The Transmuted Weibull Frechet Distribution: Properties and Applications Текст научной статьи по специальности «Науки о Земле и смежные экологические науки»

The Transmuted Weibull Frechet Distribution: Properties

and Applications

Joseph Thomas Eghwerido •

Department of Statistics, Federal University of Petroleum Resources, Effurun, Delta State, Nigeria

[email protected]

Abstract

The behaviour of everyday real life processes played a greater role in distribution theory. Thus, this article proposes a transmuted Weibull Frechet (TWFr) distribution for modeling real life datasets. Of most important, the statistical properties of the TWFr distribution such as the hazard, survival functions, order statistic, quantile, odd, cumulative functions were derived and examined. A simulation study to examine the performance of the TWFr distribution was also conducted. A glass fiber data and breaking stress of carbon data real life application were used to showcase the performance of the proposed model. The results showed that the TWFr distribution competes favourably well with other types of continuous distributions in the Frechet family of distributions.

Keywords: Frechet distribution, Hazard rate function, Order statistics, Transmutation, Weibull distribution.

1. Introduction

Modeling the distributions of real life processes poses greater challenges despite the numerous distributions that have been proposed in literature. However, there is a growing interest in developing newer classes of classical univariate distributions for modeling variety of data sets that arise from our daily scenarios. Thus, it becomes necessary to model these processes either by compounding one or more distributions to address these complex situations.

The Weibull distribution proposed by a famous statistician called Weibull in 1951 [27] has a wide range of applications in modeling failure time processes, lifetime processes, mechanical and electrical systems. More so, the Weibull distribution has been found to be better for modeling the minimum of large number of independent positive random variables in extreme value theory. On the other hand, the Frechet distribution is a special case of the Weibull distribution used to model extreme value scenarios like earthquakes, horse racing, floods, rainfall, wind speed, queues in supermarkets and sea waves (see [2]). The Frechet distribution has been widely used to model extreme value scenarios because of its stochastic phenomena. However, the Frechet distribution is used for modeling maximum of a large number of independent random variables from a particular class of distributions ([1]). Hence, because of its usefulness, improving the flexibility of the Frechet distribution becomes necessary by adding a transmuted parameter that can reflect the true characteristics of the data set(see [13]).

Several statistical distributions have been proposed in literature. For example, [2] proposed the Weibull Frechet distribution, [12] proposed the generalized odd Weibull generated family of distributions. Recently, [26] proposed the gamma extended Frechet distribution. [17] proposed the beta Frechet distribution. [16] proposed the exponentiated Frechet distribution. [14] proposed Kumaraswamy Frechet distribution. The generalized transmuted Frechet (GTFr) distribution was proposed in [18]. [14] estimated the Frechet type 11 parameters and [23] proposed a

new generalization of the Frechet distribution. In this study, the transmuted Weibull Frechet distribution is introduced.

A random variable X with a scale parameter a, and j8, t, v the shape parameters has a Weibull Frechet distribution if the cumulative density function is given as

G(x) = 1 - exp^—^exp

a\ß

<X)

1

(1)

The corresponding probability density function is given as

g(x) =Tvßaßx ß 1 exp

—vO

a\ß

1 - exp

— ( - ) x

—v—1

x exp —t<exp

( a )ß

x

1

(2)

However, the pdf and cdf of a random variable X can be transmuted with a transmutation parameter |A| < 1 as

f(x;A) = g(x)

1 + A — 2AG(x)

and

F(x;A) = 1 + A G(x) — A

G(x)

(3)

(4)

with G(x) and g(x) as the cdf and pdf of the baseline/parent distribution respectively.

