A new lifetime model by mixing gamma and geometric distributions with fitting hydrologic data

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1 A new lifetime model by mixing gamma and geometric distributions with fitting hydrologic data Hassan Bakouch, Abdus Saboor, Christophe Chesneau, Anwaar Saeed To cite this version: Hassan Bakouch, Abdus Saboor, Christophe Chesneau, Anwaar Saeed. A new lifetime model by mixing gamma and geometric distributions with fitting hydrologic data. 7. <hal-53694> HAL Id: hal Submitted on Jun 7 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 A new lifetime model by mixing gamma and geometric distributions with fitting hydrologic data Hassan S. Bakouch a,, Abdus Saboor b, Christophe Chesneau c, Anwaar Saeed b a Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt b Department of Mathematics, Kohat University of Science & Technology, Kohat, Pakistan c LMNO, University of Caen, France Abstract In this paper, we propose a new distribution obtained by mixing gamma and geometric distributions. We discuss different shapes of the probability density function and the hazard rate functions. We study several statistical properties. The maximum likelihood estimation method is performed for estimating the parameters. We determine the observed information matrix and discuss inference. Illustrative three hydrologic data sets are given to show the flexibility and potentiality of the proposed distribution. Keywords: Gamma distribution, Geometric distribution, Order statistics, Record values, Estimation. Mathematics Subject Classification: MSC : E, 6E5, 6F. Corresponding author addresses: hnbakouch@yahoo.com Hassan S. Bakouch, saboorhangu@gmail.com; dr.abdussaboor@kust.edu.pk Abdus Saboor, christophe.chesneau@unicaen.fr Christophe Chesneau, Anwaarsaeedqureshi@gmail.com Anwaar Saeed Preprint submitted to Elsevier June 6, 7

3 . Introduction An important aspect of statistics is the determination of flexible distributions to elaborate useful models for lifetime data. Among the existing approaches, new distributions can be obtained by mixing discrete and continuous distributions. Those using geometric distributions include the exponential geometric distribution [], the exponential-power series distribution [9], the extended exponential geometric distribution [], the complementary exponential geometric distribution [4], the Weibull-geometric distribution [6], the Weibull-power series distribution [8], the generalized exponential-power series distribution [6], the complementary exponentiated exponential geometric distribution [5], the extended Weibull-power series distribution [], the complementary extended Weibull-power series [], the exponentiated extended Weibull-power series distribution [3], the G-geometric distribution [3], the alternative G-geometric distribution [8], the generalized linear failure rate-geometric distribution [] and the linear failure rate-power series distribution [7]. We also refer to the review of [], and the references therein. On the other hand, among the continuous distributions, the gamma distribution is one of the most commonly used in modeling life-time data. In practice, it has been shown to be very flexible in modeling various types of lifetime distributions. To the best of our knowledge, the mixing of the geometric distribution with the Gamma distribution not reduced to the exponential one has not ever been considered in the literature. Based on such a mixing, this paper offers a new distribution with two parameters, called the gamma-geometric GG distribution. The formulation and motivations of such distribution are as follows. Let λ >, θ, and θ = θ. We say that a random variable X follows the GG distribution with parameters θ, λ, denoted by GGθ, λ, if it has the probability density function pdf given by fx = θλ xe + θe θe 3, x >. The corresponding cumulative distribution function cdf is given by + λx θe F x = θe θe, x >. The GG distribution arises from the following stochastic representation. Let X be a random variable having the following stochastic representation: X {N = n} G am, λn, N θ G trunc θ, 3 that is N is a random variable having the truncated geometric distribution with parameter θ: P N = n = θ θ n, n =,,... and the distribution of X conditionally to {N = n} is the gamma distribution G am, λn, with a conditional pdf given by f X {N=n} x = λ n xe λnx, x >. Then X follows the GGθ, λ distribution; using the geometric series expansions: X is given by f X x = n= n= n= n x n = x+x x 3, x <, the pdf of f X {N=n} xp N = n = θ θ λ x n θe n = θλ xe + θe θe 3. 4 The stochastic representation 3 can be viewed as a natural extension of the stochastic representation X {N = n} E xp λn = G am, λn, with pdf corresponding to the one of the G-geometric class proposed by [3] applied with the exponential distribution. Ratio of two independent variables. An example of simple model using the GG distribution is given by the ratio of two independent variables as described as follows. Let Y G am, λ and N G trunc θ. Suppose that Y and N are independent. Then X = Y N,

