Estimation of the density and cumulative distribution functions of the exponentiated Burr XII distribution

Hassan, Amal S.; Assar, Salwa M.; Ali, Kareem A.; Nagy, Heba F. Article Estimation of the density and cumulative distribution functions of the exponentiated Burr XII distribution Statistics in Transition New Series Provided in Cooperation with: Polish Statistical Association Suggested Citation: Hassan, Amal S.; Assar, Salwa M.; Ali, Kareem A.; Nagy, Heba F. (2021) : Estimation of the density and cumulative distribution functions of the exponentiated Burr XII distribution, Statistics in Transition New Series, ISSN 2450-0291, Exeley, New York, Vol. 22, Iss. 4, pp. 171-189, https://doi.org/10.21307/stattrans-2021-044 This Version is available at: https://hdl.handle.net/10419/266288 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ STATISTICS IN TRANSITION new series, December 2021 Vol. 22, No. 4 pp. 171–189, DOI 10.21307/stattrans-2021-044 Received – 30.05.2020; accepted – 26.03.2021 Estimation of the density and cumulative distribution functions of the exponentiated Burr XII distribution Amal S. Hassan1, Salwa M. Assar2, Kareem A. Ali3, Heba F. Nagy4, ABSTRACT The exponentiated Burr Type XII (EBXII) distribution has wide applications in reliability and economic studies. In this article, the estimation of the probability density function and the cumulative distribution function of EBXII distribution is considered. We examine the maximum likelihood estimator, the uniformly minimum variance unbiased estimator, the least squares estimator, the weighted least squares estimator, the maximum product spacing estimator, the Cramér–von-Mises estimator, and the Anderson–Darling estimator. We derive analytical forms for the bias and mean square error. A simulation study is performed to investigate the consistency of the suggested methods of estimation. Data relating to the wind speed and service times of aircraft windshields are used with the studied methods. The simulation studies and real data applications have revealed that the maximum likelihood estimator performs more efficiently than its remaining counterparts. Key words: exponentiated Burr Type XII model, least squares estimator, maximum likelihood estimator, uniform minimum variance unbiased estimator, weighted least squares estimator. Mathematical Subject Classification: 62F10. 1. Introduction The Burr Type XII (BXII) distribution has gained special attention in physics, actuarial studies, reliability and applied statistics. Characteristics of the BXII distribution are near to several distributions like exponential, normal, lognormal, etc. Extra properties about the BXII distribution can be found in Headrick et al. (2010). 1 Department of Mathematical Statistics, Cairo University, Faculty of Graduate Studies for Statistical Research, Egypt. E-mail: [email protected]. ORCID: https://orcid.org/0000-0003-4442-8458. 2 Department of Mathematical Statistics, Cairo University, Faculty of Graduate Studies for Statistical Research, Egypt. E-mail: [email protected]m. ORCID: https://orcid.org/0000-0001-7450-7486. 3 Department of Mathematical Statistics, Cairo University, Faculty of Graduate Studies for Statistical Research, Egypt. E-mail: [email protected]. ORCID: https://orcid.org/0000-0002-2522-7733. 