Hassan S Bakouch

Farrukh Jamal

Christophe Chesneau

Arslan Nasir

A new transmuted family of distributions: Properties and estimation with applications

Keywords: Transmuted distribution, Generated family, T-X transformation, Quantile function, Rényi and Mathai-Hanboldand entropy, Estimation. 2000 Mathematics Subject Classification:-60E05, 62E15, 62F10

In this paper, a new transmuted general family of distributions is introduced and studied. In particular, we obtain explicit expressions for the moments, the incomplete moments, the moment generating function, entropies measures and order statistics. Further, we introduce a bivariate extension of the new family. Some sub-models of the family are introduced, involving Burr, Gompertz, Weibull and gamma distributions. We discuss the different methods of estimation of the model parameters and a simulation study is investigated for one of the special models to check the asymptotic behavior of the estimates under all estimation methods. Further, the potentiality of the family is illustrated by fitting two real data sets to the mentioned sub-models.

Introduction

Literature of lifetime distributions is rich with various continuous univariate distributions and still growing rapidly. Several extensions of some well-known lifetime distributions have been developed during the last two decades for modeling and analysis many types of real life data that having different random nature. This development is followed by many approaches for generating new families of distributions because that the family contains many distributions and hence increase the chance of modeling a large number of real data. Among these families, there are the Exponential-G by [START_REF] Gupta | Modeling failure time data by Lehmann alternatives[END_REF], the Marshal-Olkin-G (MO-G) by [START_REF] Marshall | A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families[END_REF], the transmuted distributions by [START_REF] Shaw | The Alchemy of Probability Distributions: beyond Gram-Charlier Expansions, and a Skew-kurtotic-normal Distribution from a Rank Transmutation Map[END_REF], the beta-G by [START_REF] Eugene | Beta-normal distribution and its applications[END_REF], the Kumaraswamy-G (Kw-G) by [START_REF] Cordeiro | A new family of generalized distributions[END_REF], the McDonald-G (Mc-G) by [START_REF] Alexander | Generalized beta-generated distributions[END_REF], the gamma-G (type 1) by [START_REF] Zografos | On families of beta and generalized gamma-generated distributions and associated inference[END_REF], the gamma-G (type 2) by [START_REF] Ristić | The gamma-exponentiated exponential distribution[END_REF], the log-gamma-G by [START_REF] Amini | Log-gamma-generated families of distributions[END_REF], the logistic-G by [START_REF] Tahir | The Logistic-X family of distributions and its applications[END_REF], the exponentiated generalized-G by [START_REF] Cordeiro | The exponentiated generalized class of distributions[END_REF], the Transformed-Transformer (T-X) by [START_REF] Alzaatreh | A new method for generating families of continuous distributions[END_REF], the exponentiated (T-X) by [START_REF] Alzaghal | Exponentiated T-X family of distributions with some applications[END_REF], the Kumaraswamy odd log-logistic-G (KwOLL-G) by [START_REF] Alizadeh | Kumaraswamy odd log-logistic family of distributions: Properties and applications[END_REF], the generalized transmuted-G by [START_REF] Nofal | The generalized transmuted-G family of distributions[END_REF], the beta transmuted-H families by [START_REF] Afify | The beta transmuted-H family of distributions: properties and applications[END_REF] and the cubic rank transmuted distributions by [START_REF] Granzotto | Cubic rank transmuted distributions: inferential issues and applications[END_REF].

Genesis of the family

Here, we introduce a new family of distribution called the transmuted general (T-G) family of distributions. In order to motivate its interest, let us first present a simple case of the well-know T-X family introduced by [START_REF] Alzaatreh | A new method for generating families of continuous distributions[END_REF]. For a general baseline cumulative distribution function (cdf) denoted by G(x), a new cdf is given by

F (x) = G(x) 0 p(t)dt, x ∈ R, (1.1)
where p(t) denotes a probability density function (pdf) with support on [0, 1]. As noticed in [START_REF] Alizadeh | Generalized transmuted family of distributions: properties and applications[END_REF], the famous transmuted distributions family with cdf given by F (x) = (1+λ)G(x)-λG 2 (x), λ ∈ [-1, 1], [START_REF] Shaw | The Alchemy of Probability Distributions: beyond Gram-Charlier Expansions, and a Skew-kurtotic-normal Distribution from a Rank Transmutation Map[END_REF], corresponds to the cdf given by (1.1) defined with the pdf: p(t) = 1 + λ -2λt. As a new remark, one observe that p(t) can be expressed as generalized mixture of two pdfs with support on [0, 1]:

p(t) = (1 -λ)p 1 (t) + λp 2 (t),
where p 1 (t) = 1 and p 2 (t) = 2(1 -t). This fact motivates the consideration of a new family of distributions derived to (1.1) and generalized mixture of two pdfs; for any pdfs p 1 (t) and p 2 (t) with support on [0, 1] and λ such that inf

t∈[0,1]
[(1 -λ)p 1 (t) + λp 2 (t)] > 0, the following function F (x) is a cdf:

F (x) = G(x) 0 [(1 -λ)p 1 (t) + λp 2 (t)] dt, x ∈ R.
A new family of distributions arises by keeping p 1 (t) = 1 for the sake of simplicity but choosing a new pdf for p 2 (t), with different nature. In this paper, we focus our attention on the decreasing convex pdf given by p 2 (t) = 2 (1+t) 2 . The corresponding cdf is given by

F (x) = G(x) 0 (1 -λ) + λ 2 (1 + t) 2 dt = 1 -λ + 2λ 1 + G(x) G(x), λ ∈ [-1, 1], x ∈ R. (1.2)
An equivalent expression of (1.2) is given by

F (x) = G(x) + λ G(x) [1 -G(x)] 1 + G(x) , x ∈ R.
To the best of our knowledge, this family of distributions is new in the literature.

