Reflected Generalized Beta Inverse Weibull Distribution: definition and properties

Transcription

1 arxiv: v2 [math.st] 28 Jan 214 Refleted Generalized Beta Inverse Weibull Distribution: definition and properties Ibrahim Elbatal Institute of Statistial Studies and Researh, Department of Mathematial Statistis Cairo University - Egypt. i elbatal@staff.u.edu.eg Franesa Condino* and Filippo Domma Department of Eonomis, Statistis and Finane University of Calabria - Italy. f.domma@unial.it and franesa.ondino@unial.it * Corresponding author: franesa.ondino@unial.it

2 Refleted Generalized Beta Inverse Weibull Distribution: definition and properties Abstrat In this paper we study a broad lass of distribution funtions whih is defined by means of refleted generalized beta distribution. This lass inludes that of Beta-generated distribution as a speial ase. In partiular, we use this lass to extend the Inverse Weibull distribution in order to obtain the Refleted Generalized Beta Inverse Weibull Distribution.For this new distribution, moments, entropy, order statistis and a reliability measure are derived. The link between the Inverse Weibull and the Dagum distribution is generalized. Then the maximum likelihood estimators of the parameters are examined and the observed Fisher information matrix provided. Finally, the usefulness of the model is illustrated by means of an appliation to real data. Key words: Moments, entropy, reliability, bimodal distribution, ompound distribution. 1 Introdution In the literature various tehniques are ommonly used to extend a family of distribution funtion; some of these are based on simple transformations of the distribution funtion or of the survival funtion. In partiular, let X be a non-negative random variable (rv), representing the lifetime of an individual or unit, with umulative distribution funtion (df ), survival funtion (sf ) and probability density funtion (pdf ), respetively, denoted by G(x; τ ), Ḡ(x; τ ) 1 G(x; τ ) and g(x; τ ), where τ Ξ R p with p 1 and p N. The transformations F (x; η) [G(x; τ )] a and F (x; η) [ Ḡ(x; τ ) ] b define the proportional reversed hazard model (or Lehmann type I distribution) and proportional hazard model (or Lehamann type II distribution), respetively. Usually, the distribution G(x; τ ) is denoted as a parent or baseline distribution. In this ontext, Marshall and Olkin [27] point out that the parameters a and b assume the meaning of resiliene and frailty parameters, respetively. In this ontext, it is important to observe that one a resiliene or frailty parameter has been introdued, the reintrodution of the same kind of parameter does not expand the family of distribution funtion; for example, in the ase of the Weibull distribution, the introdution of the frailty 1

3 parameter does not expand the distribution beause the Weibull is already a proportional hazard family. Further extensions of the families of distribution funtions may be obtained by a ombination of the two methods outlined above; a remarkable example in this diretion is the Beta generated methods whih, in brief, define the following pdf f B G (x; η) 1 B(a,b) [G(x; τ )]a 1 [1 G(x; τ ))] b 1 g(x; τ ), where η (a, b, τ ) and B(, ) is the Beta funtion. Thus, in a sense, the ideas of both frailty and resiliene are embodied in this formulation of the pdf [Marshall and Olkin, 27, pag.238]. The lass of Beta-generated distributions has reeived onsiderable attention in reent years. In partiular, after the studies of Eugene et al. [22] and Jones [24], many Beta-generated distributions have been proposed. Among these, we reall the Beta-Gumbel [Nadarajah and Kotz, 24], the Beta-Exponential [Nadarajah and Kotz, 26] and the Beta-Weibull [Famoye et al., 25]. The reader is referred to Barreto-Souza et al. [21], Paranaíba et al. [211] and Domma and Condino [213] for the Beta-Generalized Exponential, the Beta-Burr XII and Beta-Dagum distribution, respetively, for more reent developments. A more omplete list of Beta-generated distributions and an interesting review of this method for generating families of distributions an be found in Lee et al. [213]. Moreover, the Beta generated method an be obtained by means of a transformation of a random variable, i.e. if U Be(a, b) then the pdf of X G 1 (U; η) is f B G (x; η). Reently, in order to inrease the flexibility of the generator, Alexander et al. [212] used the Generalized Beta of first type (GB1), first proposed by MDonald [1984], instead of the lassi Beta funtion as generator. The pdf of a GB1 is f(u; a, b, ) ua 1 (1 u ) b 1 (1) for u (, 1) and a, b, >. With this generator, the random variable X is said to have a generalized beta of first type (GB1) generated distribution, with pdf and df, respetively, given by and where η (a, b,, τ ). f GB1 G (x; η) F GB1 G (x; η) [G(x; τ )]a 1 (1 [G(x; τ )] ) b 1 g(x; τ ) (2) 1 [G(x;τ )] u a 1 (1 u) b 1 du (3) In effet, as disussed in Alexander et al. [212], (3) inreases the flexibility of the generator for the presene of parameter. Nevertheless, there are ases where this does not our; in partiular, when G(x; τ ) is already a reversed proportional hazard distribution. In suh ases, to give effet to the expansion of the distribution funtion it is possible to refer to the refleted version of (1) obtained by 2