4 Corresponding Author, Department of Mathematical Statistics, Cairo University, Faculty of Graduate Studies for Statistical Research. Egypt, E-mail: [email protected]. ORCID: https://orcid.org/0000-0003-0262- 205X. 172 Amal S. Hassan et al.: Estimation of the density and cumulative … The exponentiated Burr Type XII distribution is a generalization to the BXII distribution through adding a new shape parameter. The cumulative distribution function (CDF) of the EBXII distribution is defined as follows:  () 1 1 ; ,, , 0, k c Gx x xkc         (1) where, k, c and  are shape parameters. The probability density function (PDF) of the EBXII corresponding to (1) is specified by    β1 1 11 1 1 ; , , , 0. kk cc c gx ck x x x xkc           (2) Statistical developments on the EBXII model have been studied by several authors. Among them, AL-Hussaini and Hussein (2011) studied maximum likelihood (ML) and Bayesian estimation to the parameters of the EBXII distribution under Type II censored data. Kumar et al. (2017) established several explicit expressions and recurrence relations for single and product moments of r-th lower record values from the EBXII distribution. Statistical inference is one of the most popular topics in research and scientific studies whether from the theoretical or applied aspects. Most traditional studies have been focused on inferring the parameter(s) involved in the distribution. The importance of statistical distributions is not limited to the characterization of statistical phenomena, but rather to the calculation of many population metrics such as moments, probability weighted moments, failure rate function, etc. However, it would be more useful to study the efficient estimation of the PDF and CDF. The estimation of the PDF and the CDF is important for many reasons. For instance, the best estimators for the PDF can be used to estimate functionals of the PDF such as estimation of Kullback-Leibler divergence, as provided by Hurvich et al. (1990), estimation of Fisher information (see Mielniczuk and Wojtyś (2010), the estimation of the differential entropy (see Nilsson and Kleijn (2007), and estimation of the Rényi entropy. Similarly, the best estimators for the CDF can be used to estimate functionals of the CDF like estimation of quantiles (see Saleh et al. (1988) and estimation of the Lorenz curve (see Woo and Yoon (2001)). Some studies on the estimation of PDF and CDF have appeared in recent literature for some continuous distributions, for instance, Pareto distribution by Asrabadi (1990) and (Dixit and Nooghabi (2010), Dixit and Nooghabi (2011)), exponentiated Pareto distribution by Jabbari (2010), generalized Rayleigh distribution by Alizadeh et al. (2013), generalized exponential Poisson distribution by Bagheri et al. (2014), exponentiated Weibull by Alizadeh et al. (2015a), generalized exponential distribution by Alizadeh et al. (2015b), exponentiated Gumbel distribution by Bagheri et al. (2016a), Weibull extension distribution by Bagheri et al. (2016b), Lindley distribution by Maiti and Mukherjee (2018), generalized logistic distribution by Tripathi et al. (2017), STATISTICS IN TRANSITION new series, December 2021 173 Frechet distribution by Maleki and Deiri (2017) and Topp-Leone distribution by Benkhelifa (2017), exponentiated gamma distribution (Rasekhi (2018)), and Gompertz distribution (Dey et al. (2018)). Our objective here is to investigate the efficient estimation of the PDF and the CDF of the EBXII model due to its wide statistical applications and developments. Different parametric methods of estimation, namely ML, uniformly minimum variance unbiased (UMVU), least squares (LS), weighted least squares (WLS), Cramér-von-Mises (CvM), Anderson–Darling (AD) and maximum product spacing (MPS) are considered. This paper is organized as follows. Sections (2) and (3) provide ML and UMVU estimators of the PDF and CDF with their mean square errors (MSEs). Section (4) includes other parametric methods of estimation. Section (5) comprises a simulation study in order to compare different suggested estimators. A real data set is analyzed for illustrative purpose in Section (6). The article ends with concluding remarks in Section (7). 