ESTIMATION OF DENSITY AND DISTRIBUTION FUNCTIONS OF A BURR X DISTRIBUTION

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1 Journal of Statstcal Research 2018, Vol. 52, No. 1, pp ISSN X ESTIMATION OF DENSITY AND DISTRIBUTION FUNCTIONS OF A BURR X DISTRIBUTION AMULYA KUMAR MAHTO Department of Mathematcs, Indan Insttute of Technology Patna, Bhta , Inda Emal: amulya.pma15@tp.ac.n YOGESH MANI TRIPATHI Department of Mathematcs, Indan Insttute of Technology Patna, Bhta , Inda Emal: yogesh@tp.ac.n SANKU DEY Department of Statstcs, St. Anthonys College, Shllong , Meghalaya, Inda Emal: sankud66@gmal.com SUMMARY Burr type X dstrbuton s one of the members of the Burr famly whch was orgnally derved by Burr 1942) and can be used qute effectvely n modellng strength data and also general lfetme data. In ths artcle, we consder effcent estmaton of the probablty densty functon PDF) and cumulatve dstrbuton functon CDF) of Burr X dstrbuton. Eght dfferent estmaton methods namely maxmum lkelhood estmaton, unformly mnmum varance unbased estmaton, least square estmaton, weghted least square estmaton, percentle estmaton, maxmum product estmaton, Cremér-von-Mses estmaton and Anderson-Darlng estmaton are consdered. Analytc expressons for bas and mean squared error are derved. Monte Carlo smulatons are performed to compare the performances of the proposed methods of estmaton for both small and large samples. Fnally, a real data set has been analyzed for llustratve purposes. Keywords and phrases: Maxmum lkelhood estmator, Unformly mnmum varance unbased estmator. AMS Classfcaton: 62F10 1 Introducton Burr 1942) ntroduced twelve dfferent forms of cumulatve dstrbuton functons for modelng lfetme data or survval data. Of these twelve dstrbuton functons, Burr type X and Burr type XII were extensvely used by the researchers. The cumulatve dstrbuton functon of a Burr X dstrbuton as proposed by Surles and Padgett 2001) s gven by F x; α, γ) = 1 e γx)2) α, x > 0, α > 0, γ > 0, 1.1) Correspondng author c Insttute of Statstcal Research and Tranng ISRT), Unversty of Dhaka, Dhaka 1000, Bangladesh.

