Truncated Weibull-G More Flexible and More Reliable than Beta-G Distribution

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1 International Journal of Statistics and Probability; Vol. 6 No. 5; September 217 ISSN E-ISSN Published by Canadian Center of Science and Education Truncated Weibull-G More Flexible and More Reliable than Beta-G Distribution Hossein Naarzadegan 1 Mohammad Hossein Alamatsaz 1 & Saied Hayati 1 1 Department of Statistics University of Isfahan Isfahan Iran Correspondence: M. H. Alamatsaz Department of Statistics University of Isfahan Isfahan Iran. alamatho@sci.ui.ac.ir Received: May Accepted: June Online Published: July doi:1.5539/isp.v6n5p1 Abstract URL: Our purpose in this study includes introducing a new family of distributions as an alternative to beta-g (B-G distribution with flexible hazard rate and greater reliability which we call Truncated Weibull-G (TW-G distribution. We shall discuss several submodels of the family in detail. Then its mathematical properties such as expansions probability density function and cumulative distribution function moments moment generating function order statistics entropies unimodality stochastic comparison with the B-G distribution and stress-strength reliability function are studied. Moreover we study shape of the density and hazard rate functions and based on the maximum lielihood method estimate parameters of the model. Finally we apply the model to a real data set and compare B-G distribution with our proposed model. Keywords: Entropy Moment Moment generating function Stress-strength reliability function Weibull distribution 1. Introduction In the past decades many authors have shown interest in obtaining new classes of probability distributions with higher flexibility in applications. Exponentiated exponential family by Gupta and Kundu (21 exponentiated Weibull family by Nassar and Eissa (23 exponentiated beta distribution by Nadaraah (25a exponentiated Pareto distribution by Nadaraah (25b and generalized Gumbel distribution by Cooray (21 are examples of such interests. Eugene Lee and Famoye (22 proposed a family of beta-g distributions by F(x 1 B( G(x t 1 (1 t 1 dt (1 where > > G(x is a distribution function and B( 1 t 1 (1 t 1 dt is the beta function. Recently several new distributions have been introduced following the proposed wor by Eugene et al. (22 including the beta- Gumble and beta-exponential distributions by Nadaraah and Kots (2426 respectively beta-weibull distribution by Lee Famoye and Olumolade (27 beta-generalized exponential distribution by Barreto-Souza Santos and Cordeiro (21 beta-modified Weibull distribution by Silva Ortega and Cordeiro (21 beta-weibull-geometric distribution by Cordeiro Silva and Ortega (213 beta-generalized gamma distribution by Cordeiro et al. (213 beta-lindley distribution by MirMostafaee Mahdizadeh and Nadaraah (215 and beta-generalized Marshall-Olin-G family by Handique and Charaborty (216. Weibull distribution plays an important role in Reliability Theory. In fact Weibull distribution is the first candidate for modeling cases with monotone hazard rates but it is not a good candidate for modeling cases with non-monotone hazard rates. Thus we can see several authors attempting to modify the Weibull distribution for applying in cases with nonmonotone hazard rates. The truncated Weibull distribution is one of such distributions with bathtub shaped hazard rate function. The truncated Weibull distribution has been applied in several engineering fields partly because it has more flexible hazard rate than Weibull distribution. As seen in Eq. (1 the baseline pdf is beta distribution defined on the interval [ 1]. This fact motivated us to introduce a new class of truncated Weibull-G distributions by inserting the truncated Weibull distribution with support [ 1] into Eq. (1 instead of the beta distribution. Truncated Weibull-G distribution compared to Beta-G distribution is more reliable and has explicit forms for its cumulative distribution function (cdf and hazard rate function (hrf. Therefore in Section 2 first propose the new truncated Weibull-G (TW-G family of distributions. Density hazard rate and quantile functions of this model are discussed in this section. In Section 3 its several submodels are studied. Several mathematical characteristics of the model such as cdf and pdf moments and moment generating function Renyi and Shannon entropies order statistics unimodal property and stress-stregnth reliability function for TW-G distributions are 1