4 transformation v 1 u, with pdf and df, respetively, given by F (v; a, b, ) 1 f(v; a, b, ) (1 v)a 1 (1 (1 v) ) b 1 (4) 1 v w a 1 (1 w ) b 1 dw 1 B((1 v) ; a, b) for v (, 1), a, b, >, where B(z; a, b) is the inomplete beta funtion. Putting V G(X; τ ), the random variable X is said to have a refleted generalized beta of first type (rgb1) generated distribution, with pdf and df, respetively, given by and f rgb1 G (x; η) F rgb1 G (x; η) 1 [1 G(x; τ )]a 1 (1 [1 G(x; τ )] ) b 1 g(x; τ ) (6) 1 1 [1 G(x;τ )] [1 G(x;τ )] v a 1 (1 v ) b 1 dv w a 1 (1 w) b 1 dw 1 B ([1 G(x; τ )] ; a, b). (7) The lass of generated distributions defined in (6) is alled Beta-Exponential-X family by Lee et al. [213]. Clearly, the baseline distribution G(x; τ ) is a speial ase of (7) when a b 1. If 1 we obtain the Beta-generated distribution. Moreover, the exponentiated generalized lass of distributions proposed by Cordeiro et al. [213a] and Cordeiro et al. [213b] turns out to be a speial ase of F rgb1 G (x; η) setting a 1. We thus observe that the latter lass of distributions an be thought of as refleted Kumaraswamy-generated distributions. For b > real non-integer, using the power series representation (5) where p j1,b (1 [1 G(x; τ )] ) b 1 p j1,b[1 G(x; τ )] j 1 (8) j 1 ( 1)j 1 Γ(b) Γ(b j 1 )Γ(j 1, we an write the pdf (6) as +1) f rgb1 G (x; η) p j1,b[1 G(x; τ )] (a+j1) 1 g(x; τ ) (9) j 1 with df F rgb1 G (x; η) 1 j 1 p j1,b { } 1 [1 G(x; τ )] (a+j 1 ). (1) (a + j 1 ) 3

5 The expansions of the pdf shown in (9) an be used to determine ertain properties of rbg1-g distribution as, for example, the k-th moment, E ( X k) j 1 p j1,b x k [1 G(x; τ )] (a+j 1) 1 g(x; τ )dx { p j1,be G X k [1 G(X; τ )] } (a+j 1) 1 (11) j 1 where E G (.) denotes the expetation with respet to baseline distribution G(X; τ ). An alternative expression for the k-th moment is E ( X k) j 1 p j1,b { p j2,(a+j 1 )E G X k [G(X; τ )] } j 2. (12) j 2 In this paper, we use the generator defined by (7) to extend the Inverse Weibull distribution. The rest of the paper is organized as follows. In Setion 2, we define the Refleted Generalized Beta of first type Inverse Weibull distribution and derive the expansions for its distribution and probability density funtions. The quantile, the moments, the entropy, the order statistis and a measure of reliability are derived in Setion 3. In Setion 4, the link between the Inverse Weibull distribution and the Dagum distribution is generalized. The maximum likelihood estimation is disussed in Setion 5. An appliation on real data set is reported in Setion 6. Finally, some of the mathematial results are reported in the Appendix. 2 Refleted Generalized Beta of first type Inverse Weibull Distribution (rgb1-iwei) Several authors have highlighted the fat that there are a onsiderable number of ontexts in whih the inverse Weibull (IWei) distribution may be an appropriate model, mainly beause the empirial hazard rate is unimodal (see, for example, Kundu and Howlader [21]; Singh et al. [213]). Reently, Erto [213] has shown ertain peuliar properties of the IWei distribution; in partiular, he formulated three real and typial degenerative mehanisms whih lead exatly to the IWei random variable. The umulative distribution funtion and probability density funtion of the IWei random variable, respetively, are: G IW ei (x; τ ) e γx θ (13) and g IW ei (x; τ ) θγx θ 1 e γx θ, (14) 4

