on modified kies distribution and its applications

C. SATHEESHKUMAR^?

Department of Statistics, University of Kerala, Trivandrum-695 581, India Email: drcsatheeshkumar@gmail.com

S. H. S. DHARMAJA

Department of Statistics, Govt. College for Women, Trivandrum-695 014, India Email: shs dharmaja@yahoo.com

SUMMARY

In this paper, we consider a class of bathtub-shaped hazard function distribution through modifying the Kies distribution and investigate some of its important properties by deriving expressions for its percentile function, raw moments, stress-strength reliability measure etc.

The parameters of the distribution are estimated by the method of maximum likelihood and discussed some of its reliability applications with the help of certain real life data sets. In addition, the asymptotic behavior of the maximum likelihood estimators of the parameters of the distribution is examined by using simulated data sets.

Keywords and phrases:Failure rate; Fisher information matrix; Maximum likelihood esti- mation; Model selection; Percentile measures; Simulation.

1 Introduction

The Weibull distribution and its extended models have found wide applications in almost all areas of sciences especially in engineering sciences, hydrological sciences, meteorological sciences, social sciences etc. For details of some of these applications, see Meekar and Escobar (1998), Murthy et al.

(2004), Rinne (2009) and references therein. Some well-known extended Weibull models studied in the literature are the beta Weibull distribution (BWD) (see Almalki and Nadarajah, 2014; Cordeiro et al., 2013; Famoye et al., 2005), the beta generalized Weibull distribution (BGWD) (see Singla et al., 2012) and the exponentiated Weibull distribution (EWD) (see Mudholkar et al., 1995). All these families of distributions possess increasing, decreasing and/or bathtub-shaped hazard rate functions.

Kumar and Dharmaja (2014) studied the Kies distribution (KD) as an alternative to these extended Weibull models and shown that it gives better fit to certain real life data sets compared to both the BWD and the BGWD.

Consider the the following probability density function (pdf) of Kies distribution, in which0 ≤ x ≤ α < ∞,λ >0 andβ >0.

g1(x) =g1(x;α, λ, β) =α λ β x^β−1exp

−λ x/(α−x)β

/(α−x)^β+1 (1.1)

?Corresponding author

c Institute of Statistical Research and Training (ISRT), University of Dhaka, Dhaka 1000, Bangladesh.

A distribution with pdf (1.1) hereafter we denote as KD(α, λ, β). The cumulative distribution function (cdf)G₁(x)of theKD(α, λ, β)is given by

G1(x) = 1−exp

−λ x/(α−x)^β

, forx∈(0, α).

Recently exponentiated type distributions have received much attention in the literature due to its flexibility in modelling certain applications. Analogous to the EWD model, through this paper we propose an exponentiated form of theKD(α, λ, β)as a bathtub-shaped hazard function distribution and name it as “the modified Kies distribution (MKD)”, which can be viewed as a generalization of the exponentiated reduced Kies distribution,ERKD(β, δ), of Kumar and Dharmaja (2016). We study several important properties of the MKD in Section 2 and it is important to note that the MKD possess increasing or decreasing or bathtub-shaped hazard functions depending on the parameters of the distribution. The maximum likelihood estimation of the parameters of the distribution have been discussed in Section 3 and certain real life data applications in reliability studies are considered in Section 4 for illustrating the usefulness of the proposed class of distributions and also compared the proposed model with the existing models based on fitted values of the distribution and Weibull prob- ability plots. Further, in Section 5 we examine the asymptotic behavior of the maximum likelihood estimators of the parameters of the distribution by using certain simulated data sets.

We need the following integral/series representations in the sequel. For details regarding these representations see Gradshteyn and Ryhzik (2007). ForRe(ν)>0,Re(µ)>0,

Z u 0

x^ν−1exp(−µx)dx=µ^−νγ(ν, µu), (1.2) in which

γ(a, u) =

∞

X

i=0

(−1)ⁱ i!

u^a+i

a+i, (1.3)

Z ∞ 0

x^ν−1 exp(−µx)dx=µ^−νΓ(ν), Z ∞

u

x^−ν exp(−x)dx=u^−ν/2exp (−ν/2)W₋ν

2,(^1−ν2 ) (u), (1.4) and for|arg(−x)|<3π/2

Wk1, k2(x) = Γ(−2k2) Γ ¹₂−k2−k1

Mk1, k2(x) + Γ(2k₂) Γ ¹₂+k2−k1

Mk1,−k2(x) (1.5) with

Mk1, k2(x) = exp(−x/2) x^k2+12

∞

X

n=0

( ₁

2−k1+k2

n

(1 + 2k2)_n ·xⁿ n!

) ,

where(c)_n =c(c+ 1)· · ·(c+n−1), forn≥1 and(c)₀= 1.

We need the following series representations in the sequel. For any real valued functionA(·,·),

∞

X

s=0

∞

X

r=0

A(r, s) =

∞

X

s=0 s

X

r=0

A(r, s−r). (1.6)

r ϑ!

r! (ϑ−r)!, for anyϑ ≥ rwithr = 0,1,2, . . . , ϑandϑ ∈ N, the set of all positive integers and [ϑ]_j =_Γ(ϑ+1−j)^Γ(ϑ+1) , forϑ∈[0,∞)−N,for eachj= 0,1,2, . . .