This article is organized as follows: The introduction was given in section 1, Section 2 is the formulation of the transmuted Weibull Frechet distribution. Section 3 discussed the maximum likelihood of model parameters. In Section 4, we derived some properties of the TWFr distribution. Section 5 is the simulation study and real life application to validate the proposed model and Section 6 is the conclusion.

v

v

2

2. The Transmuted Weibull Frechet Distribution

Let X be random variable. Then, the pdf of the TWFr is defined as

frwFr(x) =Tvßaßx ß 1 exp

—v()

a\ß

1 - exp

x exp[ —^ exp

( a )ß

x

-1

1 — A + 2Aexp| _W exp

^\ß

<x>

_( a )ß

x

1

v1

The corresponding cdf is given as

FTWFr(x) = ^ 1 + A^ — exp^—^ exp

a\ß

(-)

x

1

A

1 — exp —r< exp

ft* xß

1

(5)

(6)

Figure 1 shows the plot of the pdf for TWFr distribution with different parameters values. The plot of the TWFr distribution shows that it could be increasing, decreasing and skewed to the right and left depending on the values of the parameters.

v

v

x

v

2

v

a=3.0, b=2.0, a=1.0, b=0.5, a=1.5, b=2.3, a=1.5, b=1.9, a=1.2, b=1.5, a=3.5, b=1.0, a=2.0, b=2.0, a=1.0, b=0.5,

t=1.5, v=2.2, 1=0.5, t=3.5, v=2.5, 1=-0.5 t=4.2, v=2.2, 1=0.8, t=0.5, v=1.2, 1=0.5, t=1.1, v=2.2, 1=-0.3 t=2.0, v=2.5,1=-0.2 t=2.0, v=2.2, 1=1.0, t=2.5, v=2.5, 1=0.0,

x

(a)

a=0.75, b=0.5, t=1.5, v=1.3, 1=0.5, a=0.50, b=0.2, t=2.5, v=3.5, 1=-0.2, a=4.5, b=1.3, t=4.2, v=2.2, 1=-0.22, a=1.5, b=1.0, t=0.5, v=1.2, 1=0.5, a=3.2, b=1.5, t=2.1, v=2.0, 1=-0.3, a=1.5, b=1.0, t=2.0, v=2.5, 1=-0.2, a=0.75, b=4.5, t=0.5, v=0.5, 1=0.75, a=1.5, b=1.5, t=1.5, v=1.5, 1=1.0,

(b)

Figure 1 The plots of the TWFr pdf for some parameter values

The reliability or survival function (rf) of the random variable X is given as

rfTWFr(x) =exp[ -t|exp

a sp

+A exp[ —t<exp

—v

-U 1(1-A

a

-v\ \ 2

1

(7)

x

Its hazard rate function (hrf) is given as

hrTWFr (x) =TvpaPx p 1exp x exp[ —t<exp

x

1 — exp

_(a )P

x

-v-1

(x) x

— 1

1 — A + 2Aexp[ —^ exp

<i>

x <exp\ —t<exp

(j)

exp

(x /

1

—1 2 —1

—1

1 — A ) + A( exp I —t

(8)

Figure 2 shows the plot for the hazard rate function of the TWFr distribution. The plot shows that the TWFr model is decreasing and bathtub depending on the values of the associated parameters.

v

v

x

v

1=0.7, v=1.6, 3=0.7, a=0.7, 1=0.5, t=0.6, v=1.4, 3=0.5, a=0.5, 1=0.5, 1=0.8, v=1.0, 3=0.9, a=0.7, 1=0.5, 1=1.1, v=1.0, 3=0.5, a=0.5, 1=1.0, 1=0.7, v=1.6, 3=0.8, a=0.7, 1=0.4, 1=0.7, v=1.2, 3=0.5, a=0.7, 1=0.5, 1=0.5, v=1.0, 3=0.5, a=0.5, 1=0.2, 1=0.7, v=1.6, 3=0.5, a=0.7, 1=0.1,

Figure 2 The plots of the TWFr hrf for some parameter values The cumulative hazard rate function (chrf) of the TWFr model is given as

chrfTWFr (x) = — lnj exp^—^ exp +A(expi —t<exp

(x (x

— 1 1 — A

! -V XX 2

1

(9)