6 where τ (γ, θ), with k-th moment about zero given by ( E(X k ) γ k θ Γ 1 k ) θ (15) where Γ( ) is the Gamma funtion. Reently, Gusmão et al. [211] have tried to generalize the IWei distribution by introduing the resilene parameter in (13), i.e. G(x) e γx θ but, as highlighted by Jones [212], the [ ] ɛ, proposed distribution is only a reparametrization of G IW ei (x; τ ). In other words, they do not expand the lass of distributions beause the authors introdue a resilene parameter in a reversed proportional hazard model. In order to extend the IWei distribution, given that this distribution is a reversed hazard rate model, we use the refleted Generalized Beta generator defined in (7). So, the rgb1-iwei umulative distribution funtion is defined by inserting (13) in (7), i.e. [1 GIW ei (x;τ )] F rgb1 IW ei (x; η) 1 1 v a 1 (1 v) b 1 dv 1 I ([1 GIW ei (x;τ )] )(a, b) (16) where η (γ, θ, a, b, ) and I G (a, b) is the inomplete beta funtion ratio. The orresponding pdf of the new distribution an be written as f rgb1 IW ei (x; η) γθx θ 1 e γx θ [1 e γx θ] a 1 ( 1 [1 e γx θ ] ) b 1. (17) Fig.1 shows various behaviours of the density f rgb1 IW ei (x; η), inluding the bimodal one, obtained for different values of parameters γ, θ, a, b and. The hazard rate for rgb1-iwei distribution, given by h rgb1 IW ei (x; η) f rgb1 IW ei(x;η), is a I ([1 GIW ei (x;τ )] )(a,b) omplex funtions of x. However, from the plots reported in Fig. 2, we an state that h rgb1 IW ei (x; η) is muh more flexible than the hazard rate of the IWei distribution. In partiular, by denoting the stritly inreasing (dereasing) failure rate as IFR (DFR) and the hazard rate with a minimum or a maximum as Bathtub (BT) and Upside-down Bathtub (UBT), respetively, we an observe that the rgb1-iwei model enables us to obtain IFR and DFR hazard rates or hazard rates with a non-monotoni behaviour, suh as UBT, BT-UBT or UBT-BT-UBT hazard rate. This wide range of different behaviours of the hazard rate makes the rgb1-iwei distribution a suitable model for reliability theory and survival analysis. 5

7 1 γ.6; θ 5; a.3; b 5; 2 γ.6; θ 5; a 1.2; b 5; 2 γ.6; θ 5; a 2; b 5; 2 f(x ; η) f(x ; η) γ.55; θ.8; a 1.2; b 5; 2 γ.29; θ 1.2; a 1.2; b 5; 2 γ.11; θ 1.8; a 1.2; b 5; x x (a) (b) γ 1.9; θ.2; a.2; b.2;.3 γ 1.9; θ.2; a.2; b.2; 15 γ 1.9; θ.2; a.2; b.2; 5 f(x ; η) f(x ; η) γ 7e 4; θ 8; a.2; b.1; 1 γ 7e 4; θ 8; a.2; b.15; 1 γ 7e 4; θ 8; a.2; b.3; x x () (d) Figure 1: rgb1-iwei density for ertain values of the parameters. 2.1 Expansions for the Cumulative Distribution Funtion and for the Probability Density Funtion In this subsetion, we present some representations of df and pdf of rgb1-iwei in terms of infinite sums. Using the relations (9), (13) and (14), the pdf of rgb1-iwei an be written as X frgb1 IW ei (x; η) pj,b [1 GIW ei (x; τ )](a+j) 1 giw ei (x; τ ) j (18) with df given by FrGB1 IW ei (x; η) 1 X pj,b 1 [1 GIW ei (x; τ )](a+j) j (a + j) o 1 X pj,b n θ 1 [1 e γx ](a+j) j (a + j) 6 (19)

8 h(x ; η) γ 1; θ 2; a.5; b.1; 1 γ 1; θ 2; a.5; b.1; 3 γ 1; θ 2; a.5; b.1; 5 h(x ; η) γ.2; θ 5; a.5; b.1; 5 γ.2; θ 5; a.5; b.8; 5 γ.2; θ 5; a.5; b 1.8; x x (a) (b) h(x ; η) γ 2; θ 5; a 1; b 1.8;.9 γ 2; θ 5; a 1; b 1.8; 3 γ 2; θ 5; a 1; b 1.8; 5 h(x ; η) γ 1.9; θ.2; a.2; b.2;.3 γ 1.9; θ.2; a.2; b.2; 15 γ 1.9; θ.2; a.2; b.2; x x () (d) Figure 2: rgb1-iwei hazard rate for ertain values of the parameters. Another expansion of f rgb1 IW ei (x; η) may be derived from the following relation between pdf and df of a IWei distribution, i.e. g IW ei (x; τ ) γθx θ 1 G IW ei (x; τ ). After algebra, we an write f rgb1 IW ei (x; η) in term of an infinite sum of IWei density funtions, i.e. ( ) p j2,(a+j f rgb1 IW ei (x; η) p 1 ) j1,b (j 2 + 1) g IW ei(x; γ(j 2 + 1), θ). (2) j 1 j 2 The linear ombination (2) enables us to obtain some mathematial properties of the rgb1-iwei distribution diretly from those of the IWei distribution suh as the moments, moment generating funtion and reliability. 7