(1− x)^ϑ=

∞

X

j=0

(−1)^j Γ (ϑ+ 1) j! Γ (ϑ+ 1−j)x^j=











ϑ

P

j=0

(−1)^{j ϑ}C_jx^j, ϑ∈N

∞

P

j=0

(−1)^j[ϑ+1]_j

j! x^j, ϑ∈ R⁺−N

Ei(x) =











γ+ ln (−x) +

∞

P

k=1 x^k

k×k!, forx <0 γ+ ln (x) +

∞

P

k=1 x^k

k×k!, forx >0,

(1.7)

ψ(x) =−γ− 1 x+x

∞

X

k=1

1

k(x+k), forx >0, (1.8)

whereψ(x)is the di-gamma function given by ψ(x) = d{ln [Γ (x)]}/dx, for Re(z) > 1 and q6= 0,−1,−2, . . .

ζ(z, q) =

∞

X

n=0

1

(q+n)^z (1.9)

and forRe (z) < 0and0< q ≤ 1, ζ(z, q) =2 Γ (1−z)

(2π)^1−z

"

sinzπ 2

X^∞

n=0

cos (2πqn)

n^1−z + coszπ 2

X^∞

n=0

sin (2πqn) n^1−z

#

, (1.10) in whichγ≈0.577215is the Euler’s constant.

2 Definition and Properties

Here we present the definition of the modified Kies distribution and discuss some of its important properties.

Definition 2.1. A random variableX is said to have a modified Kies distribution with parameters α,λ,β,δ∈ R⁺=(0,∞), written as “M KD(α, λ, β, δ)” if its cdfF(x)is of the following form.

Forx∈(0, α),

F(x) =n

1− exph

−λ x/(α−x)βio^δ

. (2.1)

On differentiating (2.1) with respect to x, we get the pdff(x)of theM KD(α, λ, β, δ)as in the following, forx∈(0, α).

f(x) =

α λ β δ x^β−1 exph

−λ x/(α−x)^βi n

1− exph

−λ x/(α−x)^βioδ−1

(α−x)^β+1 (2.2)

Clearly when δ = 1, theM KD(α, λ, β, δ)reduces to theKD(α, λ, β)with pdf as given in (1.1). Whenα=δ= 1, theM KD(α, λ, β, δ)reduces to the exponentiated reduced Kies distribution of Kumar and Dharmaja (2016). Now, we have the following results.

Result2.1. For anyα,λ,βandδ ∈ R⁺, ifXfollows theM KD(α, λ, β, δ), then

(i) Z1= (X/(α−X))^βfollows the generalized exponential distribution of Gupta et al. (1998), with cdfF1(z) ={1−exp (−λ z)}^δ.

(ii) Z2 = X/(α−X)follows the exponentiated Weibull distribution[EW D(λ, β, δ)]of Mud- holkar and Srivastava (1993) with cdfF₂(z) =

1−exp −λ z^{β δ}, which reduces to the Weibull distributionW D(λ, β), whenδ= 1.

PROOF. Proof is straight forward and hence omitted

Result2.2. The survival functionS(x)and the hazard functionh(x)of theM KD(α, λ, β, δ)are respectively,

S(x) = 1−F(x) and h(x) =f(x)/S(x) for anyx∈(0, α)andα,λ,β,δ ∈ R⁺.

PROOF. Proof follows from the definition of survival function, hazard rate function and Definition 2.1.

Result2.3. The hazard function h(x) of theM KD(α, λ, β, δ)is

(i) a bathtub-shaped function in the sense that it is a decreasing function ofxforx < x0and an increasing function ofxforx > x₀, in whichx₀is the solution of the equation

(β−1)α+ 2xn

F(x)^(1/δ)

− F(x)^(δ+1)/δo

−α β λ(x/(α−x))^βn

1−δexph

−λ(x/(α−x))^βi

−F(x)o

= 0, (2.3) when(a) β <1,δ≤1 (b) β <1,δ≥1 (c) β≥1,δ <1

(ii) an increasing function ofxforβ≥1,δ≥1

PROOF. Proof follows by taking the derivative of the hazard rate function with respect toxand on simplification we get,

h⁰(x) h(x) =

n

1−exph

−(x/(α−x))^βio−1

x(b−x)

1−n

1−exph

−(x/(α−x))^βioδ

×

[(β−1)α+ 2x]

F(x)¹_δ

− F(x)^(δ+1)_δ

(2.4)

−αβλ x

α−x ^β(

1−δexp

"

−λ x

α−x ^β#

−F(x) )!

,

Figure 1: The probability density function plots of the M KD(5,0.5,0.75, δ) (Left) and M KD(5,1.5,2.5, δ)(Right).

which leads to the result.

For graphical illustration, we have plotted the pdf f(x)of theM KD(5,0.5,0.75, δ)and the M KD(5,1.5,2.5, δ)for particular choices ofδin Figure 1, and the hazard rate functionh(x)of the M KD(5,0.5,0.75, δ)and theM KD(5,1.5,2.5, δ)for particular choices ofδin Figure 2.

Now we obtain certain percentile measures of theM KD(α, λ, β, δ)through the following re- sults.

Result2.4. For anyα,λ,β,δ∈ R⁺and for anyx_P ∈(0, α)such thatP =F(x_P), the percentile functionxP of theM KD(α, λ, β, δ)with cdf (2.1) isxP =α ηP (1 +ηP)⁻¹, in which

ηa=h

−(1/λ) ln

1−a^1δi_β¹ , for anya∈(0,1).

PROOF. Proof follows by inverting the cdfF(xP) =P ofM KD(α, λ, β, δ).

Next we obtain the expression forr^thraw moment of theM KD(α, λ, β, δ)through the follow- ing result.

