe−θω×1+ξ1+ξ31+θψ1+θe−θψe−θψ+ξ2+ξ31+θω1+θe−θωe−θω,where θ>0, ψ>0, and ω>0.Definition 1.
A random vectorΨ,Ω is said to have a modified Lindley distribution with parameter θ,ξ1,ξ2,ξ3, where θ>0, −1<ξ1+ξ3<1,−1<ξ2+ξ3<1, if its cumulative function is given by (8) for ψ>0 and ω>0 and will be denoted by Ψ,Ω∼BIMLθ,ξ1,ξ2,ξ3.


## 3. Statistical Properties

In this section, statistical properties of the new distribution are discussed.Lemma 1.
LetΨ,Ω∼BIMLθ,ξ1,ξ2,ξ3. Then, its probability density function is given by(9)gΨ,Ωψ,ω=θ2e−θψ+ω1−1−2θψ1+θe−θψ1−1−2θω1+θe−θω×1+ξ1+ξ321+θψ1+θe−θψe−θψ−1+ξ2+ξ321+θω1+θe−θωe−θω−1.Lemma 2.
LetΨ,Ω∼BIMLθ,ξ1,ξ2,ξ3. Then, its marginals are given by(10)GΨψ=1−1+θψ1+θe−θψe−θψ×1+ξ1+ξ31+θψ1+θe−θψe−θψ,GΩω=1−1+θω1+θe−θωe−θω×1+ξ2+ξ31+θω1+θe−θωe−θω,gΨψ=θe−θψ1−1−2θψ1+θe−θψ1+ξ1+ξ321+θψ1+θe−θψe−θψ−1,(11)gΩω=θe−θω1−1−2θω1+θe−θω1+ξ2+ξ321+θω1+θe−θωe−θω−1.Lemma 3.
LetΨ,Ω∼BIMLθ,ξ1,ξ2,ξ3. Then,(12)gψ/Ωψ/ω=θe−θψ1−1−2θψ1+θe−θψ1+Δξ1,ψ1+Δξ2,ω,(13)gΩ/Ψω/ψ=θe−θω1−1−2θω1+θe−θω1+Δξ2,ω1+Δξ1,ψ,where(14)Δξ,z=ξ+ξ31+θz1+θe−θze−θz−1.Lemma 4.
LetΨ,Ω∼BIMLθ,ξ1,ξ2,ξ3. Then,(15)μΨ/Ωrω=EΨr/Ω=ω=1+ξ1+ξ31+Δξ2,ω1−12r+111+θΓr+1θr+12r+111+θΓr+2θr−2ξ1+ξ31+Δξ2,ω1−12r+1θ+21+θ+13r+111+θΓr+1θr+12r+111+θ−13r+111+θ+14r+211+θ2Γr+2θr−122r+511+θ2Γr+3θr,(16)μΩ/Ψrψ=EΩr/Ψ=ψ=1+ξ2+ξ31+Δξ1,ψ1−12r+111+θΓr+1θr+12r+111+θΓr+2θr−2ξ2+ξ31+Δξ1,ψ1−12r+1θ+21+θ+13r+111+θΓr+1θr+12r+111+θ−13r+111+θ+14r+211+θ2Γr+2θr−122r+511+θ2Γr+3θr.Lemma 5.
LetΨ,Ω∼BIMLθ,ξ1,ξ2,ξ3. Then,(17)μr,s=EΨrΩs=1+ξ1+ξ2+2ξ3×1−12r+111+θΓr+1θr+12r+111+θΓr+2θr×1−12s+111+θΓs+1θs+12s+111+θΓs+2θs−2ξ1+ξ31−12s+111+θΓs+1θs+12s+111+θΓs+2θs×1−12r+1θ+21+θ+13r+111+θΓr+1θr+12r+111+θ−13r+111+θ+14r+211+θ2Γr+2θr−14r+321+θ2Γr+3θr+ξ2+ξ31−12r+111+θΓr+1θr+12r+111+θΓr+2θr×1−12s+1θ+21+θ+13s+111+θΓs+1θs+12s+111+θ−13s+111+θ+14s+211+θ2Γs+2θs−14s+321+θ2Γs+3θs.Lemma 6.
LetΨ,Ω∼BIMLθ,ξ1,ξ2,ξ3. Then, the bivariate reliability function is given by(18)Rψ,ω=1−1−1+θψ1+θe−θψe−θψ×1+ξ1+ξ31+θψ1+θe−θψe−θψ−1−1+θω1+θe−θωe−θω×1+ξ2+ξ31+θω1+θe−θωe−θω+e−θψ+ω1−1+θψ1+θe−θψ1−1+θω1+θe−θω×1+ξ1+ξ31+θψ1+θe−θψ+ξ2+ξ31+θω1+θe−θω.Lemma 7.
LetΨ,Ω∼BIMLθ,ξ1,ξ2,ξ3. Then, the bivariate hazard rate function is given by(19)hψ,ω=θ2e−θψ+ω1−1−2θψ1+θe−θψ1−1−2θω1+θe−θω×1+ξ1+ξ321+θψ1+θe−θψe−θψ−1+ξ2+ξ321+θω1+θe−θωe−θω−1×1−1−1+θψ1+θe−θψe−θψ×1+ξ1+ξ31+θψ1+θe−θψe−θψ−1−1+θω1+θe−θωe−θω×1+ξ2+ξ31+θω1+θe−θωe−θω+1−1+θψ1+θe−θψe−θψ1−1+θω1+θe−θωe−θω×1+ξ1+ξ31+θψ1+θe−θψ+ξ2+ξ31+θω1+θe−θω−1.The Copula function is of the most important quantities that is used to describe the stochastic dependence between continuous random variables. LetΨ,Ω∼BMLθ; since GΨ,Ωψ,ω=PΨ≤ψ,Ω≤ω, GΨψ=PΨ≤ψ, and GΩω=PΩ≤ω, then its copula function can be defined as GΨ,Ωψ,ω≜Cu,v, where u=GΨψ, v=GΩω, and u,v∈0,1 [50–52].Lemma 8.
LetΨ,Ω∼BIMLθ,ξ1,ξ2,ξ3. Then, its copula function cu,v=∂2Cu,v/∂u∂v is given by(20)cu,v=1+Δξ1,ψ+Δξ2,ω1+Δξ1,ψ1+Δξ2,ω,where Δξ,z is given by (14).