9 3 Statistial Properties This setion is devoted to studying the statistial properties of the rgb1-iwei distribution, speifially quantile funtion, moments and moment generating funtion, entropy, order statistis and reliability. 3.1 Quantile of rgb1-iwei The quantile x q of the rgb1-iwei distribution an be easily obtained onsidering that q F rgb1 IW ei (X x q ; η) 1 I ([1 GIW ei (x q;τ )] )(a, b) from whih and [1 G IW ei (x q ; τ )] [1 e γ(xq)θ ] I 1 (1 q; a, b) x q { 1γ ln ( 1 [I 1 (1 q; a, b)] 1/)} 1/θ. (21) The median an be derived from (21) be setting q 1, that is, 2 Med (rgb1 IW ei) { 1γ ln ( 1 [I 1 (.5; a, b)] 1/)} 1/θ. (22) 3.2 Moments and Moment Generating Funtion In this subsetion we disuss the r th moment for the rgb1-iwei distribution. Moments are neessary in any statistial analysis, espeially in appliations. They an be used to study the most important features and harateristis of a distribution (e.g., tendeny, dispersion, skewness and kurtosis). Proposition 1. If X has rgb1-iwei distribution, then the r th moment of X is given by the following ( p j2,(a+j µ r (η) p 1 ) j1,b (j 2 + 1) [(j 2 + 1)γ] r θ Γ (1 r ) ) (23) θ j 1 j 2 Proof. It is an immediate onsequene of the expansions of the pdf defined in (2) and by equation (15). Based on the first four moments the measures of skewness A(η) and kurtosis k(η) of the rgb1-iwei distribution an obtained as A(η) µ 3(η) 3µ 1 (η)µ 2 (η) + 2µ 3 1(η), [µ 2 (η) µ 2 1(η)] 3 2 8

10 and k(η) µ 4(η) 4µ 1 (η)µ 3 (η) + 6µ 2 1(η)µ 2 (η) 3µ 4 1(η) [µ 2 (η) µ 2 1(η)] 2. Plots of the skewness and kurtosis measures of the rgb1-iwei distribution for seleted values of parameters are reported in Fig. 3 and Fig. 4. η 2, θ,1.3, 1.2, 2 η.5, 1.1, a, 1.2, 2 A(η) A(η) θ a (a) (b) A(η) η 5, 1.1, 1.8, b, 2 A(η) η.5, 1.1, 1.8, 5, b () (d) Figure 3: Skewness measures of the rgb1-iwei distribution for ertain values of parameters. Now, we shall derive the moment generating funtion of rgb1-iwei distribution. Proposition 2. If X has rgb1-iwei distribution, then the moment generating funtion M X (t) has the following form M X (t) ( ) t r r! r j 1 p j1,b ( j 2 p j2,(a+j 1 ) (j 2 + 1) ) ( [γ(j 2 + 1)] r θ Γ 1 r ). (24) θ 9

11 k(η) η 2, θ,1.3, 1.2, 2 k(η) η.5, 1.1, a, 1.2, θ a (a) (b) k(η) η5, 1.1, 1.8, b, 2 k(η) η.5, 1.1, 1.8, 5, b () (d) Figure 4: Kurtosis measures of the rgb1-iwei distribution for ertain values of parameters. Proof. Using the series expansion e tx (tx) r r, we an write r! M X (t) r (t) r r! x r f rgb1 IW ei (x; η)dx r (t) r r! µ r(η). (25) Substituting (23) into (25) we get the expression of M X (t). 3.3 Entropy The onept of entropy plays a vital role in information theory. The entropy of a random variable is defined in terms of its probability distribution and an be shown to be a valid measure of randomness or unertainty. In this setion we present Rényi entropy I R (ρ) and Shannon entropy H(f rgb1 IW ei (x; η)). 1

12 Proposition 3. If X has rgb1-iwei distribution, then the Rényi entropy is given by ( ) ρ I R (ρ) 1 ρ log + 1 ( ) 1 ρ log Γ (ρ 1) ρ + log(γθ) θ ( ) ρ log p j1,ρ(b 1)+1p j2,ρ(a 1)+j 1 +1 (ρ 1) ρ+ [(j 2 + ρ)γ] θ j 1 j 2 (26) Proof. The Rényi entropy is defined as I R (ρ) 1 1 ρ log [ ] f rgb1 IW ei (x; η) ρ dx, where ρ >,and ρ 1. For the rgb1-iwei distribution, using the power series representation, the integral in I R (ρ) an be redued to f rgb1 IW ei (x; η) ρ dx ( γθ ) ρ j 1 j 2 p j1,ρ(b 1)+1p j2,ρ(a 1)+j 1 +1 Setting γ(j 2 + ρ)x θ w, after algebra, we obtain ( ) ( ) ρ γθ Γ ρ + (ρ 1) f rgb1 IW ei (x; η) ρ θ dx γθ through simple algebra we obtain (26). j 1 j 2 x ρ(θ+1) e (j 2+ρ)γx θ dx. p j1,ρ(b 1)+1p j2,ρ(a 1)+j 1 +1 (ρ 1) ρ+ [(j 2 + ρ)γ] θ Proposition 4. If X has rgb1-iwei distribution, then the Shannon entropy is given by ( ) θγ H(f rgb1 IW ei (x; η)) log + j 1 j 2 { ( ) (θ + 1) log [(j2 + 1)γ] Γ (1) θ (j 2 + 1) 1 + (j 2 + 1) (b 1) 2 Proof. The Shannon entropy is given by i p i,j2 +1 (i + 1) H(f rgb1 IW ei (x; η)) E [log f rgb1 IW ei (x; η)] p j1,bp j2,(a+j 1 ) (a 1) (j 2 + 1) [ψ(1) ψ(j 2 + 2)] + ( i )] } (27) [ ψ(1) ψ log(γθ) + log + (θ + 1)E [log(x)] +E [ γx θ] { (a 1)E log [1 e γx θ]} { ( (b 1)E log 1 [1 e γx θ] )}. 11