Result2.5. IfX follows theM KD(α, λ, β, δ)with cdf (2.1), then ther^thraw momentµ⁰_rof the M KD(α, λ, β, δ)is the following, in whichγ(a, u)is as given in (1.3) and for anyi≥0,j≥0,

Λ⁽¹⁾_j (k−i, λ, β) = γ

r+k−j+β

β , λ(j+ 1)

λ^r+k−jβ (j+ 1)^r+k−j+ββ

(2.5)

Λ⁽²⁾_j (k−i, λ, β) =λ^k2−βi (j+ 1)

k−i−2β

2β exp

−λ(j+ 1) 2

W₋k−i

2β, ^β−_2β^(k−i)(λ(j+ 1)), (2.6) whereWk₁, k₂(x)is the Whittaker function as defined in (1.5). For anyδ ∈N,

µ⁰_r=δ α^r

∞

X

k=0 δ−1

X

j=0

(−1)^{j+k (δ−1)}Cj k!

h

Λ⁽¹⁾_j (k, λ, β) + Λ⁽²⁾_j (k, λ, β)i

(2.7)

Figure 2: The hazard rate function plots of the M KD(5,0.5,0.75, δ) (Left) and M KD(5,1.5,2.5, δ)(Right).

and forδ∈R⁺−N

µ⁰_r=δ α^r

∞

X

k=0 k

X

j=0

(−1)^{j+k (δ−1)}Cj k!

h

Λ⁽¹⁾_j (k−j, λ, β) + Λ⁽²⁾_j (k−j, λ, β)i

(2.8)

PROOF. By definition, ther^thraw moment ofM KD(α, λ, β, δ)with pdf (2.2) is

µ⁰_r=

α

Z

0

x^rα λ β δ x^β−1 (α−x)^β+1 exp

"

−λ x

α−x β# (

1−exp

"

−λ x

α−x

β#)δ−1

dx (2.9)

On substitutingu=

x α−x

β

in (2.9), we get

µ⁰_r=α^rλ δ

∞

Z

0









 u

1 β

1 +u

1 β





r

exp (−λ u) [1−exp (−λ u)]^δ−1







du (2.10)

Now we have the following cases.

Case (i) Forδ ∈N, expanding{1−exp (−λ u)}^δ−1, we get the following from (2.10).

µ⁰_r=α^rλ δ

δ−1

X

j=0







(−1)^{j (δ−1)}Cj

∞

Z

0

u^rβ

1 + u^1βrexp [−λ(j+ 1)u]du







On splitting the integral and expanding(1 +u^β1 )^−r, we get the following.

µ⁰_r=α^rλδ

∞

X

k=0 δ−1

X

j=0

(−1)^{j+k (δ−1)}C_j(r)_k k!







1

Z

0

u^r+kβ exp [−λ(j+ 1)u]du







+α^rλδ

∞

X

k=0 δ−1

X

j=0







(−1)^{j+k (δ−1)}Cj

(r)_k k!







∞

Z

1

exp [−λ(j+ 1)u]

u^βk

du













(2.11)

On substitutingλ(j+ 1)u=vin (2.11), we get

µ⁰_r=α^rλδ

∞

X

k=0 δ−1

X

j=0







(−1)^j+k

(δ−1)Cj (r)_k k!







λ(j+1)

Z

0

v

r+k

β exp (−v) λ

r+k β (j+ 1)

r+k+β β

dv













+α^rλδ

∞

X

k=0 δ−1

X

j=0







(−1)^j+k

(δ−1)C_j (r)_k k!

∞

Z

λ(j+1)

λ

k

β(j+ 1)

k β−1

exp (−v) v

k β

dv







, (2.12)

which leads to (2.7) in the light of (1.2), (1.4), (2.5) and (2.6).

Case (ii) Forδ∈R⁺−N, on expanding{1−exp(−u)^δ−1}in (2.11),

µ⁰_r=α^rλ δ

∞

X

j=0







(−1)^j(δ−1)_j j!

∞

Z

0

u^βr

1 + u^β1rexp [−λ(j+ 1)u]du







(2.13)

On splitting the integral and expanding(1 +u^β1 )^−rin (2.13) to get the following.

µ⁰_r=α^r δ

∞

X

k=0

∞

X

j=0







(−1)^j+k (δ−1)_j (r)_k j!k!







1

Z

0

u^r+kβ exp [−λ(j+ 1) u]du













+α^rδ

∞

X

k=0

∞

X

j=0







(−1)^j+k(δ−1)_j (r)_k j!k!







∞

Z

1

exp [−λ(j+ 1)u]

u^βk

du













(2.14)

On substitutingλ(j+ 1)u=vin (2.14),

Figure 3: The plots of moment measures of skewness (Left) and kurtosis (Right) of M KD(α, λ, β, δ)for particular values of the parameters.

µ⁰_r=α^rδ

∞

X

k=0

∞

X

j=0







(−1)^j+k (δ−1)_j(r)_k j!k!







λ(j+1)

Z

0

v

r+k

β exp (−v) λ

r+k β (j+ 1)

r+k+β β

dv













+α^rδ

∞

X

k=0

∞

X

j=0







(−1)^j+k(δ−1)_j(r)_k j!k!







∞

Z

λ(j+1)

λ

k

β exp (−v) (j+ 1)

(β−k)

β v

k β

dv













µ⁰_r=α^rδ

∞

X

k=0

∞

X

j=0

(−1)^j+k(δ−1)_j (r)_k j!k!

h

Λ⁽¹⁾_j (k, λ, β) + Λ⁽²⁾_j (k, λ, β)i

which leads to (2.8) in the light of (1.2), (1.4), (1.6), (2.5) and (2.6).