4 follows the GGθ, λ distribution. It is enough to note that X {N = n} = Y/n G am, λn. This ratio representation will be useful to determine statistical properties of the GG distribution. Note that, from the ratio representation, the cdf of X given by can be expressed directly: using the cdf of Y : F Y x = e λxe, x >, and the geometric series expansions: + x n = x and + n= nx n = x, x <, the cdf of X is given by F X x = P Y xn = F Y xnp N = n n= = θe θe n λxθe n= + λx θe = θe θe. Some limit properties for fx are given as: n= n θe n fx λ θ θ x, x, fx θλ xe, x +. Moreover, one can show that fx has a unique maximum on, + given by fx where x satisfies the equation: 4 θλx e λx + λx e λx + θ λx + =. Some plots of fx are given in Figure for several values of θ, λ. The rest of the paper is organized as follows. In Section, we give some n=..5. Λ=.5 Θ=.75 Θ=.9 Θ=. Θ=.3 Θ= Λ= Θ=.75 Θ=.9 Θ=. Θ=.3 Θ= Figure : Plots of the GG density function. properties of the GG distribution. The estimation by maximum likelihood is discussed in Section 3 with illustrative real data examples.. Properties of the GG distribution In this section, we propose many features and statistical properties of the GG distribution... The survival function and hazard rate functions The survival function sf of X is given by + λx θe Sx = F x = θe θe, x >, 5 3

5 and the associated hazard rate function hrf of X is θe hx = fx Sx = + λ x θe + λx θe, x >. 6 Observe that hx λ θ θ x, x, hx λ, x +. Some plots of hx are given in Figure for several values of θ, λ...5 Λ=.5 Θ=.75 Θ=.9 Θ=. Θ=. Θ= Λ= Θ=.75 Θ=.9 Θ=. Θ=.3 Θ= Figure : Plots of the GG hazard rate function... Quantile function The quantile function of X is determined by inverting the cdf F x. The p-th quantile x p of X is the real solution of the nonlinear equation:.3. Moments F x p = p θe p + λx p θe p = p θe p. Some key features of a distribution, like mean and variance, can be investigated through its r-th moments EX r. For finding EX r, we can use the ratio representation of X: X has the same distribution of the ratio of random variables: Y/N with Y G am, λ and N G trunc θ, Y and N independent. Therefore, considering the Gamma function: Γν = + x ν e x dx, ν >, E Y r = Γ+r λ and the polylogarithm function: Li r r x = + x n, r >, x <, we have nr Y EX r r = E N r = E Y r Γ + r E N r = λ r n= In particular, by taking r =, since Li x = log x, we obtain n= n EX = θ λ θ logθ. λ r r P N = n = Γ + r θ θ Li r θ. The variance of X can be explicit in some cases. For instance, if θ =, since Li = log and Li = [π 6log ], we have V arx = EX [EX] = Γ4 λ Li [ Γ3 λ Li ] = π λ 7log. 4

6 .4. Moment generating function As for the moments, the moment generating function of X can be obtain via the ratio representation Y/N. Using Ee ty λ =, t < λ, and the conditional expectation, we get λ t Mt = Ee tx = Ee t Y N = E E e t Y N N N = E = λ + n= n λn t P N = n = n λ θ θ n n= λn t..5. Conditional and reversed moments The r-th conditional moments of X is given by EX r X > t = EX r St and the r-th reversed moments of X is given by EX r X t = F t λ λ t = λ E N x r fxdx, t >, x r fxdx, t >. λn t The integral term can be expressed using the expansion 4. Indeed, introducing the lower incomplete gamma function Γt, ν = xν e x dx, ν >, t >, we have x r fxdx = θ θ + λ n θn n= x r+ e nλx dx = θ θ λ r n= θ n Γnλt, r +. 7 nr.6. Rényi entropy An entropy plays a central role in information theory. It provides a suitable measure of randomness or uncertainty of X. For continuous distributions, Rényi entropy see [] can be determined as follows: I R γ = + γ log [fx] γ dx, γ >, γ. We have [fx] γ = θ γ λ γ x γ e λγx + θe γ θe 3γ. Using the generalized binomial series: +x α = + we have Hence, Therefore, I R γ = = + θe γ = k= γ θ k e λkx, k [fx] γ = θ γ λ γ + l= k= 3γ l k= [ + γ logθ + γ logλ + log γ l= [ + γ logθ + γ logλ + log γ α k x k, α C, x <, α k = αα α...α k+ k!, + θe 3γ = l= 3γ γ l k θl+k x γ e λl+k+γx. k= l= k= 3γ γ l k 3γ γ l k l l θl+k l θl e λlx. + x γ e λl+k+γx dx l θl+k Γγ + λ γ+ l + k + γ γ+ ] ]. 5

