A new method for constructing continuous distributions on the unit interval
AA new method for constructing continuous distributions on theunit interval
Aniket BiswasDepartment of StatisticsDibrugarh UniversityDibrugarh, Assam, India-786004Email: [email protected]
Subrata ChakrabortyDepartment of StatisticsDibrugarh UniversityDibrugarh, Assam, India-786004Email: subrata [email protected]
Abstract
A novel approach towards construction of absolutely continuous distributions over the unitinterval is proposed. Considering two absolutely continuous random variables with positivesupport, this method conditions on their convolution to generate a new random variable in theunit interval. This approach is demonstrated using some popular choices of the positive randomvariables such as the exponential, Lindley, gamma. Some existing distributions like the uniformand the beta are formulated with this method. Several new structures of density functionshaving potential for future application in real life problems are also provided. One of the newdistributions having one parameter is considered for parameter estimation and real life mod-elling application and shown to provide better fit than the popular one parameter Topp-Leonemodel.
Keywords:
Topp-Leone, Uniform, Beta, Exponential, Gamma, Lindley, Convolution, Ferlie-Gumble- Morgenstern copula.
MSC 2010:
Consider a departmental store where customers arrive independently of each other. Supposethe store opens at 10:00 a.m. and the second customer arrives at 11:00 a.m. Arrival time ofthe first customer was not observed. Then the store manager may be interested in finding theprobability of the first customer arriving within half an hour from opening the store. If the cus-tomers arrive according to a Poisson process, then the probability of the first customer arrivingwithin 10:30 a.m. is 0 .
5. In fact, for given arrival time of the second customer, arrival timeof the first customer is conditionally uniform over the duration that is uniform over 10:00 a.mto 11:00 a.m. This follows from the assumption that the inter-arrival times of the customersare independently and identically distributed (iid) exponential random variates. However, theconditional distribution does not remain uniform when any one of the above assumptions isrelaxed. Still the resulting conditional distribution has support (0 , a r X i v : . [ m a t h . S T ] J a n mportance of distributions with support (0 ,
1) is well established for modelling proportions,scores, rates, indices etc. Responses in the unit interval are often focus of investigation in manyfields of application including finance (G´omez-D´eniz et al. 2014, Biswas et al. 2020), publichealth (Mazucheli et al. 2019, Biswas and Chakraborty 2019), demography (Andreopoulos etal. 2019). The first model that comes to the mind of practitioners is obviously the well knownBeta distribution (Johnson et al. 1995). Besides this, the Kumaraswamy’s distribution (Ku-maraswamy 1980, Jones 2009) and the Topp-Leone distribution (Topp and Leone 1955) alsoreceived attention. In recent times, a number of new models have been proposed as alternativesto the Beta and Kumaraswamy’s distributions to enhance modelling flexibility as par context(Grassia 1977, Tadikamalla and Johnson 1982, Barndorff-Nielsen and Jorgensen 1991, Lemonteet al. 2013, Pourdarvish et al. 2015, Mazucheli et al. 2018, 2019, 2020). This recent surge inthe number of research papers devoted to proposing new distribution in the unit interval clearlyexhibits their growing relevance.Most of the above distributions are generated through a suitable variable transformation ofa baseline distribution with positive support. For X being a positive valued random variable,the transformations U = X/ (1 + X ), U = 1 / (1 + X ), U = exp( − X ) are popularly used toderive a distribution with support (0 , ,