3. Parameter Estimation of the Transmuted Weibull Frechet

Distribution

Let X be random variable with TWFr distribution function. Then, the log-likelihood £ of the distribution for parameter vector (A,p,t, v, a)t is given as,

£ = n log(TVpaP) + s + m + z + p

(10)

X

v

and

d ' n dA = pA = °

da = J - S'a + m'* - + P'a = ° dt n , ,

dT = n - zT +=

dt n , , , ,

3- = - - s'v + m'v - z'v + p'v = 0,

dv v

dt n kß , , , , dß = ß + k - Sß + mß - Zß + Vß = °.

where / denotes partial derivative and subscript the respective parameter and

k = aß ; s = £

i=1

-vO

a\ß

; m = £ log< 1 - exp

i=1

-( ^ ) x

v1

(11) (12)

(13)

(14)

(15)

i=1

£ -T1 exP

( ^ ) x

1

p = £ log

i=1

1 - A + 2Aexpi -r<{ exp

a\ß

1

The Equations (11), (12), (13), (14) and (15) are nonlinear and can not be easily obtained in closed form. Thus, the solutions to the parameter vector are obtained numerically using the Newton-Raphson algorithm and with various statistical and mathematical softwares like R, Mathematical, Maple and Matlab.

4. Some Statistical Properties of the Transmuted Weibull Frechet

Distribution

This section investigates some statistical properties of the TWFr distribution. These include, quantile and random number generation and order statistics.

4.1. Quantile Function and Median

Let X be a random variable such that X ~ TWFr (a, A, ß, a, b). Then, the quantile function of X for u € (0,1) is real solution of the following equation given as

Thus,

where

Q(u) = F-1 (x).

xu — a

log 1 +

(-T-1 ) log(1 - $(u))

0 < u < 1

(1+a)-v2a+a)2-4au, if a < °,

$(u) = <! (1+A)+V(1+A)2-4Au if A > °

u,

otherwise A = °.

(16)

(17)

v

Z

v

n

ß

V

By setting u = 0.5 in Equation (17), we have the median (M) of X as

M = a

log 1 +

(-t-1) log(1 - 0(0.5))

with

0(u) = <

q+^VCpi2-2*, if A < 0,

(1+A)+V(1+A)2-2A

2A

0.5,

-, if A > 0,

otherwise A = 0.

However, the 25th and 75th percentile for the random variable X is obtained as

with

with

Q1 = a

log 1 +

(-T-1) log(1 - 0(0.25))

a+^Vöw-!, if a < 0,

0(u)= W^/pEA, if A > 0,

2A ' '

Q3 = a

0.25,

log 1 +

otherwise A = 0.

(-T-1) log(1 - 0(0.75))

0(u) = <

, if A < 0,

(i+a)+^/ÄW-3A , if A > 0,

0.75,

otherwise A = 0.

(18)

(19)

(20)

4.2. Reversed Hazard Function and Odds Functions

The reversed hazard function (rhf) of the TWFr distribution is given as

rhfjwFr (x) =Tvßaßx ß 1 exp

a\ß

-v( X)

x

1 - exp

x exp[ -tv jexp

(a )ß

x

1

1 - A + 2Aexpy-TV 1 exp

' ( —tvI t

a\ß

x 11 + exp^-t jexp - Af expi -2t< exp

a\ß

Q

a>

<x>

1

-(^) x

1

-v- 1

<x>

1

A-1 1

(21)

ß

V

ß

V

ß

V

x

V

V

<x>

a )p

( a )-

—V

-n in-A

The Odd function that corresponds to the TWFr distribution is given as 0TWFr (x) =1 ^ exp^-t| exp + A^ exp^—2t| exp

4.3. Distribution of the Order Statistics

1

(22)

Let X1, X2, ■ ■ ■ , X n be a random sample of size n from the fjwFr (x) distribution and X(i), X(2), ■ ■ ■ , X(n)be the corresponding order statistics, Then, probability density function of the kth order statistics Xk, say fk(x) is given as

fk (x)

n!