Aim and organization of the paper

The aim of this paper is to explore the mathematical and practical aspects of the T-G family, and to show its importance and its flexibility using applications to real data sets. In particular, we explicit expressions for the moments, incomplete moments, generating function, mean deviation and residual and reversed residual life moments. Stochastic ordering and reliability parameter are studied. General results for order statistics are derived. Entropies measures (Rényi entropy and Mathai-Hanboldand entropy) are determined.Some sub-models of the family are introduced, involving Burr, Gompertz, Weibull and gamma distributions. A bivariate extension of the new family is also introduced. We provide simulation results of different estimation methods (maximum likelihood, least square, minimum spacing, weighted least square and Cramer Von Mises) for one of the special models to check the asymptotic behavior of the estimates under those methods. Further, applications on two different real data sets are given, with fair comparisons with recent flexible distribution. This paper is unfolded as follows. Section 2 is devoted to the mathematical properties of the T-G family. Special models are introduced in Section 3. An extension to the bivariate case is explored in Section 4. Section 5 presents estimation methods for the parameters of the T-G family. Simulation results and applications to real data sets are given in Section 6.

Mathematical properties

We say that a random variable follows the T-G distribution if it has the cdf given by (1.2). Hereafter, we consider a random variable X following the T-G distribution.

Shape of the pdf

Let g(x) be the baseline pdf, the pdf of X is given by

f (x) = 1 -λ + 2λ (1 + G(x)) 2 g(x), λ ∈ [-1, 1], x ∈ R. (2.1)
An equivalent expression is given by

f (x) = {(1 + λ) + (1 -λ)G(x) [2 + G(x)]} g(x) (1 + G(x)) 2 , λ ∈ [-1, 1], x ∈ R.
The critical points of f (x) are the roots of the equation:

g (x) g(x) -2 g(x) 1 + G(x) + 2 g(x)(λ -1) [1 + G(x)] 1 + λ -(1 -λ)G(x)[2 + G(x)] = 0.

Expansions for the pdf and the cdf

By using generalized binomial expansion we can show that the pdf (2.1) of X has the expansion

f (x) = ∞ j=0 a j h j (x), (2.2)
where a 0 = 1 + λ, a j = 2λ(-1) j for j ≥ 1 and h j (x) = (j + 1)G j (x) g(x) for j ≥ 0. The cdf of X can be expressed as

F (x) = ∞ j=0 a j H j+1 (x), (2.3)
where H j+1 (x) = G j+1 (x) for j ≥ 0. Note that h j (x) is the pdf of the exp-G distribution with power parameter j + 1, and H j+1 (x) the associated cdf. Therefore (2.2) (resp. (2.3)) reveals that the pdf (resp. cdf) of the T-G distribution is a linear combination of pdfs (resp. cdfs) of the exp-G distributions. Hence some properties of the exp-G distribution can be used to determine some mathematical properties of the T-G distribution. Technical details on the the exp-G distribution can be found in [START_REF] Gupta | Modeling failure time data by Lehmann alternatives[END_REF].

Quantile function

For λ ∈ [-1, 1), the quantile function of X is given by

F -1 (x) = G -1 1 -x + λ -x 2 + x(2 -6λ) + (1 + λ) 2 2(λ -1) , x ∈ (0, 1). ( 2

.4)

For λ = 1, we have

F -1 (x) = G -1 x 2 -x , x ∈ (0, 1). (2.5)
Thus, for any random variable U following the uniform U (0, 1) distribution, the random variable X given by X = F -1 (U ) follows the T-G distribution.

Hazard rate function

The hazard rate function (hrf) of X is given by

h(x) = f (x) 1 -F (x) = {(1 + λ) + (1 -λ)G(x) [2 + G(x)]} g(x) [1 + G(x)] [1 -λ G(x) -(1 -λ) G 2 (x)] , λ ∈ [-1, 1], x ∈ R.
The critical points of h(x) are obtained from the equation:

g (x) g(x) - g(x) 1 + G(x) + [λ -2(λ -1)G(x)] g(x) 1 -λ G(x) + (λ -1)G 2 (x) + 2 (1 -λ) [1 + G(x)] g(x) 1 + λ -(1 -λ)G(x)[2 + G(x)]
= 0.

Moments, moment generating function and mean deviations

Using the expansion (2.2), the r th moment of X is given by

E(X r ) = ∞ -∞ x r f (x) dx = ∞ j=0 a j ∞ -∞
x r h j (x) dx.

(2.6)

Similarly, denoting by 1 A the indicator random variable on A, the r th incomplete moment of X can be obtained as

µ r (x) = E(X r 1 {X≤x} ) = ∞ j=0 a j T j,r (x), (2.7)
where T j,r (x) =

x -∞ t r h j (t) dt. The moment generating function of X is given by

M (t) = E(e t X ) = ∞ j=0 a j M j (t), where M j (t) = ∞ 0 e t x h j (x) dx.
The mean deviations of X about the mean and median, respectively, can be put as

D µ = E(|X -µ|) = 2µ F (µ) -2µ 1 (µ) (2.8)
and

D M = E(|X -M |) = µ -2µ 1 (M ), (2.9)
where µ = E(X) is given by (2.6), F (µ) is easily calculated from (1.2) and µ 1 (M ) is obtained from (2.7) with M = F -1 ( 1 2 ), using (2.4) or (2.5).