## 4. Estimation

Considering a random sampleψ1,ω1,ψ2,ω2,…,ψn,ωn from the bivariate modified Lindley random variable Ψ,Ω∼BIMLθ,ξ1,ξ2,ξ3, the maximum log-likelihood function for the unknown parameters Ξ=θ,ξ1,ξ2,ξ3′ is given by(21)LΞ=2nlnθ−θ∑i=1nψi+ωi+∑i=1nln1−1−2θψi1+θe−θψi+∑i=1nln1−1−2θωi1+θe−θωi+∑i=1nln1+ξ1+ξ321+θψi1+θe−θψie−θψi−1+ξ2+ξ321+θωi1+θe−θωie−θωi−1.The score vectorΞ=Ξθ,Ξζ1,Ξξ2,Ξξ3′ is given by(22)Ξθ=2nθ−∑i=1nψi+ωi+∑i=1ne−θψi1+3+θψi−2θ1+θψi21+θ1+θ+2θψi−1e−θψi+∑i=1ne−θωi1+3+θωi−2θ1+θωi21+θ1+θ+2θωi−1e−θωi−2ξ1+ξ3∑i=1nψie−θψi1−1−2θ1+θψi/1+θ2e−θψi1+Δξ1,ψi+Δξ2,ωi−2ξ2+ξ3∑i=1nωie−θωi1−1−2θ1+θωi/1+θ2e−θωi1+Δξ1,ψi+Δξ2,ωi,(23)Ξξ1=∑i=1n21+θψi/1+θe−θψie−θψi−11+Δξ1,ψi+Δξ2,ωi,Ξξ2=∑i=1n21+θψi/1+θe−θωie−θωi−11+Δξ1,ψi+Δξ2,ωi,Ξξ3=∑i=1n21+θψi/1+θe−θψie−θψi−11+Δξ1,ψi+Δξ2,ωi+∑i=1n21+θψi/1+θe−θωie−θωi−11+Δξ1,ψi+Δξ2,ωi,where Δξ,z is given by (14).


