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.
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
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
For , 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:
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 Section 2. 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
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 be a random vector with joint probability density function , where , , and . Let be a random vectors with marginals and , respectively, and let and be functions such that and and and are differentiable and monotonically nondecreasing functions and as , as as , and as . Then, the cdf of random variable is given by
Suppose the transformation function , where , and . Then, the cumulative function of the modified Lindley random vector is given bywhere , , and .
Definition 1. A random vector is said to have a modified Lindley distribution with parameter , where , , if its cumulative function is given by (8) for and and will be denoted by .
3. Statistical Properties
In this section, statistical properties of the new distribution are discussed.
Lemma 1. Let . Then, its probability density function is given by
Lemma 2. Let . Then, its marginals are given by
Lemma 3. Let . Then,where
Lemma 4. Let . Then,
Lemma 5. Let . Then,
Lemma 6. Let . Then, the bivariate reliability function is given by
Lemma 7. Let . Then, the bivariate hazard rate function is given by
The Copula function is of the most important quantities that is used to describe the stochastic dependence between continuous random variables. Let ; since , , and , then its copula function can be defined as where , , and [50–52].
Lemma 8. Let . Then, its copula function is given bywhere is given by (14).
4. Estimation
Considering a random sample from the bivariate modified Lindley random variable , the maximum log-likelihood function for the unknown parameters is given by
The score vector is given bywhere 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, , the set II, , and the set III, . 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 (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 , 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 (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 and V . The sizes of the samples are different . The parameters , and are treated as chosen parameters. The maximum likelihood method is used to estimate the unknown parameter ; 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.
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6. Real-Data Application
In this section, two data sets are used to apply the new model, .
6.1. Financial Data
In this section, we apply 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.
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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.
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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.
Data Availability
The data used to support the findings of the study are included within the article.
Conflicts of Interest
The author declares no conflicts of interest.