13 Using the power series representation, we have E [log(x)] θγ j 1 j 2 j 1 j 2 j 1 j 2 p j1,bp j2,(a+j 1 ) p j1,bp j2,(a+j 1 ) θ(j 2 + 1) p j1,bp j2,(a+j 1 ) θ(j 2 + 1) log(x)x (θ+1) e (j 2+1)γx θ dx {log [(j 2 + 1)γ] log(w)} e w dw { } log [(j 2 + 1)γ] Γ (1) also, by Proposition 1, we an write Moreover, we have E [ γx θ] γe ( X θ) j 1 j 2 p j1,bp j2,(a+j 1 ) (j 2 + 1) 2. E { [ log 1 e γx θ]} θγ log j 1 j 2 p j1,bp j2,(a+j 1 ) [1 e γx θ] x (θ+1) [ e γx θ] j 2 +1 dx, setting w 1 e γx θ, after algebra, we obtain E { [ log 1 e γx θ]} j 1 j 2 j 1 j 2 p j1,bp j2,(a+j 1 ) p j1,bp j2,(a+j 1 ) (j 2 + 1) where ψ(.) denote the digamma funtion. Finally, the last integral is given by E { ( log 1 θγ 1 1 [1 e γx θ] )} p j1,bp j2,(a+j 1 ) j 1 j 2 1 p j1,bp j2,(a+j 1 ) j 1 j 2 j 1 j 2 j 1 j 2 p j1,bp j2,(a+j 1 ) p j1,bp j2,(a+j 1 ) i i 1 log(w)(1 w) j 2 dw {ψ(1) ψ(2 + j 2 )}, ( log 1 [1 e γx θ] ) [ x (θ+1) e γx θ] j 2 +1 dx log(w)(1 w) 1 1 [1 (1 w) 1 ] j2 dw p i,j2 +1 p i,j2 +1 (i + 1) 1 log(w)(1 w) i+1 1 dw { ψ(1) ψ ( i + 1 )}

14 3.4 Order Statistis In this subsetion, we derive losed form expressions for the pdf and moments of the k th order statisti of the rgb1-iwei distribution. Let X 1, X 2,..., X n be a simple random sample from rgb1-iwei distribution with df and pdf given by (16) and (17), respetively. Let X 1:n, X 2:n,..., X n:n denote the order statistis obtained from this sample. We now give the probability density funtion of X k:n, say f k:n rgb1 IW ei(x; η) and the moments of X k:n, k 1, 2,..., n. The probability density funtion of X k:n is given by f k:n rgb1 IW ei(x; η) 1 B(k, n k + 1) [F rgb1 IW ei(x; η)] k 1 [1 F rgb1 IW ei (x; η))] n k f rgb1 IW ei (x; η) (28) where F rgb1 IW ei (x; η) and f rgb1 IW ei (x; η) are the df and pdf of the rgb1-iwei distribution given by (16) and (17), respetively. Sine < F rgb1 IW ei (x; η) < 1, for x >, by using the binomial series expansion of [1 F rgb1 IW ei (x; η)] n k, given by n k ( ) n k [1 F rgb1 IW ei (x; η)] n k ( 1) j [F rgb1 IW ei (x; η)] j, j j j we have n k ( ) n k f k:n rgb1 IW ei(x; η) ( 1) j [F rgb1 IW ei (x; η)] k+j 1 f rgb1 IW ei (x; η), (29) j substituting (16) and (17) into (29), we an express the s th ordinary moment of the k th order statistis X k:n, say E(Xk:n s ), as a liner ombination of the probability weighted moments of the rgb1-iwei distribution, i.e. 3.5 Reliability n k ( n k [ E [Xk:n] s ( 1) )E j rgb1 IW ei X s (F rgb1 IW ei (X; η)) k+j 1]. (3) j j In reliability studies, the stress-strength term is often used to desribe the life of a omponent whih has a random strength X and is subjet to a random stress Y. The omponent fails if the stress applied to it exeeds the strength, and the omponent will funtion satisfatorily whenever Y < X. Thus R P (Y < X) is a measure of a omponent reliability whih has many appliations in physis, engineering, genetis, psyhology and eonomis (Kotz et al. 23). In this subsetion, we derive the reliability R when X rgb1 IW ei(η x ) and Y rgb1 IW ei(η y ) are independent random 13