2 International Journal of Statistics and Probability Vol. 6 No. 5; 217 considered in Section 4. In this section the new family are stochastically compared with the more classic B-G distinction and it is revealed that as a lifetime distribution it has larger reliability. We then describe shapes behavior of the pdf and hrf in Section 5. Maximum lielihood estimates and the observed information matrix are considered in Section 6. In last section an application of several submodels to a real data are considered and they are compared to several corresponding submodels of beta-g distribution where it is illustrated that our proposed distribution has better reliability than the beta-g distribution in (1. 2. The New Family By replacing the truncated Weibull distribution with support [ 1] with beta distribution in Eq. (1 we have F(x G(x t 1 exp( t dt 1 exp( G(x. > > (2 Henceforth we indicate a random variable X with cdf (2 by X TW-G( where > > and it is called truncated Weibull-G distribution. Clearly density function corresponding to (2 is denoted by f (x G(x 1 exp( G(x g(x (3 where g is pdf corresponding to G. Thus the hrf of X becomes h(x G(x 1 exp( G(x g(x exp( G(x exp( (4 The quantile function (qf of the TW-G distribution can be generated by inverting cdf (2 as Q(u G 1 (( log ( 1 1 u(1 e (5 where G 1 is the quantile function of the parent distribution G and u ( 1. Recall that log(1 x x +1 x < 1 (6 + 1 (1 x r ( r ( 1 x x < 1 (7 and an equation which is defined by Gradshteyn and Ryzhi (27 we have ( n a x d n x (8 where n N d n ( a 1 m1 [m(n + 1 ]a md n m and d n a n. Thus qf Q in (5 can be expanded as 2

3 International Journal of Statistics and Probability Vol. 6 No. 5; 217 follows: Q(u G 1 (( G 1 (( 1 G 1 [( 1 G 1 [( 1 G 1 [( 1 G 1 [( 1 G 1 [( 1 G 1 [( 1 log ( 1 u(1 e 1 ( log ( 1 u(1 e ( 1 [1 + log [ 1 u(1 e ] ] ] ( 1 ( 1 (1 + log [ 1 u(1 e ] ] ( 1 ( 1 l ( 1 ( 1 ( 1 ( ( ( log [ 1 u(1 e ] ] ( ( 1 log [ 1 u(1 e ] ] ( ( ( 1 u u l (1 e l+1 l l + 1 ] ( ] u l+ d l (9 where d l (l(1 e 1 l m1 [m( + 1 l] (1 e m+1 m+1 d l m for l > and d 1. By substituting proper values for u in (9 we can obtain quantiles of interest. As a special case the median of X is given by Median(X Q G (( log ( (1 e Hence for simulating random variable X from uniform random variable U on the unit interval (1 we have (( X Q G log ( 1 1 U(1 e. 3. Special Submodels In the present section we introduce six interesting submodels of the TW-G distribution. i TW-Uniform distribution(tw-u:. Let the parent distribution G be a uniform distribution on the interval ( δ δ > i.e. G(x δ x δ x ( δ. Then We have g(x δ 1 δ and consequently the cdf and pdf of a TW-U are given by F TW U (x δ 1 exp( ( x δ f TW U (x δ ( x δ 1 exp( ( x δ ( 1 δ < x < δ. < x < δ ii TW-Weibull distribution(tw-w: Let the parent distribution G be a Weibull distribution i.e. G(x λ γ 1 exp( λx γ forλ γ >. Then we have g(x λ γ λγx γ 1 exp( λx γ and consequently the cdf and pdf of a TW-W are given by F TWW (x λ γ 1 exp( (1 exp( λxγ f TWW (x (1 exp( λxγ 1 exp( (1 exp( λx γ λγx γ 1 exp( λx γ x > x >. 3

