For the modeling of the wind speed, we propose a family of distributions in polynomial form generating the Lindley distribution. We call this distribution Lindley-Polynomial distribution. The estimation of parameters using the maximum product spacing estimation method. A real data set has been considered to illustrate the practical utility of the paper

Ghorbanzadeh, Dariush

English

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Applications and Applied Mathematics: An International Journal (AAM) Volume 16 Issue 1 Article 44 6-2021 Generation and Statistical Properties for Lindley-Polynomial Distribution Dariush Ghorbanzadeh CNAM Follow this and additional works at: https://digitalcommons.pvamu.edu/aam  Part of the Numerical Analysis and Computation Commons, and the Statistics and Probability Commons Recommended Citation Ghorbanzadeh, Dariush (2021). Generation and Statistical Properties for Lindley-Polynomial Distribution, Applications and Applied Mathematics: An International Journal (AAM), Vol. 16, Iss. 1, Article 44. Available at: https://digitalcommons.pvamu.edu/aam/vol16/iss1/44 This Article is brought to you for free and open access by Digital Commons @PVAMU. It has been accepted for inclusion in Applications and Applied Mathematics: An International Journal (AAM) by an authorized editor of Digital Commons @PVAMU. For more information, please contact hvkoshy@pvamu.edu. Available athttp://pvamu.edu/aamAppl. Appl. Math.ISSN: 1932-9466Applications and AppliedMathematics:An International Journal(AAM)Vol. 16, Issue 1 (June 2021), pp. 776 – 786Generation and Statistical Properties for Lindley-PolynomialDistributionDariush GhorbanzadehDépartement Mathématiques-StatistiquesCNAM292, Rue Saint-Martin, 75141-Paris CEDEX 03, Francedariush.ghorbanzadeh@lecnam.netReceived: October 11, 2018; Accepted: September 26, 2020AbstractFor the modeling of the wind speed, we propose a family of distributions in polynomial formgenerating the Lindley distribution. We call this distribution Lindley-Polynomial distribution. Theestimation of parameters using the maximum product spacing estimation method. A real data sethas been considered to illustrate the practical utility of the paper.Keywords: Lindley distribution; Lindley-Polynomial distribution; Maximum Product Spacingestimation method; QuantilesMSC 2010 No.: 65C99, 65C10, 62N021. IntroductionIn 1958, Lindley (1958) suggested a one parameter distribution to illustrate the difference betweenfiducial distribution and posterior distribution. The Lindley distribution has been used for modelinglifetime data and studying some stress-strength problems and has been the subject of studies byseveral authors. For example, we mention the works of Sankaran (1970) introduced the DiscretePoisson-Lindley Distribution and Ghitany et al. (2008) presents a treatment of the mathemati-cal properties for the Lindley distribution. Ghitany et al. (2012) investigated the Marshall-Olkin7761Ghorbanzadeh: Properties for Lindley-Polynomial DistributionPublished by Digital Commons @PVAMU, 2021AAM: Intern. J., Vol. 16, Issue 1 (June 2021) 777extended Lindley distribution and Rodrigues et al. (2015) study the beta exponentiated Lindleydistribution. In this work, we propose a family of distributions in polynomial form generatingthe Lindley distribution. We call this distribution Lindley-Polynomial distribution. The estimationof parameters using the maximum product spacing estimation method. The maximum product ofspacings (MPS) method for estimating parameters in continuous univariate distributions was pro-posed by Cheng and Amin (1983) and independently by Ranneby (1984). The MPS method pro-duces consistent and asymptotically efficient estimators. These estimators are consistent see Ran-neby (1984). In their work Heathcote et al. (2002) proposes a method that uses empirical quantiles.Knowledge of extreme wind speed distributions is essential for the design of wind turbines andstructures. Statistical parameters to express wind distribution speeds are very useful, and consider-able work has been carried on in recent years. Many authors have shown the fit of wind speed to theWeibull distribution step by step. However, these studies are performed through priori acceptance.Probability density function of wind speed is not always statistically accepted as Weibull pdf.Celik (2003) proposes an empirical study and Whalen et al. (2004) proposes the method