2 44 Mahto et al. and correspondng probablty densty functon s gven by fx; α, γ) = 2αγ 2 x e γx)2 1 e γx)2) α 1, x > 0, α > 0, γ > 0 1.2) where α and γ denote the shape and the scale parameters respectvely. Ths dstrbuton s also known as exponentated Raylegh or generalzed Raylegh dstrbuton. We denote ths dstrbuton as BurrXα, γ) through out ths artcle. It s observed that the two-parameter BurrXα, γ) dstrbuton has several propertes whch are qute common to the two-parameter gamma, Webull and generalzed exponental GE) dstrbutons. The one parameter Burr X dstrbuton wth γ=1) receved maxmum attenton n manay works Sartaw and Abu-Salh, 1991; Ahmad et al., 1997; Raqab, 1998; Surles and Padgett, 1998, 2001). In the recent past, Raqab and Kundu 2006) observed several nterestng propertes of Burr X dstrbuton n ther study and establshed relatons wth gamma, Webull, exponentated exponental and exponentated Webull dstrbutons. The dstrbuton functon and the densty functon of BurrXα, γ) dstrbuton have closed form. Due to ths feature, t can be used very convenently even for censored data. Unlke, gamma, Webull and GE dstrbutons t can have non-monotone hazard functon, whch can be very useful n many practcal applcatons. Raqab and Kundu 2006) observed that for α 1/2 Burr type X densty s a decreasng functon and t s a rght skewed unmodal functon for α > 1/2. They also observed that falure rates have the dfferent shapes dependng on the value of α. For α 1/2, t has ncreasng falure rate and for α > 1/2, t s bathtub shaped. For more detaled propertes of the Burr type X dstrbuton see Surles and Padgett, 2005). It s a common practce wth statstcans to focus on nferrng the parameters) nvolved n the model. However, one would fnd t more useful to study the effcent estmaton of the PDF and CDF nstead of nferrng the parameters) nvolved n the model. The problem of estmaton of the PDF and the CDF s sgnfcant for many reasons. For example, PDF can be used for estmaton of dfferental entropy, Reny entropy, Kullback-Lebler dvergence and Fsher nformaton. For example, Nlsson and Klejn 2007) consdered the problem of estmatng dfferental entropy usng the data located on embedded manfolds. Authors mentoned that such studes have found wdespread applcatons n varous areas of sgnal processng such as source codng, pattern recognton and blnd source separaton, among others. Hampel 2008) dscussed applcatons of entropy estmaton n neuroscence. The concept of dfferental entropy can be used to nfer random changes n neuron behavor under varous expermental scenaros. Recently Melnczuk and Wojtys 2010) consdered estmaton of Fsher nformaton for a probablty densty supported on fnte nterval. These applcatons suggest that estmaton of densty functon s an mportant problem n lterature. Smlarly CDF can be used for estmaton of cumulatve resdual entropy, the quantle functon, Bonferron curve, Lorenz curve and both pdf and cdf can be used for estmaton of probablty weghted moments, hazard rate functon, mean devaton about mean etc. For nstance, Bratpvrbajgyran 2012) consdered the problem of estmatng cumulatve resdual entropy for the Raylegh dstrbuton. Aucon et al. 2012) consdered estmaton of quantle of a two-parameter kappa dstrbuton usng dfferent methods. A flood data set s analyzed n support of proposed statstcal procedures and useful dscussons are presented based on ths numercal study. Longford 2012) further derved estmators

3 Estmaton of densty and dstrbuton functons of quantles of normal, log-normal and Pareto dstrbutons. The author studed a fnancal data on monthly returns and concluded that proposed estmators work qute well n such stuatons. In ths paper, our focus s to obtan based and unbased estmators of the PDF and CDF usng dfferent classcal methods of estmaton. Numerous statstcal developments and applcatons of the BurrXα, γ) dstrbuton has generated great nterest among appled statstcans to study the effcent estmaton of the PDF and the CDF of the BurrXα, γ) dstrbuton. We consder several estmaton methods: maxmum lkelhood estmaton MLE), unformly mnmum varance unbased estmaton UMVUE), least square LS) estmaton and weghted least square WLS) estmaton, percentle estmaton PC), maxmum product spacng estmaton MPS), Cramér-von-Mses CVM) method of estmaton and Anderson-Darlng AD) method of estmaton and thereby am to develop a gudelne to choose the best estmaton method for the BurrXα, γ) dstrbuton. Smlar knd of studes has appeared n the recent lterature for other dstrbutons see Asrabad, 1990; Dxt and Jabbar Nooghab, 2010; Jabbar Nooghab and Jabbar Nooghab, 2010; Dxt and Jabbar Nooghab, 2011; Bagher et al., 2014, 2016, and the references cted theren). Throughout ths paper except for Secton 3), we assume α s unknown, but γ s known. A future work s to extend the results of the paper to the case that all two parameters are unknown. In the lterature one would fnd several papers where the PDF and the CDF have been estmated when all ther parameters are unknown. For example, Duval 2013) nvestgated the nonparametrc estmaton of the jump densty of a compound Posson process from the dscrete observaton, Durot et al. 2013) obtaned least-squares estmator of a convex dscrete dstrbuton, Er 1998) evaluated the unknown parameters n the polynomal usng weghted resdual method, Dattner and Reser 2013) consdered the estmaton of dstrbuton functons when data contans measurement errors and Przyblla et al. 2013) used maxmum lkelhood estmator to estmate the cumulatve dstrbuton functon for the three-parameter Webull CDF n presence of concurrent flaw populatons. Our present work s dfferent from the exstng work because we have consdered eght methods of estmaton for estmatng pdf and cdf whereas n exstng lterature only fve methods of estmaton s consdered to the best of our knowledge. We have organzed the rest of the content of ths paper as follows. In Sectons 2.1 and 2.2 we have derved MLEs and UMVUEs of densty functon and dstrbuton functon wth ther mean squared errors MSEs) respectvely. The LSEs and WLSEs are obtaned n Secton 2.3 and percentle estmaton s dscussed n Secton 2.4. The other suggested methods of estmaton are descrbed n sectons 2.5, 2.6 and 2.7 respectvely. We have conducted a smulaton study n Secton 3 to assess the behavor of all estmators. We have analyzed a real data set for llustratve purpose n Secton 4. Fnally, n Secton 5, we conclude the paper.