(k — 1)!( n — k) !

—œ < x < œ. On substituting into Equation (23), we have

n!

k—1 n—k

] FTWFr (x) fTWFr (x) ] 1 — FTWFr (x) (23)

fk (x) —

(k — 1)!( n — k) !

x TVpaPx—p—1exp

x exp[ —t<exp

a)P

—v( ^ x

1 — exp

-( a ]}

V1

( ^ ) x

—1

1 — A + 2Aexp^—t j exp 1 + — exp(—T (exp

A

1 — exp^—t j exp

a )P

Q

a )p

(I )"

1

1

1

(24)

2 ]

1 — 1 + A) 1 — exp I — r< exp

a )P

<i>

A

1 — exp I —t< exp

a )P

<i>

1

2]

k—1

—1

n—k

The minimum and maximum order statistics are obtained when k = 1 and k = n respectively.

4.4. Simulation Study

A simulation is performed to examine the flexibility and efficiency of the TWFr distribution.

Tables i and 2 show the simulation results for different values of parameters. The simulation was performed as follows:

• Data were generated using

xu — a

log 1 +

(—T-1 ) log(1 — $(u))

0 < u < 1.

1

V

V

x

x

V

V

x

P

V

with

(1+A)—V(1+A)2—4Au, if A < 0 2

0(M) = < (1+A) + V(A+A)2—4A», ifA > 0

U,

otherwise = 0.

• The values of the parameters were purportedly set as a = 1.5, p = 0.5,t = 1.0, v = 1.0 and A = 0.5, a = 1.5, p = 0.5,t = 1.0, v = 1.0 and A = —0.5.

• The sample sizes were taken as n = 10,50,150,350,400 and 500.

• Each sample size was replicated 5000 times.

In the simulation study, we investigated the mean estimates (MEs), variance, biases and means

squared errors (MSEs) of the MLEs.

The bias was calculated as (for S = a, A, p, t, v)

1 5000 / \

Biass = ^^ £ [Si — S .

5000

1

Also, the MSE was obtained as

M SE

s =

5000 / \ 2

E s^i — s

5000 = V i=1 ^

In Tables 1 and 2, the results of the Monte Carlo study show that the MSEs decay towards zero as the sample size increases which corroborates with the first-order asymptotic theory. The mean estimates of the TWFr distribution parameter estimates tend to the true parameter values as the sample size increases which also corroborates the fact that the asymptotic normal distribution provides an adequate approximation of the estimates.

5. Real-Life Applications

A breaking stress of carbon fibers and glass fiber real life datasets were used to examine the performance and flexibility of the model based on its test statistic. Several criteria were used to determine the distribution of the best fit: Akaike Information Criteria (AIC), Consistent Akaike Information Criteria (CAIC), Bayesian Information Criteria (BIC), and Hannan and Quinn Information Criteria (HQIC).

1

Table 1 Simulation results for mean estimates, biases and root mean squared errors of T, v, a, X > 0 and ji for the TWFr distribution.