of self-determined probability-weighted moments. Carta and Ramírez (2007) present the analysis of two-component mixture Weibull statistics for estimation of wind speed distributions and Ghorbanzadehet al. (2016) proposes a change-point model.The paper examines the applicability of probability distributions commonly used to modelwind speeds to data representing the average daily wind speed (km/h) at the Orly air-port (Paris, France), from January 1, 2006 to December 31, 2015 (data is available at:https://www.wunderground.com/history/airport/).2. Construction of the Lindley-Polynomial distributionIn this section we consider g a probability density function (pdf) on (0,∞) andm a positive integer.We assume that the moments of order k of g exists:µk =∫ ∞0xk g(x) dx <∞, ∀k ∈ {0, 1, . . . ,m}.In this paper we consider the densities familyfm(x) = Cm( m∑k=0xk)g(x) l1(0,∞)(x), (1)where l1 is the indicator function and Cm =(1 + µ1 + . . .+ µm)−1.For generating a random variable from density (1), we have the following algorithm.Proposition 2.1.Consider X and N two random variables such that:(1) N is discrete values in the set {0, 1, . . . ,m} with probability mass function (pmf) : P(N =2Applications and Applied Mathematics: An International Journal (AAM), Vol. 16 [2021], Iss. 1, Art. 44https://digitalcommons.pvamu.edu/aam/vol16/iss1/44778 D. Ghorbanzadehk)= pk where pk is defined bypk =µk1 + µ1 + . . .+ µm= µkCm, (2)(2) given as N , X has the pdffX(x|N = k) =1µkxkg(x) l1(0,∞)(x). (3)Then, the random variable X has the pdf defined in (1).Proof:The unconditional density of the random variable X is given byfm(x) =m∑k=0pk fX(x|N = k) =( m∑k=0pkµkxk)g(x) l1(0,∞)(x).Using (2) we get the result. In the following, we will consider a special case for the pdf g (g(x) = θ e−θx), which correspondsto the exponential distribution with θ > 0 parameter. In this case we have the following family ofdistributions (Lindley-Polynomial distribution):fm(x, θ) = Cm(θ)( m∑k=0xk)e−θx l1(0,∞)(x), (4)where Cm(θ) is defined byCm(θ) =θm+1m∑k=0k! θm−k. (5)For generating a random variable from density (4), we have the following algorithm.Proposition 2.2.Consider X and N two random variables such that:(1) N is discrete values in the set {0, 1, . . . ,m} with probability mass function (pmf)pk =k! θm−km∑k=0k!θm−k=k!θk+1Cm(θ), (6)(2) given as N , X has the gamma distribution with parameters k + 1 and θ (γ(k + 1, θ))fX(x|N = k) =θk+1k!xke−θx l1(0,∞)(x). (7)Then, the random variable X has the pdf defined in (4).3Ghorbanzadeh: Properties for Lindley-Polynomial DistributionPublished by Digital Commons @PVAMU, 2021AAM: Intern. J., Vol. 16, Issue 1 (June 2021) 779Proof:We have µk =∫ ∞0xk e−θx dx =k!θk+1, and using Proposition (2.1), we get the result. For m = 1, we get the pdf f1(x, θ) = θ21+θ(1 + x) e−θx l1(0,∞)(x), which is the Lindley distribution.The following figures shows the pdf of Lindley-Polynomial distribution defined in (4) as a functionof variation of θ and m.Figure 1. The pdf of Lindley-Polynomial distribution defined in (4) with θ = 0.4Figure 2. The pdf of Lindley-Polynomial distribution defined in (4) with θ = 1.84Applications and Applied Mathematics: An International Journal (AAM), Vol. 16 [2021], Iss. 1, Art. 44https://digitalcommons.pvamu.edu/aam/vol16/iss1/44780 D. GhorbanzadehFor the statistical properties of the Lindley-Polynomial distribution, we have the following propo-sition.Proposition 2.3.(1) The cumulative distribution function (cdf) is defined byFm(x, θ) = 1− Cm(θ)m∑k=0k∑j=0k!j!θj−k−1 xj e−θx. (8)(2) The characteristic function is defined byϕX(t) = E[eitX ] = Cm(θ)m∑k=0k!(θ − it)k+1. (9)(3) The rth moment is defined byE[Xr] =1θrm∑k=0(k + r)! θm−km∑k=0k! θm−k. (10)(4) The hazard rate function (failure rate function) is defined byh(x) =fm(x, θ)1− Fm(x, θ)=m∑k=0xkm∑k=0k∑j=0k!j!