4 46 Mahto et al. 2 Methods of Estmaton 2.1 Maxmum Lkelhood Estmaton In ths secton, we obtan MLEs of the PDF and the CDF of a BurrXα, γ) dstrbuton. Suppose X 1, X 2,..., X n denote ndependent and dentcally dstrbuted random samples from the BurrXα, γ) dstrbuton wth known scale parameter γ. The lkelhood functon of α can be wrtten as Lα, x) = n 2αγ 2 x e γx)2 1 e γx)2) α 1, 2.1) and the correspondng log-lkelhood functon s l = lnlx, α) nlnα) + lnx ) γ 2 n x 2 + α 1) Consderng the log lkelhood functon, we fnd the MLE of α as ln1 e γx)2 ). 2.2) n ˆα = n ln1 ) = n T, 2.3) e γx)2 where T = n ln1 e γx)2 ). Now consder the transformaton Y = gx ) = ln 1 e γx)2) wth g 1 y ) = 1/γ) [ ln 1 e y)] 1/2. Then t s seen that probablty densty of Y turns out to be, f Y y ) = f X g 1 x)) d 1 gx ) = αe αy, y > 0, α > 0. dy Thus, we see that f X has a BurrXα, γ) dstrbuton then Y = ln1 e γx)2 ) has an exponental dstrbuton wth rate α,.e. Y expα). As we know that the sum of ndependently and dentcally dstrbuted exponental dstrbuton follows a gamma dstrbuton. Thus T = n ln1 ) follows gamma Gn, α) dstrbuton havng densty functon as e γx)2 f T t) = αn tn 1 e αt, t > ) Further, we fnd that the MLE ˆα = n/t ) = W has an nverse gamma IGn, αn) dstrbuton havng densty functon as f W w) = nα)n 1/w)n+1 e αn/w, w > ) Note that Eˆα) = nα/n 1) and thus the MLE ˆα s a based but consstent estmator of the parameter α. The nvarance property of maxmum lkelhood method s appled and the desred estmators of PDF and CDF are then obtaned as ˆfx) = 2ˆαγ 2 xe γx)2 1 e γx)2)ˆα 1 and ˆF x) = 1 e γx) 2)ˆα. 2.6)

5 Estmaton of densty and dstrbuton functons Here we show that the estmators ˆfx) and ˆF x) are based for the PDF fx) and the CDF F x) respectvely and also obtan ther mean square errors. For presentaton and calculaton smplcty, we rewrte ˆfx) and ˆF x) as ˆfx) = 2wxηξ) w 1 and ˆF x) = ξ) w, 2.7) where η = γ 2 e γx)2 and ξ = 1 e γx)2. Before we obtan the mean squared errors, the followng expectatons are requred. We have E ˆfx)) m = 0 [ 2wxηξ) w 1 ] m nα) n = 2xη/ξ ) m nα) n = 2 2xη/ξ ) m nα) m+n)/2 ) m E ˆF x) = ξ w ) m nα) n 0 = nα)n = 2nα) n /w)n+1 e nα w dw w m n 1 e wmln1/ξ) nα w dw [mln 1/ξ )] m n)/2 Km n 2 ) mnαln 1/ξ), 1/w)n+1 e nα w dw w n 1 e wmln1/ξ)) nα w dw [mln 1/ξ )] n/2 K n 2 ) mnαln 1/ξ). The last equalty follows from the followng dentty 0 x ν 1 e µ x ηx dx = 2 µ/η ) ν 2 K ν 2 µη), where K ν ) s modfed Bessel functon of the second knd of order ν see also, Bagher et al., 2014). Theorem 1. Desred expectatons are and E [ ˆfx) ) m ] = 2 2xη/ξ ) m nα) m+n)/2 E [ ˆF x) ) m ] = 2 nα) n/2 [ ] m n)/2km n mln 1/ξ) 2 ) mnαln 1/ξ) [ ] n/2k n mln 1/ξ) 2 ) mnαln 1/ξ). Theorem 2. The mean squared errors of ˆfx) and ˆF x) respectvely are MSE ) ) 2 ˆfx) = 8x 2 η nα) n+2)/2 ξ ) η nα) n+1)/2 ξ [ ] 2 n)/2k2 n 2ln 1/ξ) 2 ) 2nαln 1/ξ) 8xfx) [ ] 1 n)/2k1 n ln 1/ξ) 2 ) nαln 1/ξ) + f 2 x),