Residual and reversed residual life moments

Using the binomial expansion, the r th residual life moments of X can be obtained as

M r (t) = E ((X -t) r | X > t) = r j=0 r j (-1) r-j t r-j E X j | X > t = 1 1 -F (t) r j=0 r j (-1) r-j t r-j E(X j ) -µ j (t) ,
where E(X j ) is given by (2.6) and µ j (t) is obtained from (2.7).

Similarly, the r th reversed residual life moments of X can be obtained as

m r (t) = E ((t -X) r | X ≤ t) = r j=0 r j (-1) j t r-j E X j | X ≤ t = 1 F (t) r j=0 r j
(-1) j t r-j µ j (t).

Stochastic ordering

This subsection is devoted to the stochastic ordering, which is a common concept to show the ordering mechanism in life time distributions. A random variable X is said to be stochastically smaller than a random variable Y in the

• stochastic order (X ≤ st Y ) if the associated cdfs satisfy: F X (x) ≥ F Y (x) for all x.
• hazard rate order (X ≤ hr Y ) if the associated hrfs satisfy: h X (x) ≥ h Y (x) for all x.

• likelihood ratio order (X ≤ lr Y ) if the ratio of the associated pdfs given by f X (x) f Y (x) decreases in x. When the supports of X and Y have a common finite left end-point, the following chain of implications holds:

X ≤ lr Y ⇒ X ≤ hr Y ⇒ X ≤ st Y.
Further details can be found in [START_REF] Shaked | Stochastic orders and their applications[END_REF]. Let X be a random variable following the T-G distribution with parameter λ 1 with pdf f X (x) and Y be a random variable following the T-G distribution with parameter λ 2 with pdf f Y (x). Let us prove that X ≤ lr Y under some assumptions on λ 1 and λ 2 . We have

f X (x) = 1 -λ 1 + 2λ 1 (1 + G(x)) 2 g(x) = [1 + λ 1 + (1 -λ 1 )[2 + G(x)]G(x)] (1 + G(x)) 2 g(x)
and

f Y (x) = [1 + λ 2 + (1 -λ 2 )[2 + G(x)]G(x)] (1 + G(x)) 2 g(x). Hence f X (x) f Y (x) = 1 + λ 1 + (1 -λ 1 ) z 1 + λ 2 + (1 -λ 2 ) z , where z = [2 + G(x)]G(x).
Taking derivative with respect to x, we obtain

f X (x) f Y (x) = 2(λ 2 -λ 1 ) z (1 + λ 2 + (1 -λ 2 ) z) 2 ,
where

z = 2 [1 + G(x)] g(x). Since z ≥ 0 and (1 + λ 2 + (1 -λ 2 ) z) 2 > 0, if λ 2 < λ 1 , we have f X (x) f Y (x) < 0, implying that f X (x) f Y (x) decreases in x, so X ≤ lr Y .

Stress-strength reliability

In reliability theory, a common situation is that the life of a component has a random strength subjected to a random stress. The random strength can be modeled by a random variable X and the random stress can be modeled by a random variable Y . The probability that the component functions satisfactorily is given by R = P (Y < X), which is a well-known measure of component reliability with many applications. Let X be a random variable following the T-G distribution with parameter λ 1 with pdf f X (x) and Y be a random variable following the T-G distribution with parameter λ 2 with cdf F Y (x). Owing to the expansions (2.2) and (2.3), we have

R = P (Y < X) = ∞ -∞ P (Y < X | {X = x}) f X (x) dx = ∞ -∞ f X (x) F Y (x) dx = ∞ j=0 ∞ i=0 a (λ1) j a (λ2) i ∞ -∞ h j (x) H i+1 (x) dx,
where a

(λ) j is defined by a (λ) 0 = 1 + λ and a (λ) j = 2λ(-1) j for j ≥ 1 and λ ∈ {λ 1 , λ 2 }.

Order statistics

Order statistics make their appearance in many areas of statistical theory and practice. Let X 1 , X 2 , . . . , X n is a random sample of size n of X and X 1:n ≤ X 2:n ≤ . . . ≤ X n:n denote the corresponding order statistics obtained from the sample. The pdf of X i:n is given by

f i:n (x) = n! (i -1)!(n -i)! F i-1 (x) (1 -F (x)) n-i f (x) = n! (i -1)!(n -i)! n-i j=0 n -i j (-1) j F j+i-1 (x) f (x).
Using finite and generalized binomial series expansions, we have

F j+i-1 (x) = 1 -λ + 2λ 1 + G(x) j+i-1 G j+i-1 (x) = G j+i-1 (x) j+i-1 k=0 j + i -1 k (1 -λ) j+i-1-k (2λ) k [1 + G(x)] -k = j+i-1 k=0 ∞ =0 b i,j,k, H j+i+ -1 (x),
where

b i,j,k, = j + i -1 k -k (1 -λ) j+i-1-k (2λ) k .
Owing to the expansion (2.2), we can write

F j+i-1 (x) f (x) = j+i-1 k=0 ∞ =0 ∞ m=0 a m b i,j,k, H j+i+ -1 (x) h m (x), therefore f i:n (x) = n-i j=0 j+i-1 k=0 ∞ =0 ∞ m=0 c j,k, ,m H j+i+ -1 (x) h m (x),
where

c j,k, ,m = n! (i -1)!(n -i)! n -i j (-1) j a m b i,j,k, .
The r th moment of X i:n is given by

E(X r i:n ) = ∞ -∞ x r f i:n (x) dx = n-i j=0 j+i-1 k=0 ∞ =0 ∞ m=0 c j,k, ,m ∞ -∞ x r H j+i+ -1 (x) h m (x) dx.
Note that the integral term only depends on the g(x) and G(x) the pdf and cdf of base line distribution.