θj−k−1 xj. (11)3. Estimation of ParametersIn this section we use the MPS estimates techniques to estimate the parameters of the distribution.The method is based on maximization of the geometric mean of probability spacings in the datawhere the spacings are defined as the differences between the values of the cumulative distributionfunction at sequential data indices. The MPS method, originally suggested by Cheng and Amin(1983), this method was also independently developed by Ranneby (1984) as approximation to theKullback-Leibler measure of information. In their work Heathcote et al. (2002) proposes a methodthat uses empirical quantiles.Let X1, . . . , Xn be a random sample from the Lindley-Polynomial distribution (4). We supposethat a random sample of size n has been allocated in k classes: C1, . . . , Ck with observed frequencyn1, . . . , nk. The probability of class Ci ispi(m, θ) =∫Cifm(x, θ) dx. (12)5Ghorbanzadeh: Properties for Lindley-Polynomial DistributionPublished by Digital Commons @PVAMU, 2021AAM: Intern. J., Vol. 16, Issue 1 (June 2021) 781The likelihood function of a multinomial distribution defined on the classes C1, . . . , Ck, that is,Lm(θ) =k∏i=1(pi(m, θ))ni, (13)and the log-likelihood function is given as`m(θ) = log(Lm(θ))=k∑i=1ni log(pi(m, θ)). (14)The estimator of θ is defined byθ̂(m) = argsupθ>0`m(θ), (15)and the estimator of m is defined bym̂ = argmaxm≥1`m(θ̂(m)). (16)In this work, we deliberately consider the classesC1 = (0, q1), C2 = (q1, q2), . . . , Ck = (qk−1,∞), (17)with the corresponding probabilitiesp1(m, θ) = Fm(q1, θ),pi(m, θ) = Fm(qi, θ)− Fm(qi−1, θ), i = 2, . . . , k − 1,pk(m, θ) = 1− Fm(qk−1, θ),(18)where qi denotes the ith quantile: Fm(qi, θ) = i/100.4. ApplicationWe apply the model used in this work to the data representing the average daily wind speed (km/h)at the Orly airport (Paris, France), from January 1, 2006 to December 31, 2015.We consider two types of analysis. The first analysis is to do a monthly study of the data, studyingthe twelve months, the ten years, separately (the January data gather the data for the ten monthsJanuary 2006, ..., January 2015 and the same for the other months) and the two analysis consistsin doing a seasonal study considering the four seasons.In this section, we consider the following nine classes:C1 = (0, q10), C2 = (q10, q20), C3 = (q20, q35), C4 = (q35, q50), C5 = (q50, q65),C6 = (q65, q75), C7 = (q75, q85), C8 = (q85, q90), C9 = (q90,∞).(19)4.1. Results for monthly analysisThe following table represents the results of estimation of the parameters obtained by the monthlyanalysis of the data.6Applications and Applied Mathematics: An International Journal (AAM), Vol. 16 [2021], Iss. 1, Art. 44https://digitalcommons.pvamu.edu/aam/vol16/iss1/44782 D. GhorbanzadehTable 1. Monthly data analysis resultsJanuary February Marchm̂ = 4 θ̂ = 0.3748 m̂ = 4 θ̂ = 0.3606 m̂ = 4 θ̂ = 0.3699April May Junem̂ = 4 θ̂ = 0.4270 m̂ = 6 θ̂ = 0.5957 m̂ = 7 θ̂ = 0.6645July August Septemberm̂ = 8 θ̂ = 0.7618 m̂ = 8 θ̂ = 0.8350 m̂ = 3 θ̂ = 0.3595October November Decemberm̂ = 4 θ̂ = 0.4807 m̂ = 4 θ̂ = 0.4137 m̂ = 3 θ̂ = 0.3331The results obtained by Table 1 show that, whatever the month, the estimator of θ is less than 1.On the other hand, the value of the estimators for the months May, June, July and August are wellabove 0.5 and for the other months the value being less than 0.5. For the estimator θ̂, we have thefollowing statistics: mean of θ̂ = 0.4980 and standard deviation of θ̂ = 0.1658. For the estimatorm̂, we have the following statistics: mean of m̂ = 4.9167 and standard deviation of m̂ = 1.7539.We deduce that mean of m̂ ∈ {4, 5} and standard deviation of m̂ ∈ {1, 2}. In order to visualizethe graphs of the results obtained, we present the results for the months of January.Figure 3. Graph of the log likelihood function, for the months of January, defined in (15) for different values ofm7Ghorbanzadeh: Properties for Lindley-Polynomial DistributionPublished by Digital Commons @PVAMU, 2021AAM: Intern. J., Vol. 16, Issue 1 (June 2021) 783Figure 4. Graph of the pdf defined in (4) as a function of the estimated parameters and histogram of the Januarydata taking into account the classes defined in (19)4.2. Results for seasonal analysisThe following table represents the results of estimation of the parameters obtained by the seasonalanalysis of the data.Table 2. Seasonal data analysis resultsSpring Summerm̂ = 6 θ̂ = 0.5756 m̂ = 5 θ̂ = 0.5330Autumn Winterm̂ = 3 θ̂ = 0.3379 m̂ = 3 θ̂ = 0.3186The results obtained show that, whatever the season, the estimator of θ is less than 1. In particular,for the autumn and winter seasons, the estimator of m is equal to 3.The following figures represent the results obtained for the four seasons.8Applications and Applied Mathematics: An International Journal (AAM), Vol. 16 [2021], Iss. 1, Art. 44https://digitalcommons.pvamu.edu/aam/vol16/iss1/44784 D. Ghorbanzadeh(a) The log Likelihood function for spring data. (b) The pdf and histogram for spring data.Figure 5. Seasonal