7 .7. Order statistics distributions The order statistics are central tools in non-parametric statistics and inference. Let us now present the distributions of some fundamental order statistics related to the GGθ, λ distribution. Let a sample X, X,..., X n is randomly choosen from the GGθ, λ distribution and X :n X :n... X n:n are its corresponding order statistics. A pdf of X i:n is given by f Xi:n x = = n i n! i! n i! fx l= n i n! i! n i! θλ + θe xe θe 3 x > The cdf of X i:n is given by F Xi:n x = = n! i! n i! n! i! n i! l l [F x] i +l n i n i [ l θe l l= n i l n i [F x]i+l l i + l l= n i l n i [ θe l i + l l= A joint pdf of X :n,..., X n:n is given by f X:n,...,X n:n x,..., x n = n! n n fx k = n!θ n λ n x k k= A joint pdf of X i:n, X j:n, i < j, is given by < x <... < x n. k= + λx θe θe e λ x k k= + λx θe θe ] i+l, x >. n + θe k k= n, θe 3 k k= ] i +l, f Xi:n,X j:n x i, x j = = n! i! n j! j i [F x i] i [F x j F x i ] j i [Sx j ] n j fx i fx j [ n! θe + λx ] i i i i θe i! n j! j i θe i [ θe + λx i i θe i θe θe + λx ] j j i j j θe i θe j [ θe + λx ] j n j j j θe θe θ λ 4 x j i x j e i+xj i j + θe + θe θe 3 i θe 3, j < x i < x j..8. Record values distributions Record values arise in a wide varity of real-life applications as hydrology, industry, lifetesting, economics, among the others. See, for instance, [4], [5] and []. We now present important distributions related to record values using the GGθ, λ distribution as baseline. Let X, X,..., be a sequence of i.i.d. random variables having the GGθ, λ distribution. We define a sequence of record times Un as follows: U =, Un = min{j; j > Un, X j > X Un } for n. We define the i-th upper record value by R i = X Ui, with R = X. A pdf of R i is given by f Ri x = = i! [ logsx]i fx [ log θe i! + λx θe θe ] i θλ + θe xe θe 3, x >. 6

8 A joint pdf of R,..., R n is given by n f R,...,R nx,..., x n = fx n hx k = θλ n x n e n n k= k= x k n + θe k n k= + θe n θe 3 n, < x <... < x n. θe k + λxk θe k k= A joint pdf of R i, R j, i < j, is given by f Ri,R j x i, x j = = i!j i! [ logsx i] i [logsx i logsx j ] j i hx i fx j [ log θe + λx ] i i i i θe i!j i! θe i [ log θe + λx i i θe i θe log θe + λx j j θe ] j j i i θe j θe i θe j λ + x i θe i + λxi θe θλ + x j e j i θe 3, < x i < x j. j.9. Residuals life functions The residual life functions play a fundamental role in survival or reliability studies. See, for instance, [7], [3] and [9]. We now present some related mathematical objects with a potential of interest in the context of the GGθ, λ distribution. The residual life is described by the conditional random variable R t = X t X > t, t. The sf of the residual lifetime R t is given by S Rt x = Sx + t St The associated cdf is given by = e + λx + t θe +t θe λt θe +t + λt θe, x >. λt F Rt x = e + λx + t θe +t θe λt θe +t + λt θe, x >. λt Then, the corresponding pdf is given by f Rt x = λ x + te + θe +t θe λt θe +t 3 + λt θe λt, x >. The associated hrf is given by θe +t h Rt x = λ + x + t θe +t + λx + t θe +t x >, and the mean residual life is defined as Kt = ER t = EX t X > t = St The integral term can be expressed as 7 with r =. On the other hand, the variance residual life is given by V t = V arr t = V arx t X > t = EX St 7 EX xfxdx t. x fxdx t tkt [Kt].