By using Result 2.5, we have computed the values of the moment measure of skewnessγ1(=

µ3/µ^3/2₂ , in which µ3 is the third central moment) and the moment measure of kurtosis γ2(=

(µ4/µ²₂)−3, in which µ4 is the fourth central moment) for particular values of its parameters and obtained the graphs in Figure 3.

Result2.6. LetX be the strength of a system which is subjected to a stressY, and ifX follows M KD(α, λ, β, δ1)andY followsM KD(α, λ, β, δ2), providedXandY are statistically indepen- dent random variables , thenR=P(Y < X), the measure of system performance (stress-strength reliability measure) is

R =δ1/(δ1+δ2) (2.15)

PROOF. Letf₁(x)denote the pdf ofX andf₂(x)denote the pdf ofY, then P(Y < X) =

Z α 0

Z x 0

f₂(y)dy

f₁(x)

dx (2.16)

P(Y < X) =

α

Z

0

αβ δ1x^β−1 n

1−exph

−λ(x/(α−x))^βio^δ1+δ₂−1

(α−x)^β+1exph

λ(x/(α−x))^βi dx On substitutingu= 1−exph

−λ(x/(α−x))^βi

in (2.17), we getP (Y < x) =δ1 1

R

0

u^δ1+δ2−1du which gives (2.15).

Analogous to certain characteristic property enjoyed by Weibull distribution, we obtain similar characteristic property ofM KD(α, λ, β, δ)through the following theorems.

Theorem 1. IfXfollowsM KD(α, λ, β, δ)with cdfF(x)as given in (2.1), then for anyy ∈ [0, α), and for every 0≤x≤α <∞, λ > 0, β >0, andδ >0,

(i) E

1−F(x)

|X > y = 1−F(x)

/2, (2.17)

(ii) E

ln 1−F(x)

|X > y = ln [1−F(x)]− 1 (2.18) (iii) E

δ⁻¹ln [1−F(x)]|X≤ y =δ⁻¹ln [1−F(x)]−δ⁻¹. (2.19) PROOF. Since the cdfF(x)ofM KD(α, λ, β, δ)given in (2.1) has the form

F(x) =















0 forx <0

n

1−exph

−λ(x/(α−x))^βioδ

for x∈[0, α)

1 forx≥α

and φ1(x) = 1−F(x) is real valued, continuous and differentiable function on [0, α) with E[φ1(X)] = 1/2, g(k) = 0, ψ(k) = 1/2 and by Theorem 7 on Page 260 of Rinne (2009), we obtain (2.17).

Since the cdfF(x)of the theM KD(α, λ, β, δ)given in (2.1) has the form

F(x) =















0 for x <0

1−exp

ln

1−n

1−exph

−λ(x/(α−x))^βioδ

for x∈[0, α)

1 for x≥ α

andφ2(x) = ln (1−F(x))is real valued, continuous and differentiable function on (0, α)with limx↑αφ2(x) =−∞andd=−1, by Theorem 8 on Page 262 of Rinne (2009), we obtain (2.18).

Further, since the cdfF(x)of theM KD(α, λ, β, δ)given in (2.1) takes the form

F(x) =















0 forx <0

exph δlnn

1−exph

−λ(x/(α−x))^βioi

forx∈[0, α)

1 forx≥α

andφ3(x) = (1/δ) ln [F(x)]for0≤x≤α <∞is a real-valued monotone function continuously differentiable on(0, α]withlim_x↓aφ₃(x) = −∞withE[φ₃(X)] = −1/δ, d = −1/δ and by Theorem 9 on Page 264 of Rinne (2009) , we obtain (2.19).

Theorem 2. Let X₁, . . . , X_n be n independent and identically distributed (i.i.d.) random vari- ables following theM KD(α, λ, β, δ)with cdf (2.1) and letY = max(X1, . . . , Xn). ThenY fol- lows theM KD(α, λ, β, nδ). Conversely, ifY follows theM KD(a, b, η, ω), then eachX_ifollows M KD(a, b, η, ω n⁻¹),i= 1,2, . . . , n.

PROOF. IfX₁, . . . , X_n are i.i.d.M KD(α, λ, β, δ)variates each with pdf (2.2), the pdff_n(y)of Y = max(X1, . . . , Xn)is the following, for anyα >0,λ >0,δ >0andβ >0.

f_n(y) =αλβnδy^β−1 (α−y)^β+1 exph

−λ(y/(α−y))^βi n

1−exph

−λ(y/(α−y))^βionδ−1

Since the pdfgn(z)of maximum ofni.i.d. random variates each with pdf g(z)and cdfG(z)is gn(z) =ng(z)[G(z)]ⁿ⁻¹. ThusY follows theM KD(α, λ, β, nδ).

Conversely, assume thatY = max(X₁, . . . , X_n)follows theM KD(a, b, η, ω), then the cdf Fn(y)ofY is

F_n(y) =

1−exp

−b y/(a−y)^{η ω}

(2.20) in the light of (2.1). For any i.i.d. random variatesZ₁, . . . , Z_neach with cdfG(z), the cdfG_n(z)of Z= max(Z1, . . . , Zn)is

G_n(z) = [G(z)]ⁿ. (2.21)

Now we obtain the following from (2.20) in the light of (2.21).

[F(y)]ⁿ =

1−exp

−b y

a−y ^ηω

.

Thus the pdf ofX₁is f(y) =F⁰(y) =abηω

n

y^η−1 (a−y)^η+1 exp

−b y

a−y ^η

1− exp

−b y

a−y

^ηωn−1

.