analysis results graph for spring data(a) The log Likelihood function for summer data. (b) The pdf and histogram for summer data.Figure 6. Seasonal analysis results graph for summer data5. ConclusionThe wind speed distributions and energy intensity measured at the location reveals that the currenttechnology does not provide economical electricity production from wind power and that the mea-surements should be evaluated in the long term in accordance with technological developmentsand reduction in the cost of turbines. The model studied in this work is very simple to use inpractice. The proposed algorithms are very simple to implement for the simulation of the Lindley-Polynomial distribution. The results obtained show, by monthly and seasonal analysis, that overallthe data used are well adjusted to the Lindley-Polynomial distribution with a parameter θ less than1. The result of this study encourages the use of wind potential at the approximate location.9Ghorbanzadeh: Properties for Lindley-Polynomial DistributionPublished by Digital Commons @PVAMU, 2021AAM: Intern. J., Vol. 16, Issue 1 (June 2021) 785(a) The log Likelihood function for autumn data. (b) The pdf and histogram for autumn data.Figure 7. Seasonal analysis results graph for autumn data(a) The log Likelihood function for winter data. (b) The pdf and histogram for winter data.Figure 8. Seasonal analysis results graph for winter dataREFERENCESCarta, J.A. and Ramírez, P. (2007). Analysis of two-component mixture Weibull statistics for esti-mation of wind speed distributions, Renewable Energy, Vol. 32, pp. 518-531.Celik, AN. (2003). Assessing the suitability of wind speed probability distribution functions basedon wind power density, Renew Energy, Vol. 28, pp. 1563-74.Cheng, R. C. H. and Amin, N. A. K. (1983). Estimating parameters in continuous univariate dis-tributions with a shifted origin, J. Roy. Statist. Soc. Ser. B., Vol. 45, pp. 394-403.Ghitany, M.E., Al-Mutairi, D.K, Al-Awadhi, F.A. and Al-Burais, M.M. (2012). Marshall-Olkin ex-tended Lindley distribution and its application, International Journal of Applied Mathematics,10Applications and Applied Mathematics: An International Journal (AAM), Vol. 16 [2021], Iss. 1, Art. 44https://digitalcommons.pvamu.edu/aam/vol16/iss1/44786 D. GhorbanzadehVol. 25, pp. 709-721.Ghitany, M.E., Atieh, B. and Nadarajah, S. (2008). Lindley distribution and its applications, Math-ematics and Computers in Simulation, Vol. 78, No. 4, pp. 493-506.Ghorbanzadeh, D., Durand, P. and Jaupi, L. (2016). An analytical method for detecting the change-point in simple linear regression model, Application at Weibull distribution, Journal of Ap-plied Quantitative Methods, Vol. 11, Issue 1, pp. 1-10.Heathcote, A., Brown, S. and Mewhort, D. J. K. (2002). Quantile maximum likelihood estimationof response time distributions, Psychonomic Bulletin & Review, Vol. 9, pp. 394-401.Lindley, D.V. (1958). Fiducial distributions and Bayes’ Theorem, Journal of the Royal StatisticalSociety, Vol. 20, No. 1, pp. 102-107.Ranneby, B. (1984). The maximum spacing method. An estimation method related to the maximumlikelihood method, Scand, J, Statist, Vol. 11, pp. 93-112.Rodrigues, J.A., Silva, A.P.C.M. and Hamedani, G.G. (2015). The beta exponentiated Lindleydistribution, Journal of Statistical Theory and Applications, Vol. 14, No. 1, pp. 60-75.Sankaran, M. (1970). The discrete Poisson-Lindley distribution, Biometrics, Vol. 26, No. 1, pp.145-149.Whalen, T.M., Savage, G.T. and Jeong, G.D. (2004). An evaluation of the self-determinedprobability-weighted moment method for estimating extreme wind speeds, Journal of WindEngineering and Industrial Aerodynamics, Vol. 92, pp. 219-239.11Ghorbanzadeh: Properties for Lindley-Polynomial DistributionPublished by Digital Commons @PVAMU, 2021

Generation and Statistical Properties for Lindley-Polynomial Distribution

International audienceFor the modeling of the wind speed, we propose a family of distributions in polynomial form generating the Lindley distribution. We call this distribution Lindley-Polynomial distribution. The estimation of parameters using the maximum product spacing estimation method. A real data set has been considered to illustrate the practical utility of the paper

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Generation and Statistical Properties for Lindley-Polynomial Distribution

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