Rényi entropy

The Rényi entropy of X is given by

I δ = 1 1 -δ E f δ-1 (X) = 1 1 -δ ∞ -∞ f δ (x) dx (2.10)
for δ > 0 and δ = 1. Using finite and generalized binomial series expansions (noticing that λ

2 (1+G(x)) 2 -1 < 1), we have f δ (x) = 1 + λ 2 (1 + G(x)) 2 -1 δ g δ (x) = g δ (x) ∞ k=0 δ k λ k 2 (1 + G(x)) 2 -1 k = g δ (x) ∞ k=0 k =0 δ k k λ k (-1) k-2 [1 + G(x)] -2 = g δ (x) ∞ k=0 k =0 ∞ m=0 δ k k -2 m λ k (-1) k-2 H m (x).
(2.11)

Hence we have

I δ = 1 1 -δ ∞ k=0 k =0 ∞ m=0 δ k k -2 m λ k (-1) k-2 ∞ -∞ H m (x)g δ (x)dx.
Note that the integral term only depends on the g(x) and G(x) the pdf and cdf of base line distribution. Finally, let us mention that the Shannon entropy of a random variable X is defined by E (-log[f (X)]) is the special case of the Rényi entropy when δ → 1.

Mathai-Haubold entropy

Classical Shannon entropy has been generalized in many directions. One of them is the δ -generalized entropy introduced by [START_REF] Mathai | On a generalized entropy measure leading to the pathway model with a preliminary application to solar neutrino data[END_REF]. It is defined by

f M H (x) = 1 δ -1 f 2-δ (x) dx -1 .
Similar arguments to (2.11) gives

f 2-δ (x) = g 2-δ (x) ∞ k=0 k =0 ∞ m=0 2 -δ k k -2 m λ k (-1) k-2 H m (x).
Therefore

f M H (x) = 1 δ -1 ∞ k=0 k =0 ∞ m=0 2 -δ k k -2 m λ k (-1) k-2 ∞ -∞ H m (x)g 2-δ (x)dx -1 .
The solution of above integral depends on any arbitrary base line distribution.

Special models

Let us now introduce some special T-G distributions based on some well-known distributions.

Transmuted Burr (TB) distribution

If Burr XII distribution has pdf and cdf given, respectively, by g

(x) = c k x c-1 (1 + x c ) -k-1 , k, c, x > 0, and G(x) = 1 -(1 + x c ) -k
, then the cdf and pdf of T-B distribution are given by

F (x) = 1 -(1 + x c ) -k + λ 1 -(1 + x c ) -k [1 + x c ] -k 2 -(1 + x c ) -k , x > 0 and f (x) = c k x c-1 (1 + x c ) -k-1 + λ c k x c-1 (1 + x c ) -k-1 1 -1 -(1 + x c ) -k 3 -[1 + x c ] -k 2 -[1 + x c ] -k 2 , x > 0.
Using these functions, we immediatly obtain the expression of the hrf. Figure 1 shows some plots of pdfs and hrfs for TB distribution with arbitrary choices for the parameters (c, k, λ).

Transmuted Gompterz (TGz) distribution

If Gompterz distribution has pdf and cdf given, respectively, by g(x) = a exp b x -a b (e x -1) , a, b, x > 0, and G(x) = 1 -exp -a b e b x -1 , then the cdf and pdf of T-Gz distribution are given by

F (x) = 1 -exp - a b e b x -1 1 + λ exp -a b e b x -1 2 -exp -a b (e b x -1)
, x > 0 and

f (x) = a exp b x - a b (e x -1) 1 + λ 1 -1 -exp -a b e b x -1 3 -exp -a b e b x -1 2 -exp -a b (e b x -1) 2 , x > 0.
The expression of the hrf follows immediatly. Figure 2 shows some plots of pdfs and hrfs for TGz distribution with arbitrary choices for the parameters (a, b, λ).

Transmuted Weibull (TW) distribution

If Weibull distribution has pdf and cdf given, respectively, by g(x) = α β x β-1 e -α x β , α, β, x > 0, and G(x) = 1 -e -α x β , then the cdf and pdf of T-W distribution are given by

F (x) = 1 -e -α x β + λ 1 -e -α x β
2 -e -α x β e -α x β , x > 0 and

f (x) = α β x β-1 e -α x β + λ α β x β-1 e -α x β 2 -e -α x β 2 1 -1 -e -α x β 3 -e -α x β , x > 0.
The expression of the hrf follows immediatly. Figure 3 presents some plots of pdfs and hrfs for TW distribution with arbitrary choices for the parameters (α, β, λ).