2. Maximum likelihood estimators In this section we obtain the ML estimators of the PDF and the CDF of the EBXII distribution. Let X1, X2, …, Xn be a random sample with size n from the EBXII distribution with known parameters k and c. The log likelihood function of the EBXII distribution is given by     11 1 |,, 1 ln 1 ln1 1ln11 . nn c ii ii nk c i i Lxcknln nlnknlnc c x k x x                (3) The ML estimator of ,  say ˆ,  is given as  1 , ln 1 ˆ 1 nk c i i nn T x           1 ln 1 1 . nk c i i Tx          We can rewrite CDF(1) as follows: ln𝐺󰇛𝑥󰇜 𝛽𝑉, 𝑉  ln 󰇛1  󰇛1 𝑥 󰇜󰇜. It can be seen that 𝑉 has an exponential distribution with scale parameter .  Then T has a gamma 󰇛𝑛, 𝛽󰇜, random variable with density function given by  1, 0. Γ( ) nnt f ttet n     (4) 174 Amal S. Hassan et al.: Estimation of the density and cumulative … Therefore, ˆ, nS T   has an inverse gamma (, )nn  distribution with PDF given by   1, 0 . Γ( ) nn ns n f sses n      (5) Applying the invariance property of the ML method, the required PDF and CDF estimators are obtained as follows:    1 1 1 ˆ 111 , ˆ ˆkk cc c gx ck x x x          and   ˆ 11 . ˆk c Gx x       (6) Now, we show that   ˆ gx and   ˆ Gx are biased estimators of   gx and   Gx respectively. Further, the MSEs are obtained. Theorem (1) calculates  () ˆr Egx and  ( ˆ). r EG x Theorem 1: We have     11 2 21 () K(2 1, Γ( ) 1 ˆ rn rrn rrr k r rn ckbd d n n Egx n rln d nrlnd                and     2 () 2 K(2 1, Γ( ) 1 ˆ n n r n nn E Gx nrln d nrlnd          where 1,(1). cck bx d x   Proof: First by using 1,(1), cck bx d x   then   ˆ gx can be rewritten as follows:   111, ˆ1s k gx cksbd d   and   1 ˆ. s Gx d Thus,     111 0 ˆ1Γ( ) nn r rrsr rrrr n ks n E gx cksbd d s e ds n              11(1) 1 0 1Γ( ) nn rrsrln d rrr rn ks n ckbd d s e e ds n             STATISTICS IN TRANSITION new series, December 2021 175    11 2 21 K(2 1 . Γ( ) 1 rn rrn rrr k rn ckbd d n nnrln d nrlnd                Here, K(.)  denotes the modified Bessel’s function of the second kind of order  (see equation (3.471.9) in (Gradshteyn and Ryzhik (2000)). Similarly,  () ˆr EGx takes the following form:     2 () 2 K(2 1. Γ( ) 1 ˆ n n r n nn EG x n rln d nrlnd          Theorem 2: The MSEs for   ˆ gx and   ˆ Gx respectively are given by           2 12 21 2 222 2 1 22 1 2 1K(221 Γ( ) 2 1 1K(211 Γ( . )1 ˆ(2 4)) n n kn n n n nn MSE g x c k b d d n ln d nlnd nn dnlndd nlnd                                 and         22 β2 2 K(22 1 41 K(2 1 1 . Γ( ) 2 1 Γ( ˆ )1 nn nn nn nn nn MSE G x n ln d d n ln d d n lnd n lnd                      Proof: Since         22 ˆˆ ˆ .2MSEgx Egx gxEgx gx  Hence,    2 ˆ Egx and     ˆ Egx can be obtained by setting r =1 and r =2 in Theorem (1), hence the     ˆ M SE g x is easily calculated. The proof of   ˆ M SE G x is similar. 3. Uniformly minimum variance unbiased estimators In this section, UMVU estimators of the PDF and CDF of the EBXII distribution are considered. In addition, the rth moment and the MSE of these estimators are derived. Let 1, . . . , n XX be a random sample of size n from the EBXII distribution. Then,  1 ln 1 1 nk c i i Tx          is complete sufficient statistic for the parameter  176 Amal S. Hassan et al.: Estimation of the density and cumulative … (assumed k and c are known parameters). Recall that T has a gamma 󰇛𝑛, 𝛽󰇜 distribution with density function (4). According to the Lehmann-Scheffe theorem, if     * 1| gxt g t is the conditional PDF of 1|XT , we have           * 111 | , , E gT xtftdt xtd xggtg  where   1, g xt is the joint PDF of X1 and T. Therefore,   * g tis the UMVU estimator of  . g x Lemma 1: The conditional distribution of V given T = t is obtained as    2 | 1 1 1 |,,ln11. nk c VT n ntv vt v t V