4 International Journal of Statistics and Probability Vol. 6 No. 5; 217 iii TW-Logisticl distribution(tw-l: Let the parent distribution G be a logistic distribution i.e. G(x η (1 + exp( ηx 1 for η >. Then we have g(x η η exp( ηx(1 + exp( ηx 2 and consequently the cdf and pdf of a TW-L are given by F TWL (x η 1 exp( (1 + exp ηx x > f TWL (x (1 + exp( ηx1 exp( (1 + exp( ηx η exp( ξx(1 + exp( ηx 2 x >. iv TW-Log Logisticl distribution(tw-ll: Let the parent distribution G be a log logistic distribution i.e. G(x s c 1 (1 + ( x s c 1 with parameters s > and c >. Then we have g(x s c cs c x c 1 (1 + ( x s c 2 and consequently the cdf and pdf of a TW-LL are given by F TWLL (x s c 1 exp( (1 (1 + ( x s c 1 x > f TWLL (x (1 (1 + ( x s c 1 1 exp( (1 (1 + ( x s c 1 cs c x c 1 (1 + ( x s c 2 x >. v TW-Burr XII distribution(tw-bxii: Let the parent distribution G be a Burr XII distribution i.e. G(x s c 1 (1 + ( x s c ; s c >. Then we have g(x s c cs c x c 1 (1 + ( x s c 1 and consequently the cdf and pdf of a TW-BXII are given by F TWBXII (x s c 1 exp( (1 (1 + ( x s c x > f TWBXII (x (1 (1 + ( x s c 1 exp( (1 (1 + ( x s c cs c x c 1 (1 + ( x s c 1 x >. vi TW-Normall distribution(tw-n: Let the parent distribution G be a Normal distribution i.e. G(x µ σ Φ[ (x µ σ ]; µ R σ2 >. Then we have (x µ ϕ[ σ g(x µ σ σ and consequently the cdf and pdf of a TW-N are given by F TWN (x µ σ 1 exp( (Φ[ (x µ σ ] x > f TWN (x (Φ[ (x µ σ ] 1 exp( (Φ[ (x µ σ ] ϕ[ (x µ σ ] σ x >. 4. Mathematical Properties Here we present several mathematical properties of the our new proposed distribution. 4.1 Expansion of the cdf and pdf of TW-G Here we express expansion of cdf (2 by applying the power series expansion for the exponential function as follows F(x 1 [ 1 exp ( ( 1 i i G(x i]. (1 i! Since G(x i ( i ( 1 (1 G(x 4

5 International Journal of Statistics and Probability Vol. 6 No. 5; 217 and where we obtain F(x [1 G(x] ( i ( ( 1 G(x (i(i 1...(i + 1! 1 [ ( 1 i i 1 exp ( i! [ Therefore we can write F(x as ( i ( 1 i i i!(1 exp ( ( ( 1 + G(x ] ( i ( ( 1 +] G(x. F(x b F (x (11 where b ( 1 i+ + i i!(1 exp ( ( i ( and F (x G(x (for > is the exponentiated-g (exp-g cdf with power parameter and F (x 1. By simple differentiation of Eq. (11 we can write f (x b +1 f+1 (x (12 where f +1 (x ( + 1g(xG(x is the exp-g density function with parameter(+1. Now since (12 is a mixture of exp- G densities thus we can derive certain mathematical attributes of the TW-G model from the exp-g distribution existing in the literature for exp-g distributions such as ordinary and incomplete moments and moment generating function (mgf which have been considered by Mudholar Srivastava and Freimer (1995 Gupta and Kundu (1999 Nadaraah and Kots (26 and others. 4.2 Moments The nth moment of X can be obtioned from (12 as E(X n b +1 E(Y n (13 where Y is a random variable with exp-g density function f+1 (x. Nadaraah and Kots (26 obtained ordinary moments of some well-nown exp-g models. such as exp-gamma exp-weibull exp-gumbel and exp-frechet distributions. These can be used to produce E(X n in (13. We can also obtain the ordinary moment E(X n using qf G 1 (u as where E(X n ξ n ( ( + 1b +1 ξ n ( (14 x n G(x g(xdx 1 [ G 1 (u ]n u du. (15 Hence the ordinary moments of TW-G distributions can be obtained from Eqs. (14 and (15. Cordeiro and Nadaraah (211 determined ξ n ( for some well-nown distributions. We also compute here ξ n ( for three cases as examples. 5