6 48 Mahto et al. and MSE ˆF [ ] x)) = 2 nα)n/2 n/2k n 2ln 1/ξ) 2 ) 2nαln 1/ξ) 4F x) nα)n/2 [ ] n/2 ln 1/ξ) K n 2 ) nαln 1/ξ) + F 2 x). Proof. We have MSE ˆfx)) = E ˆfx) fx)) 2 = E ˆfx)) 2 2fx)E ˆfx)) + f 2 x). The requred expectatons n the above expresson can be obtaned easly by substtutng approprate choce of m n Theorem 2 to obtan the desred MSE. Smlarly we can obtan MSE ˆF x)). 2.2 Unformly Mnmum Varance Unbased Estmaton In ths secton, our am s to obtan UMVUEs of the PDF and the CDF of the specfed dstrbuton. We see that T = n ln1 e γx)2 ) s complete and suffcent for estmatng α for gven γ and T follows a gamma Gn, α) dstrbuton. One may refer to Ferguson 1967) for ths useful result. Followng Lehmann-Scheffé theorem f gx 1 t) = h t) s the condtonal PDF of X 1 gven T = t. Then we have see also, Bagher et al., 2014), E [h T )] = gx 1 t)ft)dt = gx 1, t)dt = fx 1 ), where gx 1, t) denotes the jont PDF of X 1, T ). Thus h T ) s the UMVUE of fx). Lemma 2.1. The condtonal dstrbuton of V gven that T = t s obtaned as f V T =t v t) = where V = ln1 e γx1)2 ). Proof. We have n 1)t v)n 2 t n 1, v < t < ft, v) f V T =t v t) = ft) f v, ) n =2 ln1 e γx)2 ) = t v = f T t) = αe αv α n 1 1) t v)n 2 e αt v) α n tn 1 e αt n 1)t v)n 2 = t n 1, v < t <. In the theorem stated below we gve UMVUEs of fx) and F x).

7 Estmaton of densty and dstrbuton functons Theorem 3. The expresson ˆfx) = n 1)t + ln1 e γx)2 )) n 2 t n 1 2xγ2 e γx) 2 1 e γx) 2) for ln1 e γx)2 ) < t < s the UMVUE of fx) and also the UMVUE of F x) s as ˆF x) = { 1 + 1/t) ln 1 e γx)2)} n 1. Proof. The estmator ˆfx) s the UMVUE of fx) follows from the Lehmann-Scheffé theorem and the prevous lemma. Also ˆF x) s the UMVUE of F x) follows from the fact that d ˆF x) dx = d [ {1 + 1/t) ln 1 e γx) 2)} ] n 1 = dx ˆfx). Further we proceed to obtan MSEs of these UMVUE estmators. For notatonal smplcty, we take Ω = ln1 e γx)2 ). Frst we obtan the followng two expectatons: E ˆfx)) m = Ω n 1) [ t + ln1 e γx) 2 ) ] n 2 = [ n 1)2xη/ξ ] m α n = [ n 1)2xη/ξ ] m α n = [ n 1)2xη/ξ ] m α n t n 1 ) 2xγ2 e γx) 2 m α n 1 e γx) 2) tn 1 e αt dt n ) mn 2) [1/t) t n m 1 e αt ] dt lnξ Ω =0 ) mn 2) lnξ) t n m 1 e αt dt Ω ) mn 2) lnξ) Γ n m ), αω). 2.8) =0 =0 We know ˆF x) = { 1 + 1/t) ln 1 e γx)2 )} n 1, Ω < u < and so we have E [ ] [ m {1 } ] n 1 m α n ˆF x) = + 1/t) lnξ Ω tn 1 e αt dt = αn mn 1) ) ) mn 1) lnξ t n 1 e αt dt t = αn = αn Ω mn 1) =0 mn 1) =0 =0 ) mn 1) lnξ) t n 1 e αt dt mn 1) Ω ) lnξ) Γ n, αω), 2.9) where Γs, αx) = x ts 1 e αt dt denotes the upper ncomplete gamma functon.