15 variables. Using expansion (2) of the pdf of the rgb1-iwei random variable, after algebra, we an write where R x y B(a x, b x )B(a y, b y ) R j2,j 4 j 1 j 2 j 3 j 4 p j1,b y p j2, y(a y+j 1 )p j3,b x p j4, x(a x+j 3 ) R j2,j (j 2 + 1)(j 4 + 1) 4 g IW ei (x; γ x (j 4 + 1), θ x )G IW ei (x; γ y (j 2 + 1), θ y )dx (31) is the reliability between the independent Inverse Weibull random variables. Hene, the reliability between rgb1-iwei random variables is a linear ombination of the reliabilities between IWei random variables. Putting θ x θ y θ, after algebra, the R j2,j 4 is given by R j2,j 4 γ x (j 4 + 1)θ x θ 1 e x θ [γ x(j 4 +1)+γ y(j 2 +1)] dx γ x (j 4 + 1) [γ x (j 4 + 1) + γ y (j 2 + 1)]. Finally, the reliability between independent rgb1-iwei random variables, in the ase θ x θ y θ, is given by R x y B(a x, b x )B(a y, b y ) j 1 j 2 j 3 j 4 p j1,b y p j2, y(a y+j 1 )p j3,b x p j4, x(a x+j 3 )γ x (j 4 + 1) (j 2 + 1)(j 4 + 1) [γ x (j 4 + 1) + γ y (j 2 + 1)]. 4 Link between Generalizations of Inverse Weibull and Generalizations of Dagum distribution In this setion, we generalize the link between the Inverse Weibull distribution and Dagum distribution, via mixing Gamma density (see, for example,kleiber and Kotz [23, pag.192]). In the literature, it is reognized that a pdf, say p 1 (x), has a mixture representation if it an be written as p 1 (x) p Θ 2(x θ)p 3 (θ)dθ, where θ is regarded as a random variable with pdf p 3 (.), alled mixing density, and p 2 (x θ) is the onditional pdf of rv X given θ. Mixtures are also alled ompound distributions. The following Proposition shows that the Burr III distribution (or speial ase of the Dagum distribution) has a mixture representation with mixing Gamma density. Proposition 5. If the onditional rv X given γ has Inverse Weibull distribution, i.e. X γ IW ei(γ, θ), and the rv γ is Gamma distributed, i.e. γ Ga(β), then the rv X is distributed as a Burr III distribution, i.e. X BurrIII(β, θ). 14

16 Proof. See, for example, Kotz and Kleiber (23), pag Next, by means of two propositions, we shall disuss the onnetion between ertain generalizations of the IWei distribution and those of the Dagum distribution via mixing Gamma density. Proposition 6. If X γ BeIW ei(a, b, γ, θ) and γ Ga(β) then X Mixture of Dag(β, λ j, θ), with λ j (a + j). Proof. The onditional pdf of the rv X γ and the marginal pdf of the rv γ, respetively, are and Observed that γ is f(x γ; a, b, θ) γθ [ ] a 1 [ x θ 1 e γx θ e γx θ 1 e γx θ] b 1 f(γ; β) γβ 1 e γ. Γ(β) [ ] b 1 1 e γx θ ( 1) j Γ(b) j Γ(b j)j! e γjx θ, after algebra, the joint pdf between X and f(x, γ; a, b, β, θ) θx θ 1 Γ(β) j ( 1) j Γ(b) Γ(b j)j! γβ e γ[1+(a+j)x θ ] marginalizing with respet to γ, we obtain the marginal pdf of rv X, i.e. f(x, a, b, β, θ) θx θ 1 Γ(β) j j ( 1) j Γ(b) Γ(b j)j! γ β e γ[1+(a+j)(x θ ] dγ ( 1) j Γ(a + b) Γ(a)Γ(b j)j!(a + j) f Dag(x; β, (a + j), θ) where f Dag (x; β, (a + j), θ) β(a + j)θx θ 1 [ 1 + (a + j)x θ] β 1 is the pdf of a Dagum rv. Moreover, given that j ( 1) j Γ(a+b) Γ(a)Γ(b j)j!(a+j) f(x, a, b, β, θ) is a mixture of Dagum distribution. 1, see Domma and Condino (213), we an say that Proposition 7. If X γ rgb1 IW ei(a, b,, γ, θ) and γ Ga(β) then X rgb1 Da(a, b,, β, λ j, θ), with λ j (a + j). Proof. The onditional pdf of X γ is f rgb1 IW ei (x γ; a, b,, θ) j 1 p j1,b ( j 2 p j2,(a+j 1 ) (j 2 + 1) γ(j 2 + 1)θx θ 1 γ(j2+1)x θ e ). 15