6 International Journal of Statistics and Probability Vol. 6 No. 5; 217 Example. 1. For the TW-W distribution by integral equation 1 x 1 log n (xdx ( 1 n n 1 n! > (16 we have ξ n ( 1 λ n γ q l ( ( n γ ( ( 1 +l+q (17 l ( q!(l + 1 q+1 where x ( x(x (x + 1. Example. 2. For the TW-L distribution by we have n [ 1 t n ] 1 e (t+ log x t log(1 x dx t x ( x log( 1 x n ξ n ( 1 n ( η n t n B(t t (18 t where B( 1 x 1 (1 x 1 dx. Example. 3. For the TW-LL distribution we have ξ n ( s n ( n c (!( n (19 c Due to the important role of incomplete moments in measuring indices of income inequality such as Lorenz and Bonferroni curves and the associated Gini index we need to determine the nth incomplete moments as follows. m n (y ( + 1b +1 H n (G(y (2 where H n (x x [ G 1 (u ]n u du. (21 We can compute (21 for most baselina G distributions as shown in the following examples. Example. 4. For the TW-W distribution by (8 we have H n (G(y 1 λ n γ l q ( n γ ( +q [G(y]+q+l+1 ( 1 q + q + l + 1 d ql where d ql l 1 l m [m(q + 1 l] d ql m m+1 and G is Weibull distribution. Example. 5. For the TW-L distribution we have H n (G(y 1 n ( η n t n B G(y (t t t where B x ( x t 1 (1 t 1 dt and G is logistic distribution. Example. 6. For the TW-LL distribution where G is log logistic distribution by (16 we have H n (G(y s n ( n c ( [G(y] + + n c +1!( + + n c + 1. In what follows the probability weighted moments (PWMs of a TW-G ( random variable X is obtained. In particular when the inverse of a distribution does not have a closed form or maximum lielihood estimates are unavailable or difficult to compute PWMs moments can be used to estimate the parameters.they may also be used as starting values 6

7 International Journal of Statistics and Probability Vol. 6 No. 5; 217 for maximum lielihood estimates. The PWMs are expectations of multiplication of two certain functions of a random variable X defined as τ nr E[X n F(X r ] Thus by using (8(11 and (12 we have τ nr x n F(x r f (xdx x n F(x r f (xdx. x n( b t G(x t r b +1 ( + 1g(xG(x dx t b +1 ( + 1d rt ξ n ( + t (22 t where d rt (tb 1 t m1 [m(t + 1 t]b m d rt m and d r b r. Hence we can derive closed-form expressions for (22 by determing ξ n ( + t for different parent distributions. Indeed we have calculated ξ n ( for TW-W TW-L and TW-LL in Examples 1 2 and Moment Generating Function Let M X (t E(e tx be the moment generating function (mgf of X. Then by (12 we have M X (t b +1 E(e ty. (23 Hence we can obtain M X (t from M Y (t. The mgf of X can also be determined by qf G 1 (u as where M X (t ρ(t ( + 1b +1 ρ(t (24 e tx G(x g(xdx 1 e tg 1 (u u du. (25 We can obtain the mgfs of the special cases (i to (vi from Eqs. (24 and (25. By using integral equation 1 x 1 log σ ( 1 x dx σ 1 Γ(σ + 1 > ; σ > 1 (26 and (16 the integral (25 for the TW-W TW-L and TW-LL distributions are given by ρ(t o l ( l t ( 1 l!λ γ (l + 1 γ +1 Γ( γ + 1 and respectively. ρ(t ρ(t l ( t η (!( + + t η Renyi and Shannon Entropies (ts ( c (l!l!(l + + c + 1 An entropy is a measure of uncertainty of a random variable X. It is an important tool in Science and Engineering. There are several different entropy and information indices from which Renyi and Shannon entropies are most popular. The Renyi entropy for a random variable X with pdf f (. is defined as I F (δ 1 1 δ log f (x δ dx (27 7