8 50 Mahto et al. Theorem 4. The mean squared error of estmator ˆfx) s gven as ) 2 MSE ˆfx)) n 1)2xη α n = ξ ) 2n 2) lnξ) Γ n 2 ), αω) f 2 x) =0 and the mean square error of estmator ˆF x) s gven as MSE ˆF x)) = αn 2n 1) =0 2n 1) Proof. MSEs of estmators ˆfx) and ˆF x) are defned as and ) lnξ) Γ n, αω) F 2 x). MSE ˆfx)) = E ˆfx)) 2 2fx) ˆfx) + f 2 x) = E ˆfx)) 2 f 2 x) 2.10) MSE ˆF x)) = E ˆF x)) 2 2F x) ˆF x) + F 2 x) = E ˆF x)) 2 F 2 x), 2.11) respectvely. Usng equaton 2.8) wth m = 2, we get the requred MSE for ˆfx). Smlarly we can obtan MSE of ˆF x) usng equaton 2.9) wth m = Least Squares and Weghted Least Squares Estmators Ths secton dscusses about regresson based estmators of unknown parameters. Swan et al. 1988) frst suggested ths method to estmate the parameters of beta dstrbutons. Further many authors dscussed and used ths method for dfferent dstrbutons. Consder a random sample X 1,..., X n of sample sze n from a CDF F ). Then we observe that EF X )) = n + 1, V F X )) = n + 1) n + 1) 2 n + 2), and cov [F X ), F X j )] = n j 1) n + 1) 2 n + 2) for < j,, j = 1, 2,..., n see, Johnson et al., 1994). Further we dscuss two varants of ths method namely least square and weghted least square estmators Least Square Estmators LSEs) In ths method [ F X ) ] 2 n + 1 s mnmzed wth respect to the unknown parameters. For Burr X dstrbuton, the expresson [ 1 e γx)2) ] α 2 n + 1

9 Estmaton of densty and dstrbuton functons s mnmzed wth respect to the unknown shape parameter α when γ s known) and the least square estmator for α s denoted as ˆα ls. Then we have ˆf ls x) = 2ˆα ls γ 2 xe γx)2 1 e γx)2 )ˆα ls 1 and ˆF ls x) = 1 e γx)2 )ˆα ls as the least square estmators of the fx) and the F x), respectvely. Further, smulaton study s conducted to calculate the desred expectatons and MSE values Weghted Least Square Estmators WLSEs) To obtan WLSE of the unknown parameters, we mnmze the expresson [ w F x ) ] 2 n wth respect to the unknown parameters. Here w = Var[F X x )] s defned as the weght functon see, Johnson et al., 1994). Note that the least square estmator s obtaned under the consderaton of constant varance. If such assumpton does not hold true then weghted least square estmaton may be consdered wth nverse of varance as weght. Under such scalng the correspondng error remans fnte. For the Burr X dstrbuton, the expresson n + 1) 2 n + 2) n + 1) [ 1 e γx)2) α n + 1 = n+1)2 n+2) n +1) s mnmzed wth respect to the unknown shape parameter α when γ s known). Suppose ˆα wls denotes the WLSE of α. Then we obtan the weghted LSEs of fx) and F x) as ˆf wls x) = 2ˆα wls γ 2 xe γx)2 ) 1 e γx)2)ˆα wls 1 and ˆFwls x) = 1 e γx)2)ˆα wls, respectvely. Further we have conducted a smulaton study to obtan the requred expectatons and MSE values. 2.4 Estmators based on Percentles Ths method was orgnally suggested by Kao 1958, 1959). A well explaned explanatons, on ths topc, can be found n Mann et al. 1974); Johnson et al. 1994). Burr X dstrbuton has closed form CDF and ths method s based on nvertng the CDF. So, estmaton of parameters of ths dstrbuton can be done usng ths method. Suppose X 1,..., X n denotes an ordered random sample from Burr X dstrbuton and F X ) as the ordered dstrbuton of the sample. Let p = /n + 1) then percentles estmator of α denoted by ˆα p s the one whch mnmzes the expresson [p 1 e γx)2 ) α] 2 n [ ] 2 or equvalently lnp αln1 e γx)2 ) ] 2