n Parameter ME Bias Variance MSE

10 i 1.4627 -0.0373 0.1504 0.1518

t 0.6653 0.1653 0.0738 0.1011

T 1.0968 0.0968 0.1034 0.1128

V 0.9871 -0.0129 0.2505 0.2507

X 0.3617 -0.1383 0.1493 0.1684

50 i 1.5383 0.0383 0.0618 0.0632

t 0.5326 0.0326 0.0167 0.0177

T 1.0924 0.0924 0.0485 0.0571

V 1.0037 0.0037 0.0924 0.0924

X 0.4170 -0.0830 0.0690 0.0759

150 i 1.5540 0.0540 0.0279 0.0308

t 0.5068 0.0068 0.0050 0.0050

T 1.0529 0.0529 0.0240 0.0268

V 1.0012 0.0012 0.0322 0.0322

X 0.4742 -0.0258 0.0417 0.0423

350 i 1.5528 0.0528 0.0142 0.0170

t 0.4983 -0.0017 0.0019 0.0019

T 1.0367 0.0367 0.0116 0.0129

V 1.0063 0.0063 0.0137 0.0137

X 0.4960 -0.0040 0.0234 0.0234

400 i 1.5539 0.0539 0.0134 0.0163

t 0.4975 -0.0025 0.0016 0.0016

T 1.0334 0.0334 0.0102 0.0113

V 1.0072 0.0072 0.0115 0.0116

X 0.5001 0.0001 0.0205 0.0205

500 i 1.5500 0.0500 0.0104 0.0129

t 0.4968 -0.0032 0.0013 0.0013

T 1.0276 0.0276 0.0079 0.0087

V 1.0071 0.0071 0.0091 0.0091

X 0.5058 0.0058 0.0176 0.0176

The density functions considered include (for x > 0)

• Weibull Frechet: f (x) = abfiaPx-t-1exp -b(a x exp ^-ai^exp[(a)t] - ij ^;

• Exponentiated Frechect: f (x) = AfiaPx-t-1exp

• Kumaraswamy Frechet: f (x) = abtatx-t-1exp

• Beta Frechet: f (x) = bt) exp

• Gamma Extended Frechet: f (x) = "^ty)— exp

1 - exp

-(x)

-b-1

-(a )t

-a( a )t

-a( x )t

1 - exp

1 - exp

1 - exp b1

-(x )t

-(x)

X-1

b1

-(x)

-(x )t

1 - exp

-(x )t

a-1

(-log{1 -

Table 2 Simulation results for mean estimates, biases and root mean squared errors of t, v, a, A < 0 and p for the TWFr distribution.

n Parameter ME Bias Variance MSE

10 a 1.5887 0.0887 0.1859 0.1937

p 0.6626 0.1626 0.0532 0.0796

t 0.9814 -0.0186 0.1996 0.1999

v 0.9641 -0.0359 0.1531 0.1544

A -0.5970 -0.0970 0.7110 0.7204

50 a 1.5604 0.0604 0.0658 0.0694

p 0.5448 0.0448 0.0164 0.0184

t 0.9843 -0.0157 0.0538 0.0540

v 1.0009 0.0009 0.0683 0.0683

A -0.4541 0.0459 0.0995 0.1016

150 a 1.5587 0.0587 0.0284 0.0318

p 0.5117 0.0117 0.0060 0.0062

t 1.0013 0.0013 0.0224 0.0224

v 1.0134 0.0134 0.0272 0.0274

A -0.4521 0.0479 0.0363 0.0386

350 a 1.5561 0.0561 0.0146 0.0178

p 0.5010 0.0010 0.0024 0.0024

t 1.0100 0.0100 0.0100 0.0101

v 1.0158 0.0158 0.0114 0.0116

A -0.4636 0.0364 0.0168 0.0181

400 a 1.5582 0.0582 0.0130 0.0164

p 0.4998 -0.0002 0.0021 0.0021

t 1.0110 0.0110 0.0089 0.0091

V 1.0166 0.0166 0.0100 0.0103

A -0.4642 0.0358 0.0160 0.0173

500 a 1.5563 0.0563 0.0102 0.0134

p 0.4991 -0.0009 0.0017 0.0017

t 1.0109 0.0109 0.0074 0.0075

V 1.0159 0.0159 0.0078 0.0081

A -0.4650 0.0350 0.0129 0.0141

ax b-1

exp —(a)p Transmuted Frechet: f (x) = papx—p—1 exp Frechet: f (x) = AaAx—A—1exp

—(a )p

—(a )A

1 + A — 2 Aexp

—(a )p

• Alpha Power Inverse Weibull: f (x) = (j—)Apexp(—Ax—p)aexp(—Ax p);