3 Estimation

Here we discuss the maximum likelihood estimation of the parameters of theM KD(α, λ, β, δ).

Consider the following log-likelihood function` of a random sample X1, . . . , Xn taken from a population following theM KD(α, λ, β, δ)with pdf (2.2).

`=nln (δ) +nln (β) +nln (λ) +nln (α) + (β−1)

n

X

i=1

ln (xi)−λ

n

X

i=1

x_i α−xi

β

−(β+ 1)

n

X

i=1

ln (α−xi) + (δ−1)

n

X

i=1

ln (

1−exp

"

−λ x

α−x β#)

(3.1)

equating to zero, we can obtain the likelihood equations. On solving these likelihood equations one can obtain the maximum likelihood estimates (MLE)of the parameters ofM KD(α, λ, β, δ).

These likelihood equations do not always have a unique solution because theM KD(α, λ, β, δ)is not a regular model. By using the package‘maxLik’-R(cf. Henningsen and Toomet , 2011), we observed that the second order partial derivatives of the log-likelihood function with respect to the parameters gives negative values forα >0,λ >0,β >0andδ >0. The expressions of elements of the corresponding Fisher information matrixI(θ)are given in Appendix.

4 Applications

In this section we discuss certain applications of theM KD(α, λ, β, δ)with the help of the following data sets.

Data set 1 The data set on 30 “times of failure and running times” for a sample of devices from a field-tracking study of a larger system taken from Meekar and Escobar (1998) is:

275, 13, 147, 23, 181, 30, 65, 10, 300, 173, 106, 300, 300, 212, 300, 300, 300, 2, 261, 293, 88, 247, 28, 143, 300, 23, 300, 80, 245, 266

Data set 2 The data set on 23 “time between failures of secondary reactor pumps” taken from Bebbington et al. (2007) is:

2.160, 0.150, 4.082, 0.746, 0.358, 0.199, 0.402, 0.101, 0.605, 0.954, 1.359, 0.273, 0.491, 3.465, 0.070, 6.560, 1.060, 0.062, 4.992, 0.614, 5.320, 0.347, 1.921

Data set 3 This data set is taken from Aarset (1987) which is “on lifetimes of 50 components” and they are given as follows:

0.1, 0.2, 1, 1, 1, 1, 1, 2, 3, 6, 7, 11, 12, 18, 18, 18, 18, 18, 21, 32, 36, 40, 45, 46, 47, 50, 55, 60, 63, 63, 67, 67, 67, 67, 72, 75, 79, 82, 82, 83, 84, 84, 84, 85, 85, 85, 85, 85, 86, 86

Data set 4 The data set on 19 “initial remission times of leukemia patients” taken from Lee and Wang (2003) is:

8, 10, 10, 12, 14, 20, 48, 70, 75, 99, 103, 161, 162, 169, 195, 199, 217, 220, 245 We have fitted theM KD(α, λ, β, δ)to the data sets with the help of the package‘maxLik’-R (cf. Henningsen and Toomet, 2011). We have fitted the following models to the four data sets for comparison

(a) theKD(α, λ, β), in which0 ≤x≤α <∞,λ >0andβ > 0(cf. Kumar and Dharmaja, 2014),

(b) theBGW D(µ, θ, σ, τ, ρ), in whichx >0,µ,θ,σ,τandρ >0. (cf. Singla et al., 2012), (c) theBW D(µ, θ, σ, ρ), in whichx >0,µ,θ,σandρ >0(cf. Famoye et al., 2005) and (d) theEW D(µ, σ, ρ), in whichx >0,µ,θ,σandρ >0. (cf. Mudholkar et al., 1995).

Then we compared the fitted model theM KD(α, λ, β, δ)with that of fitted models-theKD(α, λ, β), theBGW D(µ, θ, σ, ρ, τ), theBW D(µ, θ, σ, ρ)and theEW D(µ, σ, ρ). For model comparison, we have computed the values of log-likelihood, the Akaike information criterion (AIC), Bayesian in- formation criterion (BIC) and the second order Akaike information criterion (AICc) and included in Table 1. It is seen that the values of the AIC, the BIC and the AICc of each data set in the case of theM KD(α, λ, β, δ)model is the lowest. Hence theM KD(α, λ, β, δ)can be viewed as the best model compared to other existing models considered in this section. Further, we have plotted the cdf of these fitted models against the corresponding empirical distribution in Figure 4. Figure 4 also supports the above conclusion that theM KD(α, λ, β, δ)gives a better fit to each data set compared to other generalized Weibull models considered in this paper. Moreover, using these fitted models, we have obtained the Weibull Probability Plots (WPP plots) as in Figure 5 corresponding to the two data sets for comparing the models. These plots also indicate that theM KD(α, λ, β, δ)as the best model to the data sets considered in the paper compared to other existing models.

5 Simulation

In order to assess the performance of the maximum likelihood estimators of the parameters of the M KD(α, λ, β, δ), we have carried out a brief simulation study. We, simulated datasets by adapting probability integral transform method based on the following sets of of parameters according to the nature of the skewness.

(i) α=5,λ=1.5,β= 2.5 andδ= 0.2 (positively skewed)

(ii) α=20.28,λ=27.13,β= 10.51 andδ= 0.16 (negatively skewed).

We considered 200 bootstrap samples of sizesn= 10, 25, 50 and 100 for comparison and computed average bias and mean squared errors (MSEs) in each case. The results obtained are summarized in Table 2. From Table 2, it can be observed that both the average bias and MSEs of the estimators are in decreasing order, as sample size increases.