10 52 Mahto et al. wth respect to α. Then ˆf p x) = 2ˆα p γ 2 xe γx)2 1 e γx)2)ˆα p 1 and ˆFp x) = 1 e γx)2)ˆα p are the requred percentle estmators of fx) and F x) respectvely. Snce t s dffcult to fnd the expectatons and the MSE values for these estmators analytcally, so these can be obtaned by means of smulatons. 2.5 Method of Maxmum Product of Spacng The maxmum product spacng MPS) method has been ntroduced by Cheng and Amn 1979, 1983) as an alternatve to MLE for the estmaton of the unknown parameters parameters of contnuous unvarate dstrbutons. Consder a sample of sze n be taken from a Burr X dstrbuton. Then the correspondng unform spacng s defned as D j = F x j ) F x j 1 ), j = 1, 2,..., n, where F x 0:n ) = 0, F x n+1 ) = 1 and n+1 j=1 D j = 1. The MPS estmate of α denoted by ˆα m s obtaned by maxmzng n+1 Dα, γ) = j=1 D j 1 n+1 wth respect to the unknown shape parameter α. Equvalently, the expresson D α, γ) = 1 n+1 lnd j n + 1 can be maxmzed to obtan the estmate of α as desred. It can be shown that ˆα m satsfes j=1 n D 0 x j ) D 0 x j 1 )) = 0, n + 1 D j=1 j where D 0 x j ) = 1 e γxj)2) α m ln 1 e γxj)2). Ths has been shown by Cheng and Amn 1983) that the effcency of MPS method of estmaton s very close to the ML estmaton method. Then ˆf m x) and ˆF m x) are the MPS estmators of fx) and F x) respectvely and are gven by ˆf m x) = 2ˆα m γ 2 xe γx)2 1 e γx)2)ˆα m 1 and ˆFm x) = 1 e γx)2)ˆα m, respectvely. The expectatons and the MSE of these estmators can be calculated usng smulatons.