8 International Journal of Statistics and Probability Vol. 6 No. 5; 217 for δ > and δ 1. Shannon entropy is the expectation of { log[ f (X] } where f is a density function of a random variable X which is a special case of Renyi entropy when δ 1. Now we shall compute the Renyi and Shannon entropies for a TW-G random variable X. Since f (x δ ( 1 e δ e δg(x G(x δ( 1 g(x δ ( 1 e δ g(x δ ( 1 i (δ i (G(x i (G(x δ( 1 i! i ( 1 e δ g(x δ ( 1 i (δ i (G(x i+δ( 1 i! i ( 1 e δ g(x δ ( 1 i (δ i ( i + δ( 1 ( 1 (1 G(x i! i ( 1 e δ g(x δ ( 1 i (δ i ( i + δ( 1 ( ( 1 ( 1 G(x i! i ( 1 e δ g(x δ ( 1 i (δ i ( ( i + δ( 1 ( 1 + G(x i! i [ ( ( 1 ++i (δ i ( ( ] i + δ( 1 1 e δ G(x g(x δ i! according to Eq. (28 we have where and i I F (δ 1 [ ] 1 δ log S D δ ( (29 D δ ( S i g(x δ G(x dx 1 ( ( 1 ++i (δ i ( ( i + δ( 1 1 e δ. i! (28 [g ( G 1 (u ] δ 1 u du (3 Practically it is sufficient to obtain the integral equation (3. Now for the TW-W TW-L and TW-LL distributions the integral equation (3 turns out to be where γ δ 1 δ D δ ( η δ 1 B( + δ δ ( D δ ( λ δ 1 γ (γ 1(δ 1 γ δ 1 Γ + 1 γ p ( 1 p (δ + p (γ 1(δ 1 γ 1 ( p and respectively. Thus for the Shannon entropy we have D δ ( ( c ((c 1(δ 1 s δ 1 B (δ 1(1 + 1 c c + 1 E( log f (X log( + log(1 e + E(G (X ( 1E(log G(X E(log g(x. (31 Based on Eq. (6 we can write E(G (X b +1 ( + 1 g(xg + (xdx b +1 (

9 International Journal of Statistics and Probability Vol. 6 No. 5; 217 and where E(log G(X E(log g(x W 1 b +1 ( + 1 b +1 ( + 1 ( log G(x G (xg(xdx ( log g(x G (xg(xdx b +1 ( + 1 b +1 ( + 1W ( log g ( G 1 (u u du. (32 The closed-form expressions for (32 can be obtained for deriving the Shannon entropy as shown in the following examples. For the TW-W distribution using (6 and integral equation we obtain 1 log log 1 x xµ 1 dx 1 (C + log µ [µ > C ] µ W log(λ 1 γ γ + 1 For the TW-L distribution we have W log η l + γ 1 γ ( 1 l+1 (l Finally for the TW-LL distribution we have 4.5 Order Statistics l ( l ( 1 l+1 l ( W log( c s ( + 1 c 1 c( (c c [ ] ( C + log(l l l ( 1 l+1 (. (l l p ( 1 p+1 (. (p p Order statistics are important tools especially in non-parametric statistics and inference. Here we shall consider order statistics properties of the class of TW-G distributions. Suppose that X 1 X 2... X n is a random sample from the TW-G distribution and X i:n is the corresponding ith order statistic. Then we can express the pdf of X i:n as: n! f i:n (x (i 1!(n i! f (xf(xi 1 (1 F(x (n i n! (i 1!(n i! n! (i 1!(n i! and since by (8 [ ] i+ 1 b G(x n i n i ( 1 ( n i ( 1 ( n i d i+ 1 G(x f (xf(x +i 1 [ ][ ] i+ 1 b r+1 (r + 1g(xG(x r b G(x (33 r where d i+ 1 (b 1 i+ 1 m1 [m(i + 1 ]b m d i+ 1 m and d i+ 1 b we obtain f i:n (x n! (i 1!(n i! n i r n i ( 1 ( n i n! (i 1!(n i! ( 1 [ ][ b r+1 (r + 1g(xG(x r d +i 1 G(x ] r ( n i br+1 (r + 1 r d +i 1h r+ 1 (x m r h r+ 1 (x (34 r 9