1) from conditional convolution approach.Rest of the article is organized as follows. The new method of deriving distributions on (0 , Consider a two component absolutely continuous random vector (
X, Y ) with joint cumulativedistribution function (cdf) F X,Y and joint probability density function (pdf) f X,Y . Let F X and f X are the marginal cdf and the marginal pdf of X , respectively. Similarly, F Y and f Y denote the marginal cdf and the marginal pdf of Y . The conditional cdf and pdf of X given therealization on Y are F X | Y and f X | Y , respectively.We assume that both X and Y have positive support, that is P ( X >
0) = P ( Y >
0) = 1.For Z = X + Y , the cdf of Z can be expressed as F Z ( z ) = (cid:90) z F X | Y ( z − y ) f Y ( y ) dy. (2. 1)The corresponding pdf is f Z ( z ) = (cid:90) z f X,Y ( z − y, y ) dy = (cid:90) z f X | Y ( z − y ) f Y ( y ) dy. (2. 2)In particular, when X and Y are independent, the cdf and the pdf of Z can be obtained byreplacing F X | Y ( z − y ) by F X ( z − y ) in (2. 1) and f X | Y ( z − y ) by f X ( z − y ) in (2. 2), respectively.We are interested in the conditional distribution of X given Z = 1. The correspondingrandom variable be U with support (0 , P ( U ≤ u ) = P ( X ≤ u | Z =1) = P ( X ≤ u, X + Y = 1) /f Z (1). P ( X ≤ u, X + Y = 1) can be seen as (cid:82) u f X,Y ( v, − v ) dv and hence the pdf of U is 2 U ( u ) = f X,Y ( u, − u ) f Z (1) , for 0 < u < . (2. 3)If X and Y are independently distributed, then f X,Y ( u, − u ) can be replaced by f X ( u ) f Y (1 − u )in (2. 3).It may be noted that, bivariate distributions on (0 , ∞ ) × (0 , ∞ ) are seldom naturally avail-able. The common techniques to construct bivariate distributions from given marginals involvemarginal transformation method, trivariate reduction and bivariate copula (Balakrishnan andLai 2009). To demonstrate the proposed method for constructing distributions on the unitinterval, some bivariate distributions will be considered in Section 3. The bivariate density weuse here is derived using Ferlie-Gumble-Morgenstern (FGM) copula (Gumble 1960) when themarginal densities f X and f Y are available. The joint density function of ( X, Y ) with additionalparameter α ∈ [ − ,
1] using FGM copula is f X,Y ( x, y ) = f X ( x ) f Y ( y )[1 + α (2 F X ( x ) − F Y ( y ) − . (2. 4) In this section we apply the proposed method to first derive distributions in (0 ,
1) which areknown followed by some new ones which to the best of our knowledge are not yet reported inthe literature.
As outlined in Section 1, let X and Y follow exponential distribution independently with sameparameter θ . Then f X ( x | θ ) = θ exp( − θx ) and f Y ( y | θ ) = θ exp( − θy ). Putting the the ex-pressions of f X ( x | θ ) and f Y ( y | θ ) in (2. 2), we get f Z (1 | θ ) = θ exp( − θ ). Now, putting theexpressions of f X ( x | θ ), f Y ( y | θ ) and f Z (1 | θ ) in (2. 3) we get f U ( u ) = 1 for 0 < u < G ( α, β ) denote the gamma distribution with density function α β x β − exp( − αx ) / Γ( β )for α, β >
0. Consider X ∼ G( α, β ) and Y ∼ G( α, β ) independently. Using (2. 2) it iseasy to see that Z ∼ G ( α, β + β ). Now, putting f X ( x | α, β ), f Y ( y | α, β ) and f Z (1) = α β + β exp( − α ) / Γ( β + β ) in (2. 3) we get f U ( u | β , β ) = Γ( β + β ) u β − (1 − u ) β − / Γ( β )Γ( β )which is the pdf of beta distribution with parameters β and β .While the uniform and beta cases are easily seen, generating the other distributions in theunit interval as mentioned in Section 1 using the proposed method remain a challenge. However,we do not claim that the distributions can (or cannot) be generated using the proposed method. Here we consider different distributions for (
X, Y ) to construct a number of new distributionson the unit interval.