11 Estmaton of densty and dstrbuton functons Cramér-von-Mses Method of Estmaton To motvate our choce of Cramér-von-Mses type mnmum dstance estmators, MacDonald 1971) provded emprcal evdence that the bas of the estmator s smaller than the other mnmum dstance estmators. The value of α for whch the functon ) 2. Cα, λ) = 1 12n + n j=1 F x j ) 2j 1 2n s mnmzed s defned as the Cramér-von-Mses estmator of α denoted by ˆα c. Equvalently, soluton of the equaton F x j ) 2j 1 ) D 0 x j ) = 0 2n j=1 gves the desred estmator ˆα c. Therefore Cramér-von-Mses estmators of fx) and F x) are ˆf c x) and ˆF c x) respectvely and are gven by ˆf c x) = 2ˆα c γ 2 xe γx)2 1 e γx)2)ˆα c 1 and ˆFc x) = 1 e γx)2)ˆα c, respectvely. Smulatons are used to obtan the expectatons and the MSE values due to havng dffcultes n fndng analytc solutons. 2.7 Anderson-Darlng Method of Estmaton The Anderson-Darlng test Anderson and Darlng, 1952) s as an alternatve to other statstcal tests for detectng sample dstrbutons departure from normalty. Specfcally, the AD test converge very quckly towards the asymptote Anderson and Darlng, 1954; Petttt, 1976; Stephens, 1974). The AD estmator ˆα a of the unknown parameter α s obtaned from the functon Aα, γ) = n 1 n 2j 1) lnf x j ) + ln F x n+1 j ) ), j=1 by mnmzng wth respect to α. Equvalently, the soluton of the equaton D0 x j ) 2j 1) F x j ) D ) 0x n+1 j ) = 0, F x n+1 j ) j=1 provdes the desred estmator ˆα a wth D 0 ) beng defned earler. The Anderson- Darlng estmators for fx) and F x) are presented as ˆf a x) = 2ˆα a γ 2 xe γx)2 1 e γx)2)ˆα a 1 and ˆFa x) = 1 e γx)2)ˆα a, respectvely. The desred expectatons and the MSE values of these estmators s dffcult to fnd analytcally. So smulatons can be used.

12 54 Mahto et al. 3 Smulaton Study We perform smulaton study to compare the dfferent estmators dscussed here. We have arbtrarly consdered four dfferent sets parameter values, namely α, γ) = 1, 1), 1, 2), 2, 1), 2, 2) to compare the performance of proposed methods. We menton that samples are generated from the Burr X dstrbuton usng the probablty ntegral transformaton method. We have computed results for arbtrarly selected sample szes such as n = 10, 20, 30, 40, 50. In fact devaton n MSEs of dfferent estmators of PDF and CDF from the MSEs of correspondng ML estmator of PDF and CDF are obtaned. Thus devaton of MSEs represent the dfference between MSE of an estmator from the MSE of maxmum lkelhood estmator. We have computed these values for an arbtrary value x = 1. The devatons of MSEs of dfferent estmators are presented n Tables 1 2, from whch we can easly say that MLEs are the most effcent estmators of the PDF and the CDF of a Burr X dstrbuton and UMVUEs are the second most effcent estmators for the same. Snce varous devatons are postve hence we observe that ML estmators are havng the lowest MSE values. Further vsual analyss suggests that UMVUE has the second lowest MSE values for the PDF and CDF. Table 1: Devatons of MSEs of the PDF for varous methods from the MSEs of the PDF for MLE Devatons of MSEs of fx) Para. n UMVUE LSE WLSE PCE MPS CVM ADM 1,1) ,2) ,1) ,2)

13 Estmaton of densty and dstrbuton functons Table 2: Devatons of MSEs of the CDF for varous methods from the MSEs of the CDF for MLE Devatons of MSEs of F x) Para. n UMVUE LSE WLSE PCE MPS CVM ADM 1,1) ,2) ,1) ,2) Data Analyss In ths secton, we use a real data set to compare the performance of the suggested estmators of the PDF and CDF of BurrXα, γ) dstrbuton. The data set represent the number of cycles to falure for a group of 60 electrcal tems n a lfe test. The data was obtaned from Lawless 2003, page 112). Here, for computatonal ease, we have dvded the whole data set by The data set s ftted to Burr X dstrbuton, generalzed exponental dstrbuton and generalzed logstc dstrbuton and for all these three dstrbutons the estmates for parameters α and γ together wth Kolmogrov- Smrnov values and the p-values are calculated and presented n Table 3. It can be easly observed from the Kolmogrov-Smrnov and p-values that Burr X dstrbuton fts the data better than the two compettors. Lookng at the tabulated values n Table 4, we can conclude that the most effcent estmaton method for fttng the data s the ML estmaton method. Further dfferent model secton crtera namely maxmum lkelhood, Akake nformaton crteron, corrected Akake nformaton crteron, Bayes nformaton crteron and Hannan-Qunn crteron defned by