Transmuted gamma (TGa) distribution

If Gamma distribution has pdf and cdf given, respectively, by g(x) = b a Γ(a) x a-1 e -b x , a, b, x > 0, and

G(x) = γ(a,b,x) Γ(a)
, where γ(a, b, x) = b a x 0 t a-1 e -b t dt, then the cdf and pdf of T-Ga distribution are given by We easily obtain the expression of the hrf. Figure 4 presents some plots of pdfs and hrfs for TGa distribution with arbitrary choices for the parameters (a, b, λ).

F (x) = γ(a, b, x) Γ(a)   1 + λ 1 -γ(a,b,x) Γ(a) 1 + γ(a,b,x) Γ(a)   , x > 0 and f (x) = b a Γ(a) x a-1 e -b x   1 + λ 1 -γ(a,b,x) Γ(a) 2 -γ(a,b,x) Γ(a) 1 + γ(a,b,x) Γ(a) 2    , x > 0.

Bivariate extension

We now propose a bivariate version of the T-G family of distributions. Let (X, Y ) be a bivariate random variable with the joint cdf given by

F X,Y (x, y) = 1 -λ + 2λ 1 + G(x, y) G(x, y), λ ∈ [-1, 1], (x, y) ∈ R 2 ,
where G(x, y) denoted a bivariate cdf. Let G 1 (x) and G 2 (y) be the corresponding marginal cdfs, g 1 (x) and g 2 (y) be the corresponding marginal pdfs and g(x, y) be the corresponding pdf. Then the marginal cdfs of (X, Y ) are given by

F X (x) = 1 -λ + 2λ 1 + G 1 (x) G 1 (x)
and

F Y (y) = 1 -λ + 2λ 1 + G 2 (y) G 2 (y).
The marginal pdfs of (X, Y ) are given by

f X (x) = 1 -λ + 2λ (1 + G 1 (x)) 2 g 1 (x) and f Y (y) = 1 -λ + 2λ (1 + G (y)) 2 g 2 (y).
The pdf of (X, Y ) is given by

f (x, y) = 1 -λ + 2λ (1 + G(x, y)) 2 θ(x, y), where θ(x, y) = g(x, y) - 4λ (1 + G(x, y))[(1 -λ)(1 + G(x, y)) 2 + 2λ] ∂G(x, y) ∂x ∂G(x, y) ∂y .
The conditional cdfs are given by

F X|Y (x | y) = (1 -λ)(1 + G(x, y)) + 2λ (1 -λ)(1 + G 2 (y)) + 2λ × G(x, y)(1 + G 2 (y)) (1 + G(x, y))G 2 (y)
and

F Y |X (y | x) = (1 -λ)(1 + G(x, y)) + 2λ (1 -λ)(1 + G 1 (x)) + 2λ × G(x, y)(1 + G 1 (x)) (1 + G(x, y))G 1 (x) .
The conditional pdfs are given by

f X|Y (x | y) = (1 -λ)(1 + G(x, y)) 2 + 2λ (1 -λ)(1 + G 2 (y)) 2 + 2λ × θ(x, y)(1 + G 2 (y)) 2 (1 + G(x, y)) 2 g 2 (y)
and

f Y |X (y | x) = (1 -λ)(1 + G(x, y)) 2 + 2λ (1 -λ)(1 + G 1 (x)) 2 + 2λ × θ(x, y)(1 + G 1 (x)) 2 (1 + G(x, y)) 2 g 1 (x) .

Different estimation methods

In this section, we will give the estimates of the T-G family by five different methods of estimations , i.e., Maximum Likelihood method, Least Square method, Weighted Least Square method, Maximum Product Spacing method and Cramer Von Mises method.

Maximum likelihood method (MLE)

Here, we determine the maximum likelihood estimates (MLEs) of the model parameters of T-G from complete samples only. Let x 1 , x 2 , . . . x n be observed values from the T-G distribution with parameters λ and ξ. Let Θ = (λ, ξ) T be the parameter vector. The total log-likelihood function for Θ is given by

(Θ) = n i=1 log[g(x i )] -2 n i=1 log[1 -G(x i )] + n i=1 log [(1 + λ) + (1 -λ) G(x i )] .
Maximizing (Θ) with respect to λ and ξ, we have following system of non linear equations:

∂ (Θ) ∂λ = n i=1 1 -G(x i ) {2 + G(x i )} (1 + λ) + (1 -λ)G(x i ) {2 + G(x i )} = 0, ∂ (Θ) ∂ξ = n i=1 g (x i ) ξ g(x i ) -2 n i=1 G (x i ) ξ 1 + G(x i ) + n i=1 2 (1 -λ) G (x i ) ξ {1 + G(x i )} (1 + λ) + (1 -λ)G(x i ) {2 + G(x i )} = 0, where g (x i ) ξ = ∂ ∂ξ g(x i ) and G (x i ) ξ = ∂ ∂ξ G(x i ).
This system of non-linear equations can be solved numerically by any software to obtained the estimates λMLE and ξMLE .

Least square method (LSE)

Let x 1 , x 2 , . . . , x n be the observed values from the T-G distribution with parameters λ and ξ, in increasing order. By considering the associated order statistocs X 1:n , X 2:n , . . . , X n:n , note that E [F (X i:n )] = i n+1 . Least square estimates can be obtained by minimizing the following expression

S(Θ) = n i=1 [F (x i ) -E [F (X i:n )]] 2 = n i=1 F (x i ) - i n + 1 2 .
Minimizing S(Θ) with respect to λ and ξ, we have following system of non linear equations:

∂S(Θ) ∂λ = 2 n i=1 F (x i ) - i n + 1 F (x i ) λ = 0, ∂S(Θ) ∂ξ = 2 n i=1 F (x i ) - i n + 1 F (x i ) ξ = 0, where F (x i ) λ = ∂ ∂λ F (x i ) and F (x i ) ξ = ∂ ∂ξ F (x i ).
This system of non-linear equations can be solved numerically by any software to obtained the estimates λLSE and ξLSE .