9 Again, the integral term can be expressed as 7 with r =. The reverse residual life is described by the conditional random variable R t = t X X t, t. The sf of the reversed residual lifetime R t is given by F t x S Rt x = = θe λt [ θe λt x θe λt x + λt x θe λt x ] F t θe λt x [ θe λt θe λt + λt θe λt ], < x t. The associated cdf is given by F Rt x = θe λt [ θe λt x θe λt x + λt x θe λt x ] θe λt x [ θe λt θe λt + λt θe λt ], < x t. Therefore, the corresponding pdf is given by f Rt x = and the associated hrf is given by h Rt x = θλ t xe λt x + θe λt x θe λt θe λt x 3 [ θe λt θe λt + λt θe λt ], < x t, θλ t xe λt x + θe λt x θe λt x θe λt x 3 [ θe λt x θe λt x + λt x θe λt x ], < x t. The mean reversed residual life is defined as Lt = ER t = Et X X t = t F t The integral term can be expressed as 7 with r =. The variance reversed residual life is given by xfxdx. W t = V arr t = V art X X t = tlt [Lt] t + F t Again, the last integral can be expressed as 7 with r =. x fxdx, 3. Estimation with hydrologic data examples In this section, we estimate the unknown parameters of the GG distribution using the method of maximum likelihood. Moreover, three hydrologic data sets are given to show the flexibility and potentiality of the proposed distribution. 3.. Maximum likelihood estimation Let X, X,..., X n be a random sample of size n from the GGθ, λ distribution with observed values x, x,..., x n. Set Θ = {θ, λ}. The likelihood function associated to x,..., x n is given by LΘ = n fx i = n θλ + x i e θe i n i θe 3 = θ n λ n x i i e λ x i n e λx i + θ n. e λxi + θ 3 The maximum likelihood estimators MLEs of θ and λ are obtained by maximization of LΘ, or alternatively, the log-likelihhod defined by lθ = loglθ = n logθ + n logλ + + logx i + λ loge λxi + θ 3 loge λxi + θ. 8 x i

10 It follows that the MLEs are the simultaneous solutions of the equations according to θ, λ: and lθ λ lθ θ = n θ n = n n λ + x i + n e λxi + θ 3 e λxi + θ = e λxi x i e λx i + θ 3 e λxi x i e λx i + θ =. Since we have no analytic forms, numerical methods, as the quasi-newton algorithm, can be applied to determine the estimators. The observed information matrix is given by Jθ JΘ = θ Θ J θ λ Θ, J λ θ Θ J λ λ Θ where J θ θ Θ = lθ θ J θ λ Θ = lθ θ λ J λ λ Θ = lθ λ 3 = n θ + n n e λx i + θ 3 e λx i + θ, n = e λxi x n i e λxi + θ 3 = n λ + n e λxi x i e λxi + θ, e λxi x i e λxi + θ eλxi x i e λx i x i e λx i + θ e λxi x i e λx i + θ. e λxi + θ This matrix is a key mathematical tool to obtain approximate confidence intervals or Wald tests for θ and λ in the case of a large sample. 3.. Illustrative hydrologic data examples In this section, we fit the GG distribution to three hydrologic data sets and compare with the Weibull, Gumbel, Exponentiated Exponential, Generalized Gumbel, Kappa and Weibull Geometric distributions for three data sets. Most of those distributions have received great attention for fitting hydrology data, like rainfall data, precipitation data and flood data. More precisely, the densities of the compared distributions are given as follows.: Weibull distribution with pdf: fx = k λ Gumbel distribution with pdf: x λ fx = e e k e x λ k, λ >, k >, x >. x µ σ x µ σ σ, σ >, x, µ R. Exponentiated Exponential EE distribution [4] with pdf: fx = αλ e α e, α, λ, x >. Generalized Gumbel GGu distribution [5] with pdf: x µ α e σ α e e x µ x µ σ e σ e fx =, α, σ >, µ, x R. σ 9