Table1:Fittingofvariousmodelstodatasets ModelEstimatesoftheparametersAICBICAICcEstimatesoftheparametersAICBICAICc Dataset1Dataset2 MKD(α,λ,β,δ)α=356.32,λ=8.699E-04343.7290351.3771344.6179α=6.7314,λ=5.669567.389371.931269.6115 β=3.9813,δ=0.1229β=0.1365,δ=54.5011 KD(α,λ,β)α=492.9863,λ=1.3096,363.2928370.9409364.1817α=20.8173,λ=6.6406,70.985874.392372.2489 β=0.9334β=0.7371 BGWD(µ,θ,σ,τ,ρ)µ=0.06129,θ=0.1090,348.0060357.5661349.3696µ=0.0225,θ=0.158873.696979.374477.2263 σ=0.0050,τ=2.6536τ=3.7092,σ=0.2358 ρ=5.5795ρ=5.8234 BWD(µ,θ,σ,ρ)µ=0.1446,θ=8.1399363.2318370.8799364.1207µ=47.3252,θ=14.117371.601676.143673.8239 σ=0.0021,ρ=0.2474σ=40.4547,ρ=0.1168 EWD(µ,σ,ρ)µ=0.1810,σ=0.0031362.2778368.0139362.7995µ=0.1026,σ=0.176874.097679.833774.6194 ρ=5.0619ρ=4.6524 —Dataset3Dataset2 MKD(α,λ,β,δ)α=86.0023,λ=2.0126411.6160419.2641412.5049α=245.20088,λ=4.08134212.3998216.1775559215.2569429 β=0.1341,δ=5.6589β=0.11953,δ=34.18196 KD(α,λ,β)α=86.1591,λ=0.5455416.2926422.0287416.8143α=259.54961,λ=0.84442,213.8182216.6515169215.4182 β=0.3478β=0.5766 BGWD(µ,θ,σ,τ,ρ)µ=0.587,θ=0.315,450.6730460.2331452.0366µ=0.6196,θ=17.74042,226.6459231.3680949231.2612846 σ=0.016,τ=0.136σ=0.19884,τ=0.18122 ,ρ=5.64τ=0.18122,ρ=17.88447 BWD(µ,θ,σ,ρ)µ=0.124,θ=0.313455.8860463.5341456.7749µ=0.65495,θ=0.05081,226.0321229.8098559228.8892429 σ=0.015,ρ=5.63σ=0.2542,τ=0.86686 EWD(µ,σ,ρ)µ=0.1752,σ=0.0111,466.2750472.0111466.7967σ=0.00423,τ=7.3598215.666218.4993169217.266 ρ=3.9113ρ=0.10151

Table 2: Average bias and mean squared errors (in brackets) of the MLEs of the GKD(α,λ,β,ρ) based on simulated data sets for the parameter sets (i)α = 5,λ= 1.5,β = 2.5,ρ = 0.2and (ii) α= 20.28,λ= 27.13,β = 10.51,ρ= 0.16.

Sample size

Parameter 10 25 50 100

Parameter α -0.8596 0.1329 -0.1162 -0.0939

Set (i) (1.1689) (1.1459) (0.2776) (0.0308)

λ -0.8880 0.3561 -0.2148 0.0072

(3.0252) (2.3675) (0.2435) (0.0563)

β 1.8214 -0.1367 -0.0718 0.0653

(4.4569) (1.2201) (0.0668) (0.0286)

ρ 0.2563 0.1565 0.0597 0.0287

(2.9952) (0.1266) (0.0047) (0.0011)

Parameter α -2.6349 -0.2522 -0.1579 0.1011

Set (ii) (7.9427) (0.0538) (0.0368) (0.0112)

λ -0.1548 -0.0677 0.0488 0.0226

(0.0439) (0.0068) (0.0054) (7.90E-04)

β 7.2212 0.6093 -0.1937 -0.1008

(68.3939) (0.3713) (0.0379) (0.0125)

ρ -0.0833 -0.0490 0.0107 0.0053

(0.0069) (0.0024) (8.00E-04) (2.89E-05)

6 Conclusion

In this paper, an exponentiated version of the Kies distribution namely “the modified Kies distri- butionM KD(α, λ, β, δ)” is introduced as a generalization of the exponentiated reduced Kies dis- tributionERKD(β, δ)of Kumar and Dharmaja (2016) and investigated several properties of the distribution. It can be noted that the support of the ERKD is over the range (0,1) while that of the M KD(α, λ, β, δ), is over the range(0, α), forα >0. So, in certain practical situations, the data need to be transformed while fitting theERKD(β, δ), and this drawback is investigated in the case ofM KD(α, λ, β, δ). Further the inclusion of scale parameterλhelps to create more flexibility in practical point of view and thus the practical relevance of the model is quite obvious from the fitting of the model to various data sets considered in Section 4 of the paper. Further a brief simulation study is carried out for examining the asymptotic behavior of the maximum likelihood estimates of the parameters of the distribution.

Acknowledgements

The authors would like to express their sincere gratitude to the Editor, the Associate Editor and the anonymous Referee for their valuable comments on an earlier version of the paper, which greatly improved the quality presentation of the paper.

Figure 4: Empirical and Fitted distribution function plots for the Data set 1 ( Top Left), Data set 2 (Top Right) and Data set 3 (Bottom Left) and Data set 4 (Bottom Right).