## 5. Simulation

In this section, three sets of parameters are used to discuss the behaviour of the new bivariate distribution, to show its flexibility and to investigate the goodness of fit: the set I,θ,ξ1,ξ2,ξ3=0.9,0.5,0.4,0.1, the set II, θ,ξ1,ξ2,ξ3=0.4,0.1,−0.4,0.6, and the set III, θ,ξ1,ξ2,ξ3=2,0.8,0.7,−0.4. For the set I, we can observe the heavy tail and unimodality of the joint probability density function (Figure 1(a)). The marginal densities are unimodal with right tail (Figures 1(f) and 1(h)). The joint cumulative function approaches 1 for small values ψ,ω (Figure 1(b)). It is also the case for the cumulative marginals (Figures 1(g) and 1(i)). The hazard function is approximately constant, see Figure 1(c). The level of independence increases with the increase of the value of u,v, see Figures 1(d) and 1(e). For the set II, the density function loses heavy tail property (Figure 2(a)). The marginal densities still have right tail (Figures 2(f) and 2(h)). The joint cumulative function and cumulative marginals converge the maximum value slowly in comparison with the set I, see Figures 2(b), 2(g), and 2(i). The hazard function is an increasing function (Figure 2(c)). The independence structure of the two random variables can be observed in Figures 2(d) and 2(e). For the set III, the joint probability density function and marginals have heavy tail and unimodal (Figures 3(a), 3(f), and 3(h)). The joint cumulative function and cumulative marginals approach the maximum value fast in comparison with that for set I (Figures 3(b), 3(g), and 3(i)). The hazard function changes its behaviour from increasing to decreasing (Figure 3(c)). We can observe the increase of the copula function with the increase of u,v (Figures 3(d) and 3(e)). It is clear that the model has different properties for different values of parameters and the hazard function has different shapes. To investigate the applicabilty, the performance of the maximum likelihood method, and the goodness of fit, Monte Carlo simulation is preformed to generate samples for the three sets of parameters (I, II, III) in addition to two other sets IV and V, where IV θ,ξ1,ξ2,ξ3=0.5,−0.8,−0.4,0.6 and V θ,ξ1,ξ2,ξ3=1.5,−0.8,0.4,0.3. The sizes of the samples are different n=20,50,100. The parameters ξ1,ξ2, and ξ3 are treated as chosen parameters. The maximum likelihood method is used to estimate the unknown parameter θ; the results are given in Table 1. We note that the bias and the standard error decrease with the increase of the sample size. The maximum likelihood method gives good estimates for the unknown parameter.Figure 1
Statistical quantities for the set I of parameters,θ,ξ1,ξ2,ξ3=0.9,0.5,0.4,0.1.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Figure 2
Statistical quantities for the set II of parameters,θ,ξ1,ξ2,ξ3=0.4,0.1,−0.4,0.6.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Figure 3
Statistical quantities for the set III of parameters,θ,ξ1,ξ2,ξ3=2,−0.7,−0.9,1.2.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Table 1
The average estimates for the parameterθ, for sets of data I, II, and III, with bias and standard error.
Set of dataSample sizeAverage estimateBiasSEIn = 200.8461−0.05390.0348n = 500.8626−0.30740.0219n = 1000.8200−0.08000.0180IIn = 200.3810−0.01900.0149n = 500.3836−0.01640.0110n = 1000.3885−0.01150.00740IIIn = 201.9182−0.08180.0529n = 501.9746−0.02540.0524n = 1001.9912−0.00880.0378IVn = 200.47298−0.027020.0290n = 500.46070−0.03930.00620n = 1000.4786−0.02140.00809Vn = 201.3483−0.15170.0250n = 501.3536−0.14640.0318n = 1001.4036−0.09640.0252


## 6. Real-Data Application

In this section, two data sets are used to apply the new model,BIMLθ,ξ1,ξ2,ξ3.