11 Kappa distribution [6] with pdf: fx = αθ β θ x α + β Weibull geometric WG distribution [6] with pdf: α+ αθ α x, α, θ, β, x >. β fx = αβ α px α e βxα p e βxα, p,, α, β, x >. For goodness-of-fit we have two main test statistics, i.e., information criterion and empirical distribution. The measures Akaike information criterion AIC [7], corrected Akaike information criterion AICC [8], Hannan Quinn information criterion HQIC [9], and consistent Akaike information criterion CAIC [3] are widely used information criterion for selecting the appropriate model among different others models. The Anderson-Darling A due to Anderson and Darling [3], the Cramér von Mises W due to Cramér and Mises [3] and the Kolmogorov Smirnov KS statistics due to Kolmogorov [33] with their p-values to compare the fitted models. These statistics are used to evaluate how a particular distribution with cdf, for a given data set, fits the corresponding empirical distribution. The distribution with better fit than the others will be the one having the smallest statistics and largest p-value. Some information about the data sets are given below as: The first data set is taken from engineering department and is presented by Linhart and Zucchini [34]. The data points are 3, 6, 87, 7,, 4, 6, 47, 5, 7, 46,, 4,, 5,,,, 3, 4, 7,, 4,, 6, 9,, 6, 5, 95 The second data contains of annual maximum stream amounts. The data is available in U.S. Geological Survey USGS website http : //nwis.waterdata.usgs.gov. The data points are 44, 79, 5, 8, 8, 35, 3, 48, 59, 57, 78, 35, 37, 8, 86,, 3, 35, 435, 5, 75, 6, 66, 73, 39, 667, 44, 8, 7, 68, 69, 4, 6, 45, 333, 4,, 75, 99,, 93, 87, 64, 36, 438, 63, 46, 48, 37, 4, 5, 6, 4, 5, 3, 9, 8, 3 The third data describes the maximum rainfall of Pakistan from 98 to. The data points are.7, 7.9, 69.5, 96.5,.6, 65.5, 54, 8, 4.8, 4., 74.8, 3.5, 5, 8.5, 3.8, 5.7, 6.3, 8, 6.9, 6, 9, 7.6, 47.3, 55, 9, 7, 9, 8, 3, 94 For the first data set, Table gives us estimates of the parameters of the considered models with their corresponding standard errors. Table presents their goodness-of-fit statistics. Concerning the GG model, the MLEs corresponding to the data are given by ˆθ =.554 and ˆλ =.4595, and the following information criterion are obtained: AIC = 36.5, AICC = , HQIC = 37. and CAIC = These values are the smallest in comparison to those obtained for the other models. On the other hand, we have A =.5583, W =.8935, KS =.473 with p =.89838, which are also the best. The superiority of the GG model, in terms of goodness-of-fit statistics, in comparison to the others, is also observed for second data set estimates are given in Table 3 and goodness of fit statistics in Table 4 and the third data set estimates are given in Table 5 and goodness of fit statistics in Table 6. References [] Adamidis, K. and Loukas, S. 998: A lifetime distribution with decreasing failure rate, Statistics and Probability Letters, 39, 35-4.

12 Table : Estimates of the parameters standard errors in parentheses for aeroplane data Distributions Estimates GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WG λ, a, b Table : Goodness of fit statistics for aeroplane data Distributions LogL AIC AICC HQIC CAIC GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WG λ, a, b Distributions A W KS p GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WG λ, a, b Table 3: Estimates of the parameters standard errors in parentheses for carnero data Distributions Estimates GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WG λ, a, b

13 Table 4: Goodness of fit statistics for carnero data Distributions LogL AIC AICC HQIC CAIC GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WG λ, a, b Distributions A W KS p GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WG λ, a, b Table 5: Estimates of the parameters standard errors in parentheses for jiwani data Distributions Estimates GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WG λ, a, b Table 6: Goodness of fit statistics for jiwani data Distributions LogL AIC AICC HQIC CAIC GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WG λ, a, b Distributions A W KS p GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WG λ, a, b

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