Distribution 1:
Let X and Y follow exponential distribution independently with parameter θ and θ , respectively. Putting f X ( x | θ ) = θ exp( − θ x ) and f Y ( y | θ ) = θ exp( − θ y ) in (2. 2)we get f Z (1) = (cid:40) θ θ ( e − θ − e − θ ) θ − θ for θ (cid:54) = θ θ e − θ for θ = θ = θ. (3. 5)3ow, putting f X ( x | θ ), f Y ( y | θ ) and f Z (1) in (2. 3) and replacing θ − θ by δ we obtain f U ( u ) = (cid:40) δe − δu − e − δ for δ (cid:54) = 01 for δ = 0 . (3. 6)The density function in (3. 6) reduces to that of truncated exponential distribution with regionof truncation (1 , ∞ ) when we restrict the parameter space to δ >
0. However, the new genesisdoes not require the parameter δ to be positive. Thus the density in (3. 6) provides moreflexibility with δ ∈ R , the real line despite having the same mathematical structure as thesaid truncated exponential distribution. From Figure 1 it is clear that the density can be bothincreasing and decreasing depending on δ . For δ <
0, it is increasing with mode at 1 and for δ >
0, it is decreasing with mode at 0. u (cid:72) u (cid:76) ∆(cid:61) ∆(cid:61) ∆(cid:61)(cid:45) ∆(cid:61)(cid:45) Figure 1: Plot of the density function of Distribution 1.
Distribution 2:
Let X and Y both marginally follow exponential distribution with param-eters θ . Putting f X ( x | θ ) = θ exp( − θx ), F X ( x | θ ) = 1 − exp( − θx ), f Y ( y | θ ) = θ exp( − θy ) and F Y ( y | θ ) = 1 − exp( − θy ) in (2. 4) we obtain the following bivariate exponential distributionwith parameters θ > − ≤ α ≤ f X,Y ( x, y | θ, α ) = θ e − θ ( x + y ) (cid:104) α (1 − e − θx )(1 − e − θy ) (cid:105) (3. 7)Using (3. 7) in (2. 2) we obtain f Z (1) = θ e − θ (cid:20) α (cid:26) e − θ − θ + 4 e − θ θ (cid:27)(cid:21) . (3. 8)Putting (3. 7) and (2. 2) in (2. 3) we obtain the following density function. f U ( u ) = 1 + α (1 − e − θu )(1 − e − θ (1 − u ) ) (cid:104) α (cid:110) e − θ − θ + 4 e − θ θ (cid:111)(cid:105) (3. 9)This distribution has two parameters and may prove to be useful in modelling applications. Itis clear that for α = 0, X and Y are independent and consequently (3. 9) reduces to uniform4istribution. From Figure 2, it can be seen that the distribution is symmetric about 0 . Distribution 3:
Let X and Y follow Lindley distribution independently and identically withparameter θ >
0. Putting f X ( x | θ ) = θ (1 + x ) exp( − θx ) / (1 + θ ) and f Y ( y | θ ) = θ (1 + y ) exp( − θy ) / (1 + θ ) in (2. 2) we get f Z (1) = 136 e − θ θ (1 + θ ) . (3. 10)As in previous cases, using the density functions of X and Y along with (3. 10) in (2. 3) weobtain f U ( u ) = 613 (2 − u )(1 + u ) . (3. 11)The resulting distribution is free from any parameter and thus not flexible as clear from Figure3. Clearly (3. 11) is symmetric about 0 .
5. 5 .0 0.2 0.4 0.6 0.8 1.0 u (cid:72) u (cid:76) Figure 3: Plot of the density function of Distribution 3.
Distribution 4:
Let X and Y follow Lindley distribution independently with parameters θ and θ , respectively. Here f X ( x | θ ) = θ (1 + x ) exp( − θ x ) / (1 + θ ) and f Y ( y | θ ) = θ (1 + y ) exp( − θ y ) / (1 + θ ). The density function of Z = X + Y is evaluated using (2. 2) but wedo not report it here due to its messy structure. However, the density function of U obtainedusing f X ( x | θ ), f Y ( y | θ ) and f Z (1) in (2. 3) is f U ( u ) = (cid:40) δ e − δu ( u − u )(2 − δ − δ )+ e − δ (2 δ − − δ ) for δ (cid:54) = 0 (2 − u )(1 + u ) for δ = 0 (3. 12)for δ = θ − θ ∈ R . Obviously the density in (3. 12) having one parameter is an extension of(3. 11). From Figure 4 and Figure 1, it can be seen that the Distribution 4 and Distribution 1behave quite similarly. u (cid:72) u (cid:76) ∆(cid:61) ∆(cid:61) ∆(cid:61)(cid:45) ∆(cid:61)(cid:45) Figure 4: Plot of the density function of Distribution 4.6 istribution 5:
Let X follow exponential distribution with parameter θ and Y follow Lindleydistribution with parameter θ independently. Here f X ( x | θ ) = θ exp( − θx ) and f Y ( y | θ ) = θ (1 + y ) exp( − θy ) / (1 + θ ). From (2. 3) we get f U ( u ) = 23 (2 − u ) . (3. 13)This distribution has no parameter and thus not flexible for modelling purposes . The densityin (3. 13) is decreasing and it decreases from 4 / u = 0 to 2 / u = 1 linearly as given inFigure 5. u (cid:72) u (cid:76) Figure 5: Plot of the density function of Distribution 5.