10 International Journal of Statistics and Probability Vol. 6 No. 5; 217 where h r+ 1 (x is the exp-g density function with power parameter r + 1 and m r n!b r+1 (r + 1 (i 1!(r n i ( 1!(n i! d +i 1. Moments of order statistics are more popular in reliability and quality control testing. Here we provide two formulas for moments of order statistics of a TW-G distribution. The first one follows from moment of random variable Y r with the exp-g density function h r+ 1 (x as E(X p i:n x p f i:n (xdx We can present a second formula for E(X p i:n based on the quantile function G as E(X p i:n r r m r E(Y p r. (35 m r (r x p g(xg(x r+ dx m r (r + + 1L(r (36 where L(r 1 [ G 1 (u ]p u +r du. By similar calculations as those of ordinary moments the integral L(r for the TW-W TW-L and TW-LL distributions can be expressed as and respectively. L(r 1 λ p γ +r l ( 1 l (l + 1 Γ( p ( + r p γ +1 γ + 1 l L(r 1 p ( η p t p B(t + + r t t L(r 4.6 Unimodal Property l s p ( p c (l l!(l + + r + p c + 1 Loosely speaing the concept of unimodality is usually considered as a distribution with a density f (x which has one pic. Investigation of unimodal distributions originally started by Khinchin (1938. The main interest of unimodality is in optimization theory and mathematical programing. Also this property has become a useful tool in characterization of distributions appearing in different areas of statistic. In this section we consider unimodal property of TW-G( distributions through their log-convex and log-concave properties. These will also enable as to investigate their hazard rate behaviour. Theorem 1 Suppose that g(x is twice differentiable then the pdf of a TW-G( distribution is log-convex if 1 and g(x is non-increasing and log-convex. Proof. Since g(x is twice differentiable we have d 2 log ( f (x dx 2 r (x[ g ( 1 G (x ] ( 1 g2 (x [ 1 + G (x ] + d2 log (g(x. G(x G 2 (x dx 2 Thus f (x is log-convex when 1 and g(x is non-increasing and log-convex. Theorem 2 Suppose that g(x is twice differentiable then pdf of a TW-G( distribution is log-concave when 1 and either G (x 1 and g(x is non-increasing and log-concave or G (x 1 and g(x is non-decreasing and log-concave. Proof. The proof is similar to that of Theorem 1. Now since log concave and log convex densities are unimodal then we have Corollary 1 Suppose that g(x is twice differentiable then TW-G( distributions are unimodal if either 1 and g(x is non-increasing and log-convex or 1 and either G (x 1 and g(x is non-increasing and log-concave or G (x 1 and g(x is non-decreasing and log-concave. 1