Minimum spacing method (MPS)

This method is based on an idea that the differences (spacings) of the consecutive points should be identically distributed. Let x 1 , x 2 , . . . , x n be the observed values from the T-G distribution with parameters λ and ξ, in increasing order. The geometric mean of the differences is given as

GM = n+1 n+1 i=1 D i ,
where

D i = F (x i ) -F (x i-1 ) with F (x 0 ) = 0 and F (x n+1 ) = 1. We have log(GM ) = 1 n + 1 n+1 i=1 log [F (x i ) -F (x i-1 )] .
Using the cdf in (1.2), we have

log(GM ) = 1 n + 1 n+1 i=1 log 1 -λ + 2λ 1 + G(x i ) G(x i ) -1 -λ + 2λ 1 + G(x i-1 )
G(x i-1 ) .

Minimizing log(GM ) with respect to λ and ξ, we have following system of non linear equations:

∂ log(GM ) ∂λ = 1 n + 1 n+1 i=1   -1 + 2 1+G(xi) G(x i ) --1 + 2 1+G(xi-1) G(x i-1 ) 1 -λ + 2λ 1+G(xi) G(x i ) -1 -λ + 2λ 1+G(xi-1) G(x i-1 )   = 0, ∂ log(GM ) ∂ξ = 1 n + 1 n+1 i=1   1 -λ + 2λ 1+G(xi) G (x i ) ξ + 2λ [1+G(xi)] 2 G(x i ) 1 -λ + 2λ 1+G(xi) G(x i ) -1 -λ + 2λ 1+G(xi-1) G(x i-1 ) - 1 -λ + 2λ 1+G(xi-1) G (x i-1 ) ξ + 2λ [1+G(xi-1)] 2 G(x i-1 ) 1 -λ + 2λ 1+G(xi) G(x i ) -1 -λ + 2λ 1+G(xi-1) G(x i-1 )   = 0.
This system of non-linear equations can be solved numerically by any software to obtained the estimates λMP E and ξMP E .

Weighted least square (WLS)

Let x 1 , x 2 , . . . , x n be the observed values from the T-G distribution with parameters λ and ξ, in increasing order. The likelihood function for weighted least square estimates is given by

W (Θ) = n i=1 (n + 1) 2 (n + 2) i(n -i + 1) F (x i ) - i n + 1 2 .
Minimizing W (Θ) with respect to λ and ξ, we have following system of non linear equations:

∂W (Θ) ∂λ = 2 n i=1 (n + 1) 2 (n + 2) i(n -i + 1) F (x i ) - i n + 1 F (x i ) λ = 0, ∂W (Θ) ∂ξ = 2 n i=1 (n + 1) 2 (n + 2) i(n -i + 1) F (x i ) - i n + 1 F (x i ) ξ = 0.
These equations can be solved numerically to obtained the estimates λwls and ξwls .

Cramer Von Mises (CVM)

Cramer von Mises is a type of minimum distance estimators. Let x 1 , x 2 , . . . , x n be the observed values from the T-G distribution with parameters λ and ξ, in increasing order. The likelihood function for Cramer Von Mises estimates is given by

C(Θ) = 1 12 n + n i=1 F (x i ) - 2 i -1 2 n 2 .
Minimizing C(Θ) with respect to λ and ξ, we have following system of non linear equations:

∂C(Θ) ∂λ = 2 n i=1 F (x i ) - 2 i -1 2 n F (x i ) λ = 0, ∂C(Θ) ∂ξ = 2 n i=1 F (x i ) - 2 i -1 2 n F (x i ) ξ = 0.
This system of non-linear equations can be solved numerically to obtained the estimates λCME and ξCME .

6 Simulation results and applications to real data

In this section, we present some experimental results to compare the performance of the different estimators proposed in the previous section. Then we use two real data sets set to compare TB, TW, TGa and TGz to TGE(transmuted generalized exponential distribution), TLE (Transmuted linear exponential distribution), GT-W (generalized transmuted Weibull), Burr XII, Weibull, Gamma and Gompertz distributions.

Simulation results

We perform extensive Monte Carlo simulations to compare the performance of the different estimators (CVM, LSE, WLS, MPS and MLE), mainly with respect to their biases and mean-squared errors (MSEs) for different sample sizes and for different parameter values. For obtaining different estimators, the number of replications is 10000 in all the simulations. We have considered different sample sizes n = 10, 50, 100 and 250. For obtaining frequentist estimators, we consider TGz with a = 10, b = 2.5 λ = 0.5, because it was the superior among the considered sub-models of the T-G family as it will be shown in next subsection. The simulation results are provided in Table 1. The results indicate that the maximum likelihood method performs quite well in estimating the model parameters of the proposed distribution.

Empirical reliability data examples

In this subsection, we evaluate the performance of the T-G family of distributions by fitting four sub-models of this family, namely TB, TW, TGa and TGz two reliability data sets. The data sets are described as follows.