Distribution 6:
Let X follow exponential distribution with parameter θ and Y follow Lindleydistribution with parameter θ independently. Here f X ( x | θ ) = θ exp( − θ x ) and f Y ( y | θ ) = θ (1 + y ) exp( − θ y ) / (1 + θ ). Using the density functions first in (2. 2) and then in (2. 3) weget f U ( u ) = (cid:40) δ e δ (1 − u ) (2 − u )1 − δ +(2 δ − e δ for δ (cid:54) = 0 (2 − u ) for δ = 0 . (3. 14)Here δ = θ − θ ∈ R . A few representative plots of the density function are provided in Figure 6. Distribution 7:
Consider X following exponential distribution with parameter θ and Y fol-lowing gamma distribution with parameters α and β . Putting f X ( x | θ ) = θ exp( − θx ) and f Y ( y | α, β ) = α β y β − exp( − αx ) / Γ( β ) in (2. 2) we get f Z (1) = θα β e − θ { Γ( β ) − Γ( α − θ ) } Γ( β )( α − θ ) β for α (cid:54) = θ α β +1 exp( − α )Γ( β +1) for α = θ. (3. 15)Here Γ( s, t ) denotes the incomplete gamma function (cid:82) ∞ t v s − exp( − v ) dv . Now using the densityfunctions of X and Y along with (3. 15) in (2. 3) we obtain f U ( u ) = (cid:40) δ β Γ( β ) − Γ( β,δ ) e δ (1 − u ) (1 − u ) β − for β > , δ (cid:54) = 0 β (1 − u ) β − for α > , δ = 0 (3. 16)7 .2 0.4 0.6 0.8 1.0 u (cid:72) u (cid:76) ∆(cid:61) ∆(cid:61) ∆(cid:61)(cid:45) ∆(cid:61)(cid:45) Figure 6: Plot of the density function of Distribution 6.where δ = α − θ ∈ R . The density in (3. 16) has two parameters and may prove to be acompetitor of the popular beta distribution as evident from Figure 7.Figure 7: Plot of the density function of Distribution 7. Distribution 8:
Let X be a Lindley random variable with parameter θ and Y be a gammarandom variable with parameters θ and β . Here f X ( x | θ ) = θ (1 + x ) exp( − θx ) / (1 + θ ) and f Y ( y | θ, β ) = θ β y β − exp( − θx ) / Γ( β ). Putting the densities of x and Y first in (2. 2) and thenusing f Z (1) along with the density functions in (2. 3) we obtain f U ( u ) = β (1 + β )2 + β (1 + u )(1 − u ) β − for β > . (3. 17)The parameter θ labelling the distributions of both X and Y surprisingly plays no role in (3.17). Despite being a one-parameter distribution, it is quite flexible for modelling purposes sincethe corresponding density function has a wide range of shapes as can be seen from Figure 8. Thederived distribution can also be seen as a mixture of two beta distributions where the mixingproportions are also functions of the parameter β .8 .0 0.2 0.4 0.6 0.8 1.0 u (cid:72) u (cid:76) Β(cid:61)
Β(cid:61)
Β(cid:61)
Β(cid:61) .8 Β(cid:61) .3 Figure 8: Plot of the density function of Distribution 8.