11 International Journal of Statistics and Probability Vol. 6 No. 5; 217 Consequently since a log-concave (log-convex pdf has non-decreasing (non-increasing hazard function we have the following. Corollary 2 Suppose that g(x is twice differentiable then TW-G( distributions have non-increasing hazard functions when 1 and g(x is non-increasing and log-convex and have non-decreasing hazard functions when 1 and either G (x 1 and g(x is non-increasing and log-concave or G (x 1 and g(x is non-decreasing and log-concave. 4.7 Stochastic Orders One of the most fundamental measures of comparison of the behavior of random variables is stochastic ordering. Several types of stochastic orderings with different implications and applications can be seen in Shaed and Shanthiumar (27. Here we first recall definitions of the required stochastic orders: the lielihood ratio order ( lr the usual stochastic order ( st the hazard rate order ( hr the reversed hazard rate order ( rh and the expectation order ( E. Then we shall stochastically compare TW-G( and B-G( distributions. For instance we shall reveal that as lifetime distributions B-G distribution has smaller reliability than TW-G distribution. Let U and V be two random variables with pdfs f U and f V respectively. Then we have i U lr V if f V (x/ f U (x is an increasing function in x. ii U st V if P(U x P(V x for all x. iii U hr V if f U (x/p(u x f V (x/p(v x for all x. iv U E V if E(U E(V. Theorem 3 Let U B-G( and V TW-G(. Then U lr ( hr st and E V if (1 G(x 1 (1 G(x 1. Proof. Since it easily follows that f V (x f U (x B( exp( G(x (1 e (1 G(x 1 d ( fv (x B( g(x exp( G(x (1 G(x 2[ G(x 1 (1 G(x + ( 1 ] dx f U (x (1 e (1 G(x 2 2 which is non-negative if [1 G(x 1 (1 G(x] 1. Therefore by (i U lr V if [1 G(x 1 (1 G(x] 1. Now based on the results of Shaed and Shanthiumar (27 that: U lr V U hr V U st V U E V we have the results if [1 G(x 1 (1 G(x] Stress-strength Reliability Function Let X 1 TW G( 1 1 ξ and X 2 TW G( 2 2 ξ be independent random variables with pdf s and cdf s f 1 F 1 and f 2 F 2 respectivly and ξ be the vector of parameters of the parent distribution G. Then the stress-strength reliability function R P(X 1 > X 2 is given by R P(X 1 > X 2 Thus by Eqs. (11 and (12 we have R where b (1 s+1 and b(2 are b (1 s+1 b 2 s+1 s s b (2 b(1 s+1 b (2 b(1 s+1 F X2 (x f X1 (xdx. (37 ( 1 i+ +s+1 1 i i!(1 exp ( 1 ( 1 i+ + 2 i i!(1 exp ( 2 F (x f s+1 (xdx s b (2 b(1 s+1 G(x +s (s + 1g(xdx s s + 1 (38 ( ( i1 ( i2 ( s + 1 Clearly if Eq. (38 reduces to R

12 International Journal of Statistics and Probability Vol. 6 No. 5; 217 (a (b (c (d (e (f Figure 1. Plots of the (a TW-U (b TW-W (c TW-LL (d TW-L (e TW-BXII and (f TW-N densities. 3. Shapes of the pdf and hrf Figures 1 and 2 indicate possible shapes of the pdfs and hrfs respectively of TW-G distributions for its various parameter values. As we observe Figure 1 shows that the TW-G family distributions have various shapes such as symmetrical left and right sewed bathtub and reversed-j. Figure 2 also shows that hazard rate of the new distributions have great flexibility such as decreasing increasing bathtub upside-down bathtub J reversed- J and S shapes. Hence TW-G distributions are useful for fitting to different data sets with various shapes. Infact shapes of the pdfs and hr functions can also be expressed analytically. The critical points of the NW-G density are the roots of the following equation: g(x G(x ( 1 (x G(x 1 g(x + g (39 g(x and critical points of h(x are obtained from the equation: ( 1 g(x G(x G 1 (xg(x + g (x g(x G 1 (xg(xe G(x e G (x e. (4 12

13 International Journal of Statistics and Probability Vol. 6 No. 5; 217 (a (b (c (d (e (f Figure 2. Plots of the (a TW-U (b TW-W (c TW-LL (d TW-L (e TW-BXII and (f TW-N hazard rates. By using most symbolic computation software platforms we can examine Eqs. (39 and (4 to determine the local maxima and minima and inflexion points. 4. Maximum Lielihood Estimation(MLE In this section we discuss estimation of the parameters of TW-G model by the maximum lielihood method. Let x 1... x n be n observations of a random sample from a TW-G distribution with parameters Θ ( ξ T where ξ is a K 1 parameter vector in G(.. The log-lielihood function for parameters Θ ( ξ T is given by l n (Θ n log + n log + log g(x i + ( 1 log G(x i G (x i n log (1 e. (41 13