• Transmuted Rayleigh: f (x) = exp ^ — — p + 2pexp(^ —

5.1. Breaking Stress of Carbon fibres

The first data consist of 100 breaking stress of carbon fibres as used in [19] and [5]. It consists of 100 observations taken on breaking stress of carbon fibers (in Gba). Table 3 shows the test statistics. The dataset are as follow:

2.81, 2.77, 2.17, 2.83, 1.92, 1.41, ,1.57, 0.81, 5.56, 1.73, 1.59, 2, 1.22, 1.12, 1.71, 3.11,4.42, 2.41, 3.19, 3.22,1.69, 3.28, 3.09, 1.87, 3.15, 4.9, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.53,2.67, 2.93, 3.22, 3.39, 2.81, 4.2, 3.33, 2.55, 3.31, 3.31, 2.85, 2.56, 3.56, 3.15, 2.35, 2.55, 2.59,2.38, 2.17, 1.17, 5.08, 2.48,1.18, 3.51, 2.17, 1.69,1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 4.7, 2.03, 1.8, 1.57,1.08, 2.03, 1.61, 2.12,1.89, 2.88, 3.68, 2.97, 1.36,3.7, 2.74, 2.73, 2.5, 3.6, 3.11, 3.27, 2.87, 1.47, 0.98, 2.76, 4.91, 3.68, 1.84, 1.59, 3.19 2.82, 2.05, 3.65.

Table 3 Performance rating of the TWFr distribution with breaking stress of carbon fibres dataset

Model Parameter MLEs(Std. Errors) AIC CAIC BIC HQIC -21

Transmuted Weibull Frechet T = 86.6226(41.4818) v = 0.4167(0.2361) a = 3.7977(1.3096) X = 0.6301(0.1704) t = 0.4463(0.1260) 292.0566 292.4776 302.4773 296.2740 282.0566

Weibull Frechet a = 0.6942(0.363) t = 0.6178(0.284) a = 0.0947(0.456) b = 3.5178(2.942) 294.6000 295.0211 305.0207 298.8174 286.6000

Exponentiated Frechect i = 69.1489(57.349) t = 0.5019(0.0800) X = 145.3275(122.824) 295.7000 295.8237 300.9103 297.8087 291.7000

Kumaraswamy Frechet i = 2.0556(0.0710) t = 0.4654(0.0070) a = 6.2815(0.0630) b = 224.1800(0.1640) 297.1000 297.5211 307.5207 301.3174 289.1000

Beta Frechet i = 1.6097(2.4980) t = 0.4046(0.1080) a = 22.0143(21.432) b = 29.7617(17.4790) 311.1000 311.5211 321.5207 315.3174 303.1000

Gamma Extended Frechet i = 1.3692(1.3692) t = 0.4776(0.1330) a = 27.6452(14.1360) b = 17.4581(14.8180) 312.0000 312.4211 322.4207 316.2174 304.0000

Alpha Power Inverted Weibull i = 109.8227(75.5562) t = 1.1138(0.2018) X = 2.2803(0.1420) 328.4842 328.7342 336.2997 331.6473 322.4842

Transmuted Frechet i = 1.9315(0.0971) t = 1.7435(0.0760) X = 0.0819(0.1980) 350.5000 350.7500 358.3155 353.6631 344.5

Frechet i = 1.8705(0.1120) X = 1.7766(0.113) 348.3000 348.4237 353.5103 350.4087 344.3000

Figures 3 and 4 show the empirical pdf and cdf for the breaking stress of carbon for the TWFr model.

5.2. Glass fibres data

The second data consist of 1.5 cm strengths of glass fibres obtained at the UK National Physical Laboratory. The data were used to compare the performance of the TWFr distribution as used in [25], [11], [3], [15], [24], [21], [5], [6], [4], [7], [22], [8], [9], [10], [20] and [28]. The observations are as follows:

1.53, 1.54,1.55, 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, 1.49,1.61, 1.58, 1.59,1.60, 1.61,0.55, 0.74,1.50, 1.50,1.55, 1.52,1.64, 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, 1.63, 1.61, 1.61, 1.62, 1.62, 1.67,.