The first data set (data set 1) refer to the 50 observations with hole and sheet thickness are 12 mm and 3.15 mm reported by [START_REF] Dasgupta | On the distribution of Burr with applications[END_REF]. The data are: 0.04, 0.02, 0.06, 0.12, 0.14, 0.08, 0.22, 0.12, 0.08, 0.26, 0.24, 0.04, 0.14, 0.16, 0.08, 0.26, 0.32, 0.28, 0.14, 0.16, 0.24, 0.22, 0.12, 0.18, 0.24, 0.32, 0.16, 0.14, 0.08, 0.16, 0.24, 0.16, 0.32, 0.18, 0.24, 0.22, 0.16, 0.12, 0.24, 0.06, 0.02, 0.18, 0.22, 0.14, 0.06, 0.04, 0.14, 0.26, 0.18, 0.16.

The second data set (data set 2) relating to the strengths of 1.5 cm glass fibers given by [START_REF] Oguntunde | A New Generalization of the Lomax Distribution with Increasing, Decreasing and Constant Failure Rate[END_REF]. This set was obtained by workers at the UK National Physical Laboratory. The descriptive statistics of the two data sets are given in Table 3. From this table, both data sets are under-dispersed. The data set 1 is approximately symmetric and the data set is moderately skewed. Moreover, the first data set is platykurtic while the second one is leptokurtic.

For all the compared distributions, the maximum likelihood method is used to estimate the parameters and also their standard errors are obtained. The model adequacy measures: Anderson-Darling (A*), Cramer-von Mises (W*) and Kolmogorov-Smirnov (K-S) statistic with its p-value are used to compare these distributions, where the smaller values of these statistics and larger p-value give the best fit to the data. The obtained results are given in Tables 2 and7, and we conclude that the considered sub-models of the family are good competitor to the compared distributions. Also, it is found that the smallest values of A*, W*, K-S and largest values of p-value are obtained for the TGz distribution, and by this the TGz distribution has a good performance among the compared distributions. Also, this conclusion is asserted again by Figures 5 and6. Consequently, the variance-covariance matrices of the MLEs of the TGz distribution for both data sets are obtained, noting that the diagonal entries of those matrices are the variances of the MLEs of the TGz parameters for each data set, while the other entries can led to positive and negative correlations for pairs of the estimates. Also, 95% and 99% confidence intervals of the TGz parameters are given in Tables 5 and6 for both data sets. Moreover, some measures of the TGz distribution for both data sets are obtained by using its theoretical properties and are shown in Table 4. Making use of Tables 3 and4, we can note the closeness between measures of the TGz distribution and the corresponding descriptive statistics of both data sets. The variance-covariance matrices of the MLEs of the TGz distribution for data set 1 is given as

= 5 k 3 Figure 1 :

 531 Figure 1: Plots of pdf and hrf for TB distribution with different parameter values.

1 Figure 2 :

 12 Figure 2: Plots of pdf and hrf for TGz distribution with different parameter values.

1 Figure 3 :

 13 Figure 3: Plots of pdf and hrf for TW distribution with different parameter values.

  1 b = 0.5 λ = -0.8 a = 8 b = 6 λ = -0.5 a = 5 b = 4.5 λ = -0.1 a = 1.2 b = 1.5 λ = 01 b = 0.5 λ = -0.9 a = 1 b = 1.1 λ = -0.1 a = 1.5 b = 1.5 λ = -0.1 a = 1.2 b = 1.5 λ = 0.8

Figure 4 :

 4 Figure 4: Plots of pdf and hrf for TGa distribution with different parameter values.

Figure 5 :

 5 Figure 5: The estimated pdf and cdf for data set 1.

Figure 6 :

 6 Figure 6: The estimated pdf and cdf for data set 2.

Table 1 :

 1 Simulation results for TGz.

	n	parameters CVM LSE WLS MPS MLE
	20	Bias(â)	1.334 4.044 3.495 0.385 0.275
		MSE(â)	1.643 5.158 4.291 0.949 1.191
		Bias( b)	0.300 0.240 0.264 0.017 0.111
		MSE( b)	0.300 0.336 0.400 0.388 0.480
		Bias( λ)	0.077 0.092 0.085 0.041 0.050
		MSE( λ)	0.125 0.203 0.185 0.070 0.089
	50	Bias(â)	1.121 3.777 3.870 0.580 0.033
		MSE(â)	1.233 4.153 4.096 1.296 0.799
		Bias( b)	0.300 0.268 0.368 0.201 0.034
		MSE( b)	0.300 0.294 0.438 0.347 0.339
		Bias( λ)	0.071 0.091 0.092 0.032 0.034
		MSE( λ)	0.114 0.201 0.205 0.048 0.058
	100	Bias(â)	1.166 3.459 4.179 0.535 0.010
		MSE(â)	1.869 3.685 4.313 1.265 0.589
		Bias( b)	0.307 0.249 0.427 0.179 0.035
		MSE( b)	0.465 0.285 0.472 0.326 0.269
		Bias( λ)	0.021 0.088 0.099 0.021 0.023
		MSE( λ)	0.033 0.191 0.219 0.032 0.040
	250	Bias(â)	0.747 3.452 4.263 0.290 0.028
		MSE(â)	1.510 3.560 4.337 1.049 0.438
		Bias( b)	0.198 0.261 0.440 0.102 0.024
		MSE( b)	0.346 0.281 0.462 0.240 0.221
		Bias( λ)	0.014 0.089 0.100 0.012 0.017
		MSE( λ)	0.022 0.191 0.223 0.020 0.028