Figure 5: Weibull Probability Plots for Data set 1 ( Top Left ), Data set 2 (Top Right), Data set 3 (Bottom Left) and Data set 4 (Bottom Right).

A Elements of Information Matrix

I₁₁=− 1

α² + λδ (β+ 1) α²

δ−1

X

j=0

(−1)^{j (δ−1)}C_j





 1 λ(j+ 1)+

2 Γ

1 β + 1 [λ(j+ 1)]

1 β+1

+ Γ

2 β + 1 [λ(j+ 1)]

2 β+1







− λ²β δ (β+ 1) α²

δ−1

X

j=0

(−1)^{j (δ−1)}Cj





 1 [λ(j+ 2)]² +

2 Γ

1 β + 1 [λ(j+ 2)]

1 β+1

+ Γ

2 β + 1 [λ(j+ 2)]

2 β+1







+ λ²β δ (β+ 1) (δ−1) α²

δ−2

X

j=0

(−1)^{j (δ−2)}Cj





 1 [λ(j+ 2)]² +

2 Γ

1 β+ 1 [λ(j+ 2)]

1 β+1

+ Γ

2 β + 1 [λ(j+ 2)]

2 β+1







+ λ³β²δ(δ−1) α²

δ−3

X

j=0

(−1)^{j (δ−3)}Cj





 2 [λ(j+ 3)]² +

2 Γ

1 β + 2 [λ(j+ 3)]

1 β+1

+ Γ

2 β+ 2 [λ(j+ 3)]

2 β+1





 ,

I₁₂=I₂₁ = λβδ α

δ−1

X

j=0

(−1)^{j (δ−1)}C_j





 1 [λ(j+ 1)]² +

Γ

1 β + 2 [λ(j+ 1)]

1 β+2







− λβδ(δ−1) α

δ−2

X

j=0

(−1)^{j (δ−2)}C_j





 1 [λ(j+ 2)]² +

Γ

1 β + 2 [λ(j+ 2)]

1 β+2







+ λ²β δ (δ−1) α

δ−3

X

j=0

(−1)^{j (δ−3)}Cj





 2 [λ(j+ 2)]³ +

Γ

1 β+ 3 [λ(j+ 2)]

1 β+3





 ,

I13=I31=−λδ α

δ−1

X

j=0

(−1)^{j (δ−1)}Cj





 1 [λ(j+ 1)]+

Γ

1 β+ 1 [λ(j+ 1)]

1 β+1







+λδ α

δ−1

X

j=0

(−1)^{j (δ−1)}Cj





 1 [λ(j+ 1)]² +

Γ

1 β+ 2 [λ(j+ 1)]

1 β+2







+λδ α

δ−1

X

j=0

(−1)^{j (δ−1)}C_j







{ψ(2)−ln [λ(j+ 1)]}

[λ(j+ 1)] + Γ

1 β+ 2 n

ψ

1 β+ 2

−ln [λ(j+ 1)]o [λ(j+ 1)]^β1+2







− λ²δ(δ−1) α

δ−2

X

j=0

(−1)^{j (δ−2)}C_j





 1 [λ(j+ 2)]² +

Γ

1 β+ 2 [λ(j+ 2)]

1 β+2





 ,

I₁₄=I₄₁=−λ²β δ α

δ−2

X

j=0

(−1)^{j (δ−2)}C_j





 1 [λ(j+ 2)]²+

Γ

1 β + 2 [λ(j+ 2)]

1 β+2





 ,

I₂₂=− 1

λ² −2λδ(δ−1)

δ−3

X

j=0

(−1)^{j (δ−3)}C_j [λ(j+ 2)]³ ,

I₂₃=I₃₂=−λδ β

X

j=0

(−1)^{j (δ−1)}C_j ψ(2) −ln [λ(j+ 1)]

[λ(j+ 1)]² + λδ (δ−1)

β

δ−2

X

j=0

(−1)^{j (δ−2)}Cj

(ψ(2) −ln [λ(j+ 2)]

[λ(j+ 2)]² )

− 2λ²δ(δ−1) β

δ−3

X

j=0

(−1)^{j (δ−3)}Cj

(ψ(3)−ln [λ(j+ 2)]

[λ(j+ 2)]³ )

,

I24=I42=λδ

δ−2

X

j=0

((−1)^{j (δ−2)}Cj

[λ(j+ 2)]² )

I33=−1

β² − λ²δ β²

δ−1

X

j=0

(−1)^{j (δ−1)}Cj

({ψ(2) −ln [λ(j+ 1)]}²+ζ(2,2) [λ(j+ 1)]²

)

+ λ²δ(δ−1) β²

δ−2

X

j=0

(−1)^{j (δ−2)}C_j

({ψ(2) −ln [λ(j+ 2)]}²+ζ(2,2) [λ(j+ 2)]²

)

− 2λ³δ(δ−1) β²

δ−3

X

j=0

(−1)^{j (δ−3)}Cj

({ψ(3) −ln [λ(j+ 2)]}²+ζ(2,3) [λ(j+ 2)]³

) ,

I₃₄ =I₄₃= λ²δ β

δ−2

X

j=0

(−1)^{j (δ−2)}C_j

(ψ(2)−ln [λ(j+ 2)]

[λ(j+ 2)]² )

,

I₄₄=I₄₄⁰ =− 1 δ²,

I₁₁⁰ =−1

α² + λδ (β+ 1) α²

∞

X

j=0

(−1)^j (δ−1)_j j!