### 6.1. Financial Data

In this section, we applyBIMLθ,ξ1,ξ2,ξ3 to a bivariate financial dataset. The dataset consists of the absolute normalization of the daily closing prices of NASDAQ Composite index and the Microsoft Corporation share price between 1 January 2020 and 31 December 2020. The data are available at https://finance.yahoo.com/. Fung and Seneta [53] analyzed the returns of the daily closing prices of NASDAQ Composite index and the Microsoft Corporation share price between 1 January 1996 and 31 December 2005 as a bivariate set. The maximum likelihood estimates with Akaike’s information criterion (AIC) and Bayesian information criterion (BIC) are given in Table 2. The statistical functions related to the fitted distribution are presented in Figure 4.Table 2
Estimated parameters for the financial data with AIC and BIC.
ParameterEstimateAICBICθ1.4191778697.7194711.8687ξ10.3844708ξ2−0.1507097ξ3−0.5212492Figure 4
Statistical functions fitted for the dataset: (a) joint density function, (b) cumulative function, (c) hazard function, (d) copula function, (e) contour plot of copula function, and (f)–(i) marginals.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


### 6.2. UEFA Champion’s League Data

In this section, the UEFA Champion’s League data obtained from [54] have been analyzed to show the applicability of the new model and the performance of the maximum likelihood method for different types of data. The estimated parameters with AIC and BIC are in Table 3. The statistical functions related to the fitted distribution are presented in Figure 5.Table 3
Estimated parameters for the UEFA Champion’s League data with AIC and BIC.
ParameterEstimateAICBICθ0.02873765616.8248624.2686ξ1−0.55156010ξ20.61205812ξ30.10996939Figure 5
Statistical functions fitted for the dataset: (a) joint density function, (b) cumulative function, (c) hazard function, (d) copula function, (e) contour plot of copula function, and (f)–(i) marginals.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 6.1. Financial Data

In this section, we applyBIMLθ,ξ1,ξ2,ξ3 to a bivariate financial dataset. The dataset consists of the absolute normalization of the daily closing prices of NASDAQ Composite index and the Microsoft Corporation share price between 1 January 2020 and 31 December 2020. The data are available at https://finance.yahoo.com/. Fung and Seneta [53] analyzed the returns of the daily closing prices of NASDAQ Composite index and the Microsoft Corporation share price between 1 January 1996 and 31 December 2005 as a bivariate set. The maximum likelihood estimates with Akaike’s information criterion (AIC) and Bayesian information criterion (BIC) are given in Table 2. The statistical functions related to the fitted distribution are presented in Figure 4.Table 2
Estimated parameters for the financial data with AIC and BIC.
ParameterEstimateAICBICθ1.4191778697.7194711.8687ξ10.3844708ξ2−0.1507097ξ3−0.5212492Figure 4
Statistical functions fitted for the dataset: (a) joint density function, (b) cumulative function, (c) hazard function, (d) copula function, (e) contour plot of copula function, and (f)–(i) marginals.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 6.2. UEFA Champion’s League Data

In this section, the UEFA Champion’s League data obtained from [54] have been analyzed to show the applicability of the new model and the performance of the maximum likelihood method for different types of data. The estimated parameters with AIC and BIC are in Table 3. The statistical functions related to the fitted distribution are presented in Figure 5.Table 3
Estimated parameters for the UEFA Champion’s League data with AIC and BIC.
ParameterEstimateAICBICθ0.02873765616.8248624.2686ξ1−0.55156010ξ20.61205812ξ30.10996939Figure 5
Statistical functions fitted for the dataset: (a) joint density function, (b) cumulative function, (c) hazard function, (d) copula function, (e) contour plot of copula function, and (f)–(i) marginals.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 7. Conclusion

The univariate one-parameter Lindley model is a simple probability distribution that has a lot of preferable properties and various applications. This study presented a new bivariate parametric probability distribution using the one-parameter modified Lindley model as a base line distribution, named bivariate modified Lindley (BIML) distribution. Statistical quantities of the new model have been derived in explicit forms. That strongly supports the usefulness of the new model from theoretical and also practical point of view. The new proposed model has absolutely continuous probability density function that changes according to parameters. The hazard function presented different shapes. The flexibility and applicability of the new model have been shown via simulation. The maximum likelihood estimators have been introduced. The method presented good performance via Monte Carlo simulation. The new model has been applied for two different types of data.



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