  The observations are as follows: 0.55, 0.74, 0.77, 0.81, 0.84, 1.24, 0.93, 1.04, 1.11, 1.13, 1.30, 1.25, 1.27, 1.28,1.29, 1.48, 1.36, 1.39, 1.42, 1.48, 1.51, 1.49, 1.49, 1.50, 1.50, 1.55, 1.52, 1.53, 1.54, 1.55, 1.61, 1.58, 1.59, 1.60, 1.61, 1.63, 1.61, 1.61, 1.62, 1.62, 1.67, 1.64, 1.66, 1.66, 1.66, 1.70, 1.68, 1.68, 1.69, 1.70, 1.78, 1.73, 1.76, 1.76, 1.77, 1.89, 1.81, 1.82,1.84, 1.84, 2.00, 2.01, 2.24.

Table 2 :

 2 MLEs, their standard errors (in parentheses), A*, W*, KS, and P-value for data set 1.

	Distribution		MLE's		A*	W*	KS	P-value
	TB(c,k,λ)	2.148	38.854	-0.0097 0.660 0.107 0.109	0.581
		(0.321) (16.111) (0.555)
	TW(α,β,λ)	35.904	2.108	-0.025 0.642 0.104 0.109	0.587
		(15.729) (0.334) (0.562)
	TGa(a,b,λ)	2.818	18.218	-0.293 1.045 0.173	0.236
		(0.678)	(3.924) (0.442)
	TGz(a,b,λ)	2.285	8.765	-0.421 0.445 0.072 0.097	0.736
		(1.404)	(4.719) (0.228)
	TGE(α,θ,λ)	2.693	12.493	-0.547 1.076 0.178 0.147	0.234
		(0.839)	(1.666) (0.318)
	TLE(β,θ,λ)	0.612	65.579	-0.378 0.570 0.096 0.107	0.630
		(1.895) (14.184) (0.441)
	GT-W(η,σ,λ)	0.170	1.991	-0.271 0.607 0.098 0.117	0.612
		(0.023)	(0.330) (0.430)
	Burr(c,k)	2.154	39.124	-	0.675 0.108 0.120	0.571
		(0.240) (15.013)	-
	Weibull(α,β)	36.145	2.117	-	0.649 0.108 0.110	0.569
		(14.393) (0.245)	-
	Gamma(a,b)	3.029	18.564	-	1.120 0.192 0.145	0.170
		(0.575)	(3.836)	-
	Gompertz(a,b)	1.605	10.221	-	0.475 0.0875 0.135	0.568
		(0.512)	(1.753)	-

Table 3 :

 3 Descriptive statistics for the two data sets.

	Statistics	Mean	SD	Median MD Mean MD Median Skewness Kutosis Entropy
	Data set 1	0.16	0.08	0.16	0.065	0.07	0.07	2.21	2.47
	Data set 2	1.51	0.32	1.59	0.24	0.23	-0.88	3.92	3.80

Table 4 :

 4 Some moments of the TGz distribution for the two data sets.

	Statistics	Mean	S.D	Median Variance Skewness Kurtosis Entropy MD Mean MD Median
	Data set 1	0.163	0.079	0.177	0.006	0.093	2.606	2.68	0.065	0.069
	Data set 2	1.52	0.32	1.60	0.11	-0.90	4.20	3.86	0.25	0.23

Table 5 :

 5 Confidence intervals of TGz for given data set 1.

	CI	a	b	λ
	95% [0, 5.036] [0, 18.014] [0, 0.025]
	99% [0, 5.907] [0, 20.940] [0, 0.167]

Table 6 :

 6 Confidence intervals of TGz for given data set 2.

	CI	a	b	λ
	95% [0, 0.051] [ 2.242, 4.111] [0, 0.253]
	99% [0, 0.061] [1.946, 4.407] [0, 0.532]

Table 7 :

 7 MLEs, their standard errors (in parentheses), A*, W*, KS, and P-value for data set 2.

	Distribution		MLE's		A*	W*	KS	P-value
	TB(c,k,λ)	5.889	0.492	-0.909 5.751 1.055 0.293	0.003
		(1.208) (0.115) (0.173)
	TW(α,β,λ)	0.077	5.467	-0.314 1.242 0.226 0.144	0.442
		(0.034) (0.680) (0.366)
	TGa(a,b,λ)	16.354 11.241	-0.465 2.921 0.532 0.201	0.223
		(3.177) (2.105) (0.266)
	TGz(a,b,λ)	0.020	3.177	-0.629 0.808 0.140 0.124	0.685
		(0.016) (0.477) (0.450)
	Burr(c,k)	7.482	0.320	-	6.128 1.132 0.330	0.002
		(1.280) (0.065)	-
	Weibull(α,β)	0.059	5.77	-	1.303 0.237 0.152	0.307
		(0.020) (0.575)	-
	Gamma(a,b)	17.439 11.573	-	3.117 0.568 0.216	0.105
		(3.078) (2.072)	-
	Gompertz(a,b)	0.009	3.618	-	0.842 0.144 0.130	0.437
		(0.004) (0.294)	-

  The variance-covariance matrices of the MLEs of the TGz distribution for data set 2 is given as

		5.779	-11.185 -2.841	
		-11.185	22.275	5.421	 .
		-2.841	5.421	1.509
		0.001	-0.007 -0.006	
		-0.007	0.226	0.163	 .
		-0.006	0.163	0.202