Distribution 8 is derived from the distribution of Lindley random variable conditioned on theconvolution of the same with gamma random variable. Thus a working abbreviation for thedistribution with pdf in (3. 17) is taken as
LCG ( β ) and we write U ∼ LCG ( β ). The cumulativedistribution function (cdf) of LCG ( β ) is F U ( u ) = 1 − (1 − u ) β (2 + β + uβ )2 + β , < u < . (4. 18)Figure 9 exhibits the functional form of F U for different choices of β . With increasing β , thecdf becomes more and more convex in shape. u (cid:72) u (cid:76) Β(cid:61)
Β(cid:61)
Β(cid:61)
Β(cid:61) .8 Β(cid:61) .3 Figure 9: Plot of the cumulative distribution function of LCG distribution.9he corresponding hazard rate is h ( u ) = β (1 + β )2 + β + uβ u − u , < u < . (4. 19)The LCG family has increasing failure rate as shown in Figure 10. u (cid:72) u (cid:76) Β(cid:61)
Β(cid:61)
Β(cid:61)
Β(cid:61) .8 Β(cid:61) .3 Figure 10: Plot of the hazard rate of LCG distribution.The lower order moments are very simple, the mean and the variance of
LCG ( β ) is given below. E ( U ) = 4 + β (2 + β ) and V ( U ) = β (16 + 9 β + β )(2 + β ) (3 + β ) (4. 20) To generate random observations from
LCG ( β ) it is convenient to use the cdf in (4. 18). Weemploy cdf inversion technique as the structure is not too complex. For a random observation v from uniform distribution over (0 ,
1) we equate F U ( u ) = v and find solution of the equationin u . This u is a random observation from LCG ( β ). This technique produces the followingnonlinear equation in u (1 − u ) β (2 + β + uβ ) − (1 − v )(2 + β ) = 0 . (4. 21)We solve (4. 21) by uniroot function available on basic R . Since E ( U ) is structurally simple, a method of moment estimator can easily be implemented.For a given dataset u = ( u , u , ..., u n ), we calculate the sample mean ¯ u = (1 /n ) (cid:80) ni =1 u i andform the following equation. E ( U ) = 4 + β (2 + β ) = ¯ u (4. 22)The equation in (4. 22) can be rewritten as¯ uβ + (4¯ u − β + 4(¯ u −
1) = 0 . (4. 23)10he method of moment estimator ˆ β MM is the solution of (4. 23) in β .The log-likelihood function of β given the dataset u is l ( β | u ) = n log (cid:20) β (1 + β )2 + β (cid:21) + n (cid:88) i =1 log(1 + u i ) + ( β − n (cid:88) i =1 log(1 − u i ) . (4. 24)Differentiating l ( β | u ) with respect to β and equating it with 0 yields the following. T ( u ) β + (3 T ( u ) − β + (2 T ( u ) −
4) = 0 (4. 25)Here T ( u ) = − (1 /n ) (cid:80) ni =1 log(1 − u i ). To solve both (4. 23) and (4. 25) in β , we use polyroot function available in basic R . The solution of (4. 25) is the maximum likelihood estimate namelyˆ β ML . Table 1: Bias and mean-squared error (MSE) of ˆ β MM and ˆ β ML . n β Bias( ˆ β MM ) MSE( ˆ β MM ) Bias( ˆ β ML ) MSE( ˆ β ML )20 2.30 0.07792 0.27779 -0.21959 0.334407.80 0.34196 3.74806 0.21788 3.7248415.00 0.52377 12.36922 0.46433 12.3600940 2.30 0.01869 0.12236 -0.29453 0.213427.80 0.13403 1.48331 -0.00307 1.4837715.00 0.38477 5.71176 0.30913 5.7291160 2.30 0.01719 0.08165 -0.29909 0.174217.80 0.12145 0.96437 -0.02729 0.9688315.00 0.16190 3.51346 0.08157 3.5191280 2.30 0.00449 0.06157 -0.31263 0.162307.80 0.12302 0.70574 -0.03383 0.7035515.00 0.14265 2.69120 0.05492 2.69554100 2.30 0.01120 0.04923 -0.30954 0.146397.80 0.06890 0.55969 -0.08804 0.5776115.00 0.17952 2.35752 0.08494 2.35630A simulation experiment is performed to study the relative performance of the estimators. Wegenerate 1000 samples of different sizes ( n ) using the method in Section 4.1 and compute theaverage bias and mean squared error of the estimators. From Table 1, it is seen that theperformance of both the estimators are quite similar. With increasing β , the mean squarederror of both the estimators increase. As desired the mean squared error and absolute bias ofboth the