14 International Journal of Statistics and Probability Vol. 6 No. 5; 217 The first derivatives of l n (Θ with respect to the parameters and ξ are: l n (Θ l n (Θ ξ l n (Θ n + log G(x i ġ(x g(x i + ( 1 n G (x i g(x i G 1 (x i ne 1 e Ġ(x i G(x i G(x 1 Ġ(x i (42 where ġ(x g(x i ξ and Ġ(x i G(x i ξ. Therefore solutions of these equations yeild the MLE of Θ parameters which can be obtioned by a numerical method. For interval estimation and hypothesis testings on the parameters we need the information matrix I(θ E[ 2 l n (Θ ] such that Θ 2 2 I l n (Θ1 I 12 I 13 I Θ 2 12 I 22 I 23 I 13 I 23 I 33 whose elements are I 11 2 l n (Θ 2 n 2 + ne (1 e 2 I 12 2 l n (Θ g(x i G 1 (x i I 22 2 l n (Θ 2 I 13 2 l n (Θ ξ i n g(x 2 i G 1 (x i ( 1 G 1 (x i Ġ(x i g 2 (x i G 2 (x i I 23 2 l n (Θ Ġ(x i ξ G(x i G 1 (x i Ġ(x i ( 1 g(x i G 2 (x i Ġ(x i I 33 2 l n (Θ g(x i g(x i (ġ(x i 2 G(x i G(x i (Ġ(x i 2 + ( 1 ξ 2 g 2 (x i G 2 (x i ( 1 G 2 (x i (Ġ(x i 2 G 1 (x i G(x i where g(x i 2 g(x i ξ 2 and G(x i 2 G(x i ξ 2 for If MLE of Θ ( ξ is ˆΘ ( ˆ ˆ ˆξ then by assuming the regularity conditions ( ˆ ˆ ˆξ ξ converges in distribution to (+2 multivariate normal distribution with zero means and variance-covariance matrix I Application In this section we apply an data set corresponding to remission times of a random sample of 128 bladder cancer patients (cf. Lee & Wang 23 to provide an application of six submodels of the TW-G distribution i.e. TW-U TW-W TW-LL TW-L TW-BXII and TW-N distributions. The data are given below: Table 1 indicates the MLEs of the parameters Aaie Information Criterion (AIC as AIC 2 log L + 2 the Bayesian information criterion (BIC as BIC 2 log L + log(n 14

15 International Journal of Statistics and Probability Vol. 6 No. 5; 217 Table 1. MLEs of the parameters AIC BIC HGIC and K-S (p-value of the TW-U TW-W TW-L TW-LL TW-BXII and TW-N models Model MLEs of parameters AIC BIC HQIC K-S (p-value TW-U ( ˆ ˆ ˆδ ( (.555 TW-W ( ˆ ˆ ˆλ ˆγ( (.994 TW-L ( ˆ ˆ ˆη( (.145 TW-LL ( ˆ ˆ ŝ ĉ( (.999 TW-BXII ( ˆ ˆ ŝ ĉ ˆ( (.998 TW-N ( ˆ ˆ ˆµ ˆσ( (.31 HannanCQuinn information criterion (HQIC as HQIC 2L + 2 log(log(n Kolmogorov CSmirnov (KCS distance between the empirical distribution and the fitted model (and its corresponding P-value in parentheses where n is the number of observations is number of parameters in the model and L is the maximized value of the lielihood function for all six special models of the TW-G distribution. Analysis of Table 1 indicates that TW-LL distribution has the lowest AIC BIC HQIC and K-S values and so it is the best model to fit to this data. We shall also arrive at the same results by comparing p-values for all six models. Table 2 illustrate AIC BIC and HQIC for the three submodels of B-G distributions. Comparing Tables 1 and 2 indicate that TW-W and TW-BXII distributions provide better fits in comparison with B-W and B-BXII distributions. The histogram of the data and fitted TW-U TW-W TW-L TW-BXII TW-LL and TW-N distributions are plotted in Figure 3. As we observe the TW-U TW-W TW-LL and TW-BXII models yield better fits among other models. The empirical and fitted survival functions of TW-LL distribution are shown in Figure 4. Table 2. MLEs of the parameters AIC BIC HGIC of the B-W B-LL B-BXII models Model MLEs of parameters AIC BIC HQIC B-W ( ˆ ˆ ˆλ ˆγ( B-LL ( ˆ ˆ ŝ ĉ( B-BXII ( ˆ ˆ ŝ ĉ ˆ( Figure 3. The histogram of remission times of bladder cancer data and the fitted pdfs of TW-U TW-W TW-L TW-BXII TW-LL and TW-N. 15

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