xgt        Proof: We have           22 |11 ,Γ()1 |,. Γ1 nn nt VT nn t n gvtv tv e n n tv vt v t ft nte t g               In the following theorem the UMVU estimators for   g xand   Gx are obtained. Theorem 3: The uniformly minimum variance unbiased estimators for  g xand  Gxare given by          2 1 1 * 1 1ln11 1 () , ln1 1 , 11 n k ck cc k c nk c nt x kcx x gx gt x t tx             and   1 ln 1 1 . n k c tx Gx t               Proof: The estimator () g x  is the UMVU estimator for () g x can be proved by the Lehmann-Scheffe theorem and Lemma (1). In addition,   x G  is the UMVU estimator of G(x) from the fact that  1 ln(1 (1 ) ) (). n ck dt x x gx dx t dG dx         STATISTICS IN TRANSITION new series, December 2021 177 Further, we compute the MSEs for the two UMVU estimators of  g x and   ,G x  suppose that      1 1 11 and ( ) ln 1 1 . 11 k cc k c k c nkcx x Mpxx x          Then, we obtain the following expectation:       () 2 1 () () . Γ( ) px px nr r n rr rnt nr r tpx E gx gx ftdt M t e dt nt           After some simplification, we obtain    ( 1 ) 2 0 21, Γ( ) () nr r r ri iir nir u x ip px nr r M Egx eduu i n             which () 1nir u px eduu     is the upper incomplete gamma function, so    r Egx  can be formulated as follows:    2 0 21. Γ() ( , ) () () nr r ir ri ri i nr r Egx iMpx n i rpx n             (7) Similarly, we can prove that    0 () ( ,())1. Γ( ) i nr r ri i i nr r px n ipxEGx i n             (8) Theorem 4. The mean square errors for   g x and   Gx , respectively, are given by    24 2 22 0 241(() ( 2,()) () g( )) , niii i n M MSE p x n i p x n gx x i             and    22 2 0 221(G()). Γ( ) () ( ,()) i i ni i n MSE G x x ipx i x nnp              Proof: Since        22 , M SE g x E g x g x  where    2 Egx  can be obtained by setting r =2 in (7), hence we can calculate     . M SE g x  The proof of    M SE G x is similar. 178 Amal S. Hassan et al.: Estimation of the density and cumulative … 4. Other parametric methods of estimation In this section, several methods of estimation such as LS, WLS, MPS, CvM and AD are considered. All these methods are based on the CDF. Let :,1,,, in Xi n   be the order statistics of a random sample from the EBXII distribution and assumed k and c are known parameters. Then, the LS, WLS, MPS, CvM and AD estimators of the PDF and the CDF of the EBXII distribution are derived in the following subsections. 4.1. Least squares and weighted least squares estimators The ordinary least squares and the weighted least squares (Swain et al. (1988)) are well-known methods used for estimating the unknown parameters. The LS estimator of ,  say,   and the WLS estimator of ,  say, ,  are given by minimizing the following quantities with respect to  2 : 1 () , 1 n in i i Gx n        (9)    22 : 1 12 () . 11 n in i nn i Gx in i n          (10) There is no closed form solution for ,  in minimizing Equations (9) and (10), so the numerical technique is applied to find   and  . Hence, the LS and WLS estimators of the CDF and PDF for the EBXII distribution are obtained, respectively, as follows:   11 , k c xxG            1 1 1,111 kk cc c xckx xgx            and   11 , k c Gx x          1 1 1111 . kk cc c gx ck x x x          4.2. Maximum product of spacing estimators The MPS has been proposed by Cheng and Amin (1979) as an alternative method for the ML for the estimation parameters of continuous univariate distribution. Let a sample of size n be available from EBXII, we define the corresponding uniform spacings as follows: :1: ,1,2,..