17 Assuming that the rv γ has pdf f(γ; β) γβ 1 e γ, the marginal pdf of X is given by Γ(β) f(x; η 1 ) f rgb1 IW ei (x γ; a, b,, α, θ)f(γ; β)dγ j 1 j 1 j 1 j 1 p j1,b p j1,b p j1,b p j1,b j 2 j 2 j 2 j 2 p j2,(a+j 1 ) (j 2 + 1) p j2,(a+j 1 ) (j 2 + 1) θx θ 1 Γ(β) θx θ 1 Γ(β) γ β (j 2 + 1)e γ[(j 2+1)x θ +1] dγ (j 2 + 1) [(j 2 + 1)x θ + 1] β+1 w β e w dw p j2,(a+j 1 ) (j 2 + 1) β(j 2 + 1)θx θ 1 [ (j 2 + 1)x θ + 1 ] β 1 p j2,(a+j 1 ) (j 2 + 1) f Da(x; β, (j 2 + 1), θ). 5 Estimation and Inferene In order to estimate the parameters η (a, b,, γ, θ) of the rgb1-iwei distribution, we use the maximum likelihood (ML) method. Let x (x 1, x 2,..., x n ) be a random sample of size n from the rgb1- IWei given by (17). The log-likelihood funtion for the vetor of parameters η (a, b,, γ, α, θ) an be expressed as l(η) n log() nlog() + (a 1) + (b 1) n log [1 G IW ei (x i ; τ )] i1 n log {1 [1 G IW ei (x i ; τ )] } + i1 n log {g IW ei (x i ; τ )} (32) where τ (γ, θ). In what follows, we denote with ḣy(y, z) h(y,z), ḧyy(y, z) 2 h(y,z) and y y 2 ḧ yz (y, z) 2 h(y,z) the partial derivatives of first order, of seond order and mixed of a funtion, y z say h(y, z), respetively. Moreover, to simplify the notation, we use G(x i ; τ ) G IW ei (x i ; τ ) and g(x i ; τ ) g IW ei (x i ; τ ). Differentiating l(η) with respet to a, b,, γ and θ, respetively, and setting the results equal to zero, i1 16

18 we have l(η) a n Ḃa(a,b) B(a,b) + n i1 lg [1 G(x i; τ )] l(η) b l(η) n Ḃb(a,b) B(a,b) + n i1 lg (1 [1 G(x i; τ )] ) n + a n i1 lg [1 G(x i; τ )] (b 1) n [1 G(x i ;τ )] lg[1 G(x i ;τ )] i1 (1 [1 G(x i ;τ )] ) l(η) τ j (a 1) n Ġ τj (x i ;τ ) i1 + (b 1) n [1 G(x i ;τ )] 1 Ġ τj (x i ;τ ) [1 G(x i ;τ )] i1 (1 [1 G(x i + n ġ τj (x i ;τ ) ;τ )] ) i1 g(x i ;τ ) for j 1, 2, 3 and with τ 1 γ and τ 2 θ, where Ġ...(x i ; τ ) and ġ... (x i ; τ ) are reported in the Appendix. The system does not admit any expliit solutions; therefore, the ML estimates ˆη (â, ˆb, ĉ, ˆγ, ˆθ) an only be obtained by means of numerial proedures. Under regularity onditions, the ML estimator ˆη is onsistent and asymptotially normally distributed. Moreover, the asymptoti variane-ovariane matrix of ˆη an be approximated by the inverse of observed information matrix given by J aa J ab J a J aγ J aθ.. J bb J b J bγ J bθ J(η).... J J γ J θ J γγ J γθ J θθ whose elements are reported in the Appendix. In order to build the onfidene interval and hypothesis tests, we use the fat that the asymptoti distribution of ˆη an be approximated by the multivariate normal distribution, N 6 (, [J(ˆη)] 1 ), where [J(ˆη)] 1 is the inverse of observed information matrix evaluated at ˆη. 6 Appliation In this subsetion we onsider a real data set onerning the survival times (in days) of guinea pigs injeted with different doses of tuberle bailli. These data, divided by 1, have already been used in Kundu and Howlader [21] in order to illustrate some results derived for the IWei distribution. We ompare the fit of the IWei model, resulting from the maximum likelihood estimates of the parameters reported by the authors for the unensored data ase, with that obtained by rgb1-iwei model. For the rgb1-iwe model, we obtain the following ML estimates and the orresponding standard errors: ˆγ.8176 (5.421), ˆθ.1284 (.627), â ( ), ˆb (72.825), ĉ (5.724). Both the Akaike Information Criterion (AIC) and the Kolmogorov-Smirnov 17

19 Density rgb1 IWei IWei x Figure 5: Empirial and fitted densities of the rgb1-iwei and IWei distributions. (KS) statistis suggest that the rgb1-iwei model provides a better representation of the data than the IWei model. Indeed, onsidering the different number of parameters and the log-likelihood values reported for the IWei and rgb1-iwei models (ˆl IW ei , ˆl rgb1 IW ei ), we obtain the AIC values, equal to and , respetively. The values of the KS statisti for the two models are KS IW ei.1364 and KS rgb1 IW ei.999. The assoiated p-value for the rgb1-iwei, equal to.4692, supports the hypothesis that the data are drawn from the proposed distribution. Furthermore, sine the IWei is a sub-model of the rgb1-iwei when a b 1, we an onsider the maximum values of the unrestrited and restrited log-likelihoods in order to obtain the LR statistis for testing the need of the extra parameters. Based on the LR statisti, equal to 1.932, and the orresponding p-value, equal to.121, at 5% signifiane level, we rejet the null hypothesis in favor of the new distribution. The plot of the empirial and the fitted densities are given in Fig. 5. Finally, we examine the empirial and the fitted hazard rate. In their paper, Kundu and Howlader [21] affirm that the use of the IWei model seems reasonable sine the plot of the empirial version of the saled TTT transform for the data indiates that the hazard rate is unimodal. As suggested by Bergman and Klefsjö [1984], we use this plotting proedure as a omplementary tehnique for model identifiation. If we ompare the fitted urves for the IWei and the rgb1-iwei models (see Fig. 6), the latter better illustrates the empirial behaviour of the hazard rate, again onfirming the superiority of the proposed model in desribing suh data. 18