 1 λ(j+ 1) +

2 Γ

1 β + 1 [λ(j+ 1)]

1 β+1

+ Γ

2 β+ 1 [λ(j+ 1)]

2 β+1







− λ²β δ (β+ 1) α²

∞

X

j=0

(−1)^j (δ−1)_j j!





 1 [λ(j+ 2)]² +

2 Γ

1 β + 1 [λ(j+ 2)]

1 β+1

+ Γ

2 β+ 1 [λ(j+ 2)]

2 β+1







+λ²β δ (β+ 1) (δ−1) α²

∞

X

j=0

(−1)^j (δ−2)_j j!





 1 [λ(j+ 2)]² +

2 Γ

1 β + 1 [λ(j+ 2)]

1 β+1

+ Γ

2 β+ 1 [λ(j+ 2)]

2 β+1







+ λ³β²δ(δ−1) α²

∞

X

j=0

(−1)^j (δ−3)_j j!





 2 [λ(j+ 3)]²+

2 Γ

1 β+ 2 [λ(j+ 3)]

1 β+1

+ Γ

2 β+ 2 [λ(j+ 3)]

2 β+1





 ,

I₁₂⁰ =λβδ α

∞

X

j=0

(−1)^j (δ−1)_j j!





 1 [λ(j+ 1)]² +

Γ

1 β+ 2 [λ(j+ 1)]

1 β+2







− λβδ(δ−1) α

∞

X

j=0

(−1)^j (δ−2)_j j!





 1 [λ(j+ 2)]² +

Γ

1 β + 2 [λ(j+ 2)]

1 β+2







+ λ²β δ (δ−1) α

∞

X

j=0

(−1)^j (δ−3)_j j!





 2 [λ(j+ 2)]³ +

Γ

1 β + 3 [λ(j+ 2)]

1 β+3





 ,

I₁₃⁰ =I₃₁⁰ =−λδ α

∞

X

j=0

(−1)^j (δ−1)_j j!





 1 [λ(j+ 1)]+

Γ

1 β+ 1 [λ(j+ 1)]

1 β+1







+λδ α

∞

X

j=0

(−1)^j (δ−1)_j j!





 1 [λ(j+ 1)]² +

Γ

1 β+ 2 [λ(j+ 1)]

1 β+2







+λδ α

∞

X

j=0

(−1)^j (δ−1)_j j!







{ψ(2) −ln [λ(j+ 1)]}

[λ(j+ 1)] + Γ

1 β + 2 n

ψ

1 β+ 2

−ln [λ(j+ 1)]o [λ(j+ 1)]^β1+2







−λ²δ(δ−1) α

∞

X

j=0

(−1)^j (δ−2)_j j!





 1 [λ(j+ 2)]² +

Γ

1 β+ 2 [λ(j+ 2)]

1 β+2







−λ²δ(δ−1) α

∞

X

j=0

(−1)^j (δ−2)_j j!







{ψ(2)−ln [λ(j+ 2)]}

[λ(j+ 2)]² + Γ

1 β+ 3 n

ψ

1 β + 3

−ln [λ(j+ 2)]o [λ(j+ 2)]

1 β+3







+ λ²δ(δ−1) α

∞

X

j=0

(−1)^j (δ−3)_j j!







2{ψ(3)−ln [λ(j+ 2)]}

[λ(j+ 2)]³ + Γ

1 β+ 3

{ψ(2) −ln [λ(j+ 2)]}

[λ(j+ 2)]

1 β+3





 ,

I₁₄⁰ =I₄₁⁰ =−λ²β δ α

∞

X

j=0

(−1)^j (δ−2)_j j!





 1 [λ(j+ 2)]² +

Γ

1 β + 2 [λ(j+ 2)]

1 β+2





 ,

I₂₂⁰ =−1

λ² − 2λδ(δ−1)

∞

X

j=0

(−1)^j (δ−3)_j j! [λ(j+ 2)]³ , I₂₃⁰ =I₃₂⁰ =−λδ

β

∞

X

j=0

(−1)^j (δ−1)_j j!

(ψ(2)−ln [λ(j+ 1)]

[λ(j+ 1)]² )

+ λδ (δ−1) β

∞

X

j=0

(−1)^j (δ−2)_j j!

(ψ(2) −ln [λ(j+ 2)]

[λ(j+ 2)]² )

−2λ²δ(δ−1) β

∞

X

j=0

(−1)^j (δ−3)_j j!

(ψ(3) −ln [λ(j+ 2)]

[λ(j+ 2)]³ )

,

I₂₄⁰ = I₄₂⁰ =λδ

∞

X

j=0

((−1)^j (δ−2)_j j! [λ(j+ 2)]²

) ,

I₃₃⁰ =−1

β² − 1λ²δ β²

∞

X

j=0

(−1)^j (δ−1)_j j!

({ψ(2) −ln [λ(j+ 1)]}²+ζ(2,2) [λ(j+ 1)]²

)

+ 1λ²δ(δ−1) β²

∞

X

j=0

(−1)^j (δ−2)_j j!

({ψ(2) −ln [λ(j+ 2)]}²+ζ(2,2) [λ(j+ 2)]²

)

− 2λ³δ(δ−1) β²

∞

X

j=0

(−1)^j (δ−3)_j j!

({ψ(3) −ln [λ(j+ 2)]}²+ζ(2,3) [λ(j+ 2)]³

)

I₃₄⁰ =I₄₃⁰ = 1λ²δ β

∞

X

j=0

(−1)^j (δ−2)_j j!

(ψ(2)− ln [λ(j+ 2)]

[λ(j+ 2)]² )

,

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Received: July 18, 2016 Accepted: May 17, 2017