estimators decrease with increase in sample size. Consider the following datasets from Caramanis et al. (1983) providing computation time oftwo different algorithms namely SC16 and P3.11C16:= (0.853, 0.759, 0.866, 0.809, 0.717, 0.544, 0.492, 0.403, 0.344, 0.213, 0.116, 0.116, 0.092,0.070, 0.059, 0.048, 0.036, 0.029, 0.021, 0.014, 0.011, 0.008, 0.006)P3:= (0.853, 0.759, 0.874, 0.800, 0.716, 0.557, 0.503, 0.399, 0.334, 0.207, 0.118, 0.118, 0.097,0.078, 0.067, 0.056, 0.044, 0.036, 0.026, .019, 0.014, 0.010)Mazumdar and Gaver (1984) comparison of the mentioned algorithms for estimating the unitcapacity factor. For a population level comparison one may refer to Genc (2013) where theauthors fitted Topp-Leone model to both SC16 and P3 datasets for further analysis. It is to benoted that to improve the population level comparison, one need to select a better suited model.One parameter distributions on the unit interval is rare and the Topp-Leone distribution is abenchmark in this category. The introduced LCG distribution is a new addition to this categoryand hence a real life data driven comparison with Topp-Leone is sufficient to justify its utility.Topp-Leone model fits the datasets well as can be seen in Genc (2013). However, we comparefitting of the Topp-Leone model with that of the LCG in Table 2 and find that the proposedLCG model is far better as indicated by the corresponding AIC values.Table 2: Goodness of fit.Dataset Distribution MLE Log-likelihood AICSC16 Topp-Leone 0.5943 -11.3660 25.7837LCG 1.9876 2.9459 -3.8981P3 Topp-Leone 0.6778 -10.9016 23.8032LCG 1.8646 2.3009 -2.6098
A number of new distributions with simple yet flexible structures are derived in the current work.Throughout the article we discussed about the distribution of U = X | X + Y = 1. It is obviousthat in many situations the distribution of V = Y | X + Y = 1 is different from U and hencemay be of interest fom new distribution perspective. However, we refrain from detailing suchdensity function since the same can be done by simply finding density function of V = 1 − U .We envisage that the proposed method and the derived distributions will be a useful addition inthe existing literature. In fact the LCG is shown to be a better one parameter model than thebenchmark Topp-Leone distribution. Since the LCG model possesses simple structure of themean, modelling unit responses with covariates using LCG is of interest. Moreover Section 4.2hints a productive study on the stress-strength reliability of the LCG distribution. These prob-lems are under consideration and will be reported as independent works. Detailed investigationon the other derived distributions are required and further research on this is warranted. Thisarticle is intended to motivate researchers to delve into the possibilities of exploring new dis-tributions with different choices of the joint distributions of ( X, Y ). The proposed method cannaturally be extended. One may consider the random vector ( X , X , ..., X n ) with P ( X i >
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Journal ofthe American Statistical Association , 50(269), 209-219.
Appendix: R-codes
C1: Random sample generation from LCG distribution. rGL= function ( n , beta ) { sample =0 f o r ( i i n 1 : n ) { v= r u n i f ( 1 )f= function ( u ) { (1 − u ) ˆ beta ∗ (2+ beta +u ∗ beta ) − (1 − v ) ∗ (2+ beta ) } proc = uniroot ( f , c ( 0 , 1 ) ) sample [ i ]= proc $ r o o t } return ( sample ) }
2: Method of moment estimation for LCG distribution.
MME= function ( data ) { ubar= mean ( data )z= c ( 4 ∗ ( ubar −
1) ,4 ∗ ubar − return ( max ( Re ( polyroot ( z ) ) ) ) } C2: Maximum likelihood estimation for LCG distribution.
MLE= function ( data ) { Tu= − mean ( log (1 − data ) )z= c ( 2 ∗ Tu − ∗ Tu − return ( max ( Re ( polyroot ( z ) ) ) ) }}