14 56 Mahto et al. Table 3: Goodness of ft tests for proposed models n the real data set Dstrbuton α γ KS p-value Burr X Generalzed Logstc Generalzed Exponental Table 4: Estmaton of parameters and the model selecton crtera for the real data set Estmator Estmate of α Estmate of γ ML AIC AICc BIC HQC MLE LSE WLSE PCE MPS CVM AD maxmum lkelhood = 2lnLθ), Akake nformaton crteron = 2lnLθ) + 2n p ) n Corrected Akake nformaton crteron = 2lnLθ) + 2n p n n p 1 Bayes nformaton crteron Hannan-Qunn crteron = 2lnLθ) + n plnn) and = 2lnLθ) + 2n plnlnn)), respectvely are used for assessng the behavor of the suggested methods of estmaton. Here lnlθ) denotes the log-lkelhood, n denotes the number of observatons n the data set, and n p denotes the number of parameters of the dstrbuton. The smaller values of these model selecton crtera leads to the better ft. It can easly be seen from Table 4 that the values of all the model selecton crtera for ML estmaton method are smaller than others. Thus the maxmum lkelhood method of estmaton s preferred to use n practce. 5 Concluson In ths artcle, we have consdered eght methods of estmaton of the probablty densty functon and the cumulatve dstrbuton functon for the BurrXα, γ) dstrbuton and comparsons are performed. Such comparsons can be useful to fnd the best estmators for the PDF and the CDF whch can be used to estmate functonals lke dfferental entropy, Rény entropy, Kullback-Lebler dvergence, Fsher nformaton, cumulatve resdual entropy, the quantle functon, Bonferron curve, Lorenz curve, probablty weghted moments, hazard rate functon, mean devaton about mean etc. From both smulaton study and real data analyss, we observed that MLE performs better than ther counter part. The performance of AD s farly reasonable and compettve. Also, evdence based on the MSEs n the smulaton study, the log-lkelhood values, and the model selecton crtera show that the ML estmators for the pdf and the CDF are the best. We hope our results and methods of

15 Estmaton of densty and dstrbuton functons estmaton mght attract wder sets of applcatons n the above mentoned functonals. As suggested by an anonymous revewer, t would be nterestng to nvestgate propertes of dfferent estmaton methods of PDF and CDF under some censorng technques as well, possbly usng some dfferent data sets. To the best of our knowledge, not much work has been done on ths partcular problem n lterature. Also we have obtaned results for consdered estmaton problem based on fnte sample stuatons. More work s requred to study the asymptotc behavor of such estmators. We wll try to work on these aspects n near future. Acknowledgments The authors would lke to thank a referee for careful readng and for valuable comments that greatly mproved the artcle. They also thanks the Edtor for constructve suggestons. References Ahmad, K., Fakhry, M., and Jaheen, Z. 1997), Emprcal Bayes estmaton of P Y < X) and characterzaton of Burr-type X model, Journal of Statstcal Plannng and Inference, 64, Anderson, T. and Darlng, D. 1952), Asymptotc theory of certan goodness-of-ft crtera based on stochastc processes, Annals of Mathematcal Statstcs, 23, ), A test of goodness of ft, Journal of Amercan Statstcal Assocaton, 49, Asrabad, B. 1990), Estmaton n the Pareto dstrbuton, Metrka, 37, Aucon, F., Ashkar, F., and Bayentn, L. 2012), Parameter and quantle estmaton of the 2- parameter kappa dstrbuton by maxmum lkelhood, Stochastc Envronmental Research and Rsk Assessment, 26, Bagher, S., Alzadeh, M., and Nadarajah, S. 2014), Effcent estmaton of the PDF and the CDF of the Webull extenson model, Communcatons n Statstcs - Smulaton and Computaton, DOI: / ), Effcent estmaton of the PDF and the CDF of the exponentated Gumbel dstrbuton, Communcatons n Statstcs-Smulaton and Computaton, 45: Bratpvrbajgyran, S. 2012), A test of goodness of ft for Raylegh dstrbuton va cumulatve resdual entropy, Proceedngs of the 8th World Congress n Probablty and Statstcs, Burr, I. 1942), Cumulatve frequency dstrbuton, Annals of Mathematcal Statstcs, 13,

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