()( ) ., iinin inDGx Gx    where 0: 1: ()0,( )1, nnn Gx Gx   1 1 1. n i i D     STATISTICS IN TRANSITION new series, December 2021 185 6. Application to real data Real data sets are considered to compare between ML, LS, WLS, CvM, AD and MPS methods. The first data consist of 31 observations that represent the Average Monthly Wind Speed (m/s) at Kolkata (from 1st March, 2009 to 31st March, 2009); these data were introduced by Bhattacharya and Bhattacharjee (2010). The second data set represents the data on service times of 63 aircraft windshield given by Murthy et al. (2004). For both data sets, all the three parameters are considered as unknown parameters. The parameters are estimated by ML, MPS, LS, WLS, CvM and AD methods. ML, MPS estimators are obtained by maximizing Equations (3) and (11), respectively, with respect to , k  and c. LS, WLS, CvM and AD estimators can be obtained by minimizing Equations (9), (10), (13) and (14), respectively, with respect to  , k and c. We compared the estimation methods by means of model selection criteria. The criteria like Akaike information criterion (AIC), Bayesian information criterion (BIC), and corrected Akaike information criterion (AICc) are considered. The model with the minimum AIC, BIC and AICc is chosen as the best model to fit the data. In addition, the PDF plot (estimated PDFs versus the empirical histogram for the data) and the CDF plot (estimated CDFs versus the empirical CDF for the data) are used in the model selection. Tables 3 and 4 give the parameter estimates and the values of the model selection for different methods. Table 3. Estimates of the parameters and the corresponding AIC, BIC and AICc for first data Methods c Estimate k Estimate  Estimate AIC BIC AICc ML LS WLS CvM AD MPS 2.139 3.295 3.113 3.278 3.461 1.198 1.631 0.686 0.856 0.757 0.701 1.040 1.333 0.599 0.701 0.647 0.585 0.989 56.304 57.841 56.921 57.237 57.317 79.264 60.605 62.143 61.223 61.539 61.619 83.566 56.946 58.484 57.564 57.88 57.96 79.687 Table 4. Estimates of the parameters and the corresponding AIC, BIC and AICc for second data Methods c Estimate k Estimate  Estimate AIC BIC AICc ML LS WLS CvM AD MPS 1.378 1.148 1.174 1.154 1.381 1.402 1.206 2.103 2.302 2.142 1.242 1.1 1.988 6.789 8.321 7.156 2.456 1.734 235.331 270.693 291.356 275.685 237.674 235.739 241.76 277.123 297.786 282.115 244.103 242.168 235.527 270.89 291.553 275.882 237.87 235.939 186 Amal S. Hassan et al.: Estimation of the density and cumulative … As seen from Tables 3 and 4, the ML estimates give the smallest values compared with the other estimates. Figures 9 and 10 represent plots of the CDFs and PDFs of the EBXII distribution based on the fitted ML, LS, WLS, CvM, AD and MPS methods to the data, the figures indicate the superiority of the ML method over the other methods. Figure 9. CDF and PDF plots for Wind Speed (m/s) data fitted by different methods of estimation Figure 10. CDF and PDF plots for service times of 63 aircraft windshield fitted by different methods of estimation data Densi ty 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 data ML LS WLS AD CvM MPS PDF plot for Bhattacharya and Bhattacharjee data data Densi ty 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 data CDF data ML LS WLS AD CvM MPS CDF plot for Bhattacharya and Bhattacharjee data data CDF data Densi ty 0123456 0.0 0.1 0.2 0.3 0.4 0.5 data ML LS WLS AD CvM MPS PDF plot for Murthy et al. data data Densi ty STATISTICS IN TRANSITION new series, December 2021 187 7. Conclusion In this paper, we consider seven different estimators of the PDF and CDF of the EBXII distribution when the shape parameters k and c are assumed to be known. Maximum likelihood estimator, uniformly minimum variance unbiased estimator, least squares estimator, weighted least squares estimator, maximum product spacing estimator, Cramér-von-Mises estimator and Anderson-Darling estimator are obtained. 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