WFr

TWFr

APIW

EFr

KFr

BFr

TFr

0

2

3

4

5

6

x

Figure 3 The empirical cdfs of the TWFr density for the breaking stress of carbon

Figure 4 The empirical pdfs of the TWFr density for the breaking stress of carbon

The descriptive statistics of the glass fibers dataset are showed in Table 4. Table 5 shows the measure of comparison for the various distribution under consideration.

Table 4 Descriptive statistics for the glass fibres dataset to 2 decimal points

Mean Median Mode StD IQR Variance Skewness Kurtosis 25th P. 75th P. 1.51 1.59 1.61 0.32 0.31 0.11 -0.81 0.80 1.38 1.69

Figures 5 and 6 show the empirical pdf and cdf for the glass fiber data for the TWFr model.

Table 5 Performance rating of the TWFr distribution with glass fibres dataset

Model Parameter MLEs(Std. Errors) AIC CAIC BIC HQIC -21

T = 0.5739(0.1877) v = 3.9560(2.2162)

Transmuted Weibull Frechet  = 0.0300(0.0135) a = 5.8185(4.2645) $ = 1.2345(0.1734) 35.7214 36.7740 46.4370 39.9359 25.7214

Weibull Frechet a = 0.3865(0.7990) $ = 0.2436(0.2850) a = 1.4762(4.782) b = 16.8561(20.4850) 39.0000 39.6896 47.5725 42.3716 31.0000

Beta Frechet a = 2.0518(0.9886) $ = 0.6466(0.1630) a = 15.0756(12.057) b = 36.9397(22.649) 68.6261 69.3157 77.1986 71.9977 69.300

Gamma Extended Frechet a = 1.6625(0.9520) $ = 0.7421(0.197) a = 32.1120(17.3970) b = 13.2688(9.967) 69.6237 70.3016 78.1098 72.9007 61.4503

a = 61.10(48.14)

Alpha Power Inverted Weibull $Â = 0.78(0.16) Â = 3.80(0.30) 82.5800 82.9900 89.0100 85.1100 76.5848

Frechet a = 1.2640(0.0589) Â = 2.8879(0.2340) 97.7105 97.9045 102.0078 99.3560 93.6980

a = 1.3068(0.034)

Transmuted Frechet $Â = 2.7898(0.1648) Â = 0.1298(0.2080) 100.1009 100.5078 106.4897 102.5908 94.0893

a = 1.0895(1.1e - 08)

Transmuted Rayleigh $ = 1.0e - 10(1.7e - 12) 103.5818 103.7820 107.8680 105.2676 99.5818

a = 83.4497(79.2814)

Alpha Power Inverted Exponential  = 0.3137(0.0774) 196.3253 196.5253 200.6116 198.0111 191.4580

x

Figure 5 The empirical cdfs of the TWFr density for the glass fiber data

5.3. Discussion

The performance of a model is determined by the value that corresponds to the lowest Akaike Information Criteria (AIC) or the highest Log-likelihood value is regarded as the best model. In

o

o o

ecdf APIW

Tfr WFr

TWFr

Efr Fr KFr

0.5

1.0

1.5

2.0

x

Figure 6 The empirical pdfs of the TWFr density for the glass fiber data

the two real life cases considered, the TWFr distribution has the lowest AIC values.

6. Conclusion

The Transmuted Weibull Frechet distribution has been successfully derived. Its expressions for the basic statistical properties which include the order statistics distribution, cumulative hazard function, reversed hazard function, quantile, median, hazard function, odds function have been successfully established. The shape of the distribution could be increasing (depending on the value of the parameters). An application of real life data shows that the TWFr distribution is a better competitor for some other families of distributions.

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J. T. Eghwerido RT&A, No 4 (71)

TWFr Model Volume 17, December 2022

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