20 Saled TTT Transform rgb1 IWei IWei Figure 6: Empirial and fitted Saled TTT transformation of the Guinea pigs data. 7 Final Remarks In this paper, we introdued the Refleted Generalized Beta of Inverse Weibull Distribution defined by using the Generalized Beta of first type distribution as a generator funtion, as proposed by Alexander et al. [212], onsidering the refleted version for this generator. This method allows us to obtain a muh more flexible distribution than the Inverse Weibull one. Indeed, both the density and the hazard funtion show additional behaviours ompared to the Inverse Weibull distribution. This onsiderable flexibility makes the new distribution a suitable model for different appliations, suh as, for example, reliability and survival analysis. We provided full treatment of mathematial properties, obtaining moments, entropy, order statistis and a reliability measure. We disussed maximum likelihood estimation and obtained the observed Fisher Information Matrix. Finally, an appliation to a real data set is given to illustrate the usefulness of the proposed distribution. 19

21 8 Appendix The elements of the observed information matrix J(η) for the parameters (a, b,, γ, θ) are given by J aa n ] 2 B aa (a, b) [Ḃa (a, b) [] 2 ; J ab n B ab (a, b) Ḃa(a, b)ḃb(a, b) [] 2 J a n log [1 G(x i ; τ )] ; i1 J bb n ] 2 B bb (a, b) [Ḃb (a, b) [] 2 J b n i1 [1 G(x i ; τ )] log [1 G(x i ; τ )] (1 [1 G(x i ; τ )] ) J n n (b 1) [1 G(x i ; τ )] (log [1 G(x i ; τ )]) 2 2 (1 [1 G(x i ; τ )] ) 2 i1 J aτj n i1 Ġ τj (x i ; τ ) [1 G(x i ; τ )] ; J bτj n i1 [1 G(x i ; τ )] 1 Ġ τj (x i ; τ ) (1 [1 G(x i ; τ )] ) J τj n i1 { Ġ τj (x i ; τ ) 1 + (b 1) [1 G(x i; τ )] ([1 G(x i ; τ )] } log [1 G(x i ; τ )] 1) [1 G(x i ; τ )] (1 [1 G(x i ; τ )] ) 2 for j 1, 2 and τ 1 γ and τ 2 θ; ( n Gτj τ J τj τ h (1 a) h (x i ; τ ) [1 G(x i ; τ )] + Ġτ j (x i ; τ )Ġτ h (x i ; τ ) [1 G(x i1 i ; τ )] 2 + ) g τj τ h (x i ; τ )g(x i ; τ ) ġ τj (x i ; τ )ġ τh (x i ; τ ) n [1 G(x i ; τ )] 2 [g(x i ; τ )] 2 + (b 1) i1 (1 [1 G(x i ; τ )] ) 2 { (1 [1 G(x i ; τ )] ) (Ġτj (x i; τ )Ġτ (x h i; τ ) + [1 G(x i ; τ )] G ) τj τ (x h i; τ ) } Ġτ j (x i ; τ )Ġτ h (x i ; τ ) for j 1, 2 and h 1, 2 with j h, and τ 1 γ and τ 2 θ. Where 2

22 Ġ γ (x i ; τ ) x θ i G(x i ; τ ) ; Ġ θ (x i ; τ ) γx θ i log(γx i )G(x i ; τ ) ; Gγγ (x i ; τ ) x θ i Ġ γ (x i ; τ ) G γθ (x i ; τ ) x θ i { } log(x i )G(x i ; τ ) Ġθ(x i ; τ ) } G θθ (x i ; τ ) γx θ i log(x i ) {Ġθ (x i ; τ ) log(x i )G(x i ; τ ) { } ġ γ (x i ; τ ) θx θ 1 i G(x i ; τ ) + γġγ(x i ; τ ) { } ġ θ (x i ; τ ) γθx θ 1 i [1 θ log(x i )] G(x i ; τ ) + θġθ(x i ; τ ) { g γγ (x i ; τ ) θx θ 1 i 2Ġγ(x i ; τ ) + γ G } γγ (x i ; τ ) { [ ] g γθ (x i ; τ ) x θ 1 i [1 θ log(x i )] G(x i ; τ ) + γġγ(x i ; τ ) + θ [Ġθ (x i ; τ ) + γ G ]} γθ (x i ; τ ) { g θθ (x i ; τ ) γx θ 1 i θ G θθ (x i ; τ ) [2 θ log(x i )] log(x i )G(x i ; τ )+ } 2 [1 θ log(x i )] Ġθ(x i ; τ ) 21

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