COSINE MARSHAL-OLKIN-G FAMILY OF DISTRIBUTION: PROPERTIES AND APPLICATIONS Текст научной статьи по специальности «Прочие естественные и точные науки»

COSINE MARSHAL-OLKIN-G FAMILY OF DISTRIBUTION: PROPERTIES AND APPLICATIONS

Akeem Ajibola Adepoju1, Alhaji Modu Isa2, Olalekan Akanji Bello3

department of Statistics, Faculty of Computing and Mathematical Sciences, Aliko Dangote University of Sceince and Technology, Wudil, 713281, Wudil, Kano. Nigeria.

2Department of Mathematics and Computer Science, Borno State University, Maiduguri, Nigeria.

3Department of Statistics, Ahmadu Bello University, Zaria, Nigeria. Email: [email protected] ; [email protected] ; [email protected]

Abstract

Trigonometric distributions have recently been emphasized due to it applicability and relevance for modeling different phenomena. This article contributes to the existing literature on trigonometric family by introducing and investigating new trigonometric family of distribution which is developed by compounding the cosine family of distribution with Marshall-olkin family of distribution to form a new Cosine Marshall-Olkin family of distribution (CMO). Graphical, numerical and analytical approach was explored to study the properties and applicability of the new CMO family of distribution. Special representations and important reliability properties and other statistical properties were defined. Simulation study was conducted in order to have an insight on the estimates of the three parameters model using maximum products of spacing (MPS). Emphases on the greater flexibility of the new CMO family of distribution beyond the cosine-G family and other top models of the Cosine related family was made through Weibull distribution. The results revealed the superiority of the Cosine Marshall-Olkin Weibull model (CMO-W) over others via two data sets.

Keywords: Cosine-G family, Marshall-Olkin-G family, Maximum Products of Spacing, Hazard function, Survival function.

I. Introduction

Recently, many authors have introduced various approaches to develop flexible continuous distributions from classical continuous distributions. The statisticians' attentions have been drawn to various applications of these continuous distributions in environment, physics, medicine, biology, finance, insurance, engineering and economy to mention few. The classical distributions are induced by adding parameter(s) to enhance the asymmetry, kurtosis, tails properties, central and dispersion parameters. This idea is considered as generalization of the classical distributions. These generalized distributions belong to particular families defined by transformation of the baseline cumulative distribution function (cdf). The values of the newly introduced parameter(s) can enhance the statistical capacities of the baseline distribution. for instance, families such as Weibull-G [1], Exp-G [2], Topp-Leone generated (TL-G) [3] Type I Half Logistic-G [4], new power

TL-G [5], Type II half Logistic-G [6], truncated inverted Kumaraswamy-G [7], a new alpha power transformed-G [8], a new extended alpha power transformed-G [9], type II power TL-G [10], Odd Beta prime-G [11].

A recent approach involves defining families of distributions by using the trigonometric transformation, be it parametric or not. Kumar et al. [12] and Souza [13] launched this trigonometric family exploring the use of the sine function, resulting to the sine-G family. The [14] and [15] extended the exponential and weibull distribution through sine-G family. The non trigonometric compounding families of distributions seen in the literature include but not limited to [16], [17], it extension is found in [18], [19]. The trigonometric compounded families include [20], [21], [22], [23], [24], [25] [26], [27], [28], [29], [30], [31], [32], [33].

The Marshall-olkin-G family of distribution was proposed by [34] and it was used to extended flexibility of Exponential and weibull distribution The Cosine -G family of distribution was proposed by [35]. Now, this article intends to compound the two families to form a new family of distribution called Cosine Marshall-olkin-G family of distribution.

The motivations behind CMO-G family are to develop models with improved shapes for the pdf and hazard function, improve symmetrical and asymmetrical distributions, construct heavy-tailed distributions, improve the flexibility of the baseline model through skewness, kurtosis, mean and variance, provide better fits than other Cosine family of distribution with the same baseline distribution and possibly with the same number of parameters and more complexity.

II. Methods

2.1 The Marshal-Olkin-G Family of Distribution

Definition 1: Suppose X ~ MO{x\6,£} with corresponding cdf and pdf given by:

(1)

and

eg(x;t)

M1 ~0)GMf

(2)

2.2 The Cosine-G Family of Probability Distribution

Definition 2: Suppose X ~ COS(x\KV} with corresponding cdf and pdf given by: F(x;¥) = 1 - cos

(3)

and

f (x;¥) = ^h(x)sin ^H(x)

n

n

(4)

2.3 The proposed Cosine Marshal Olkin-G family of distribution

Definition 3: Suppose X ~CMO(x-,6,%} with cdf expressed below, where 0 > 0 and 6 is a shape parameter and % is a baseline vector parameter is defined as the Cosine Marshal-Olkin-G Family

fcmo m = 1 -

cos

G(x)

в + (1 -в)й[х)

(5)

It is important to note that for any baseline distribution, signified as G^x), CMO cdf satisfy the following;

a.

dG{x)

b. }g(x)dx = 1

0

c. The survival function 1 -G(x)

Definition 4: Suppose X ~ CMO{x\6,£} with pdf expressed below, where 0 > 0 and 6 is a shape parameter and % is a baseline vector parameter is defined as the Cosine Marshal-Olkin-G Family

fCMO M =

Og{x)

(в + (1 -0)G(x))2

-sin

G(x)

в + (1-в) g(x)

(6)

2.4 Special Representation

The pdf of the proposed Cosine Marshall-olkinG family can be expanded using the tailor series and binomial expansion; thus

cmo (xe) =

eg(x)

sin

2 (в+(1 -в)G(x))' G(x)

G(x)

в + (1 -в) G(x)

в + (1 -в) G[x)

(-1)'

= 5 (27^ F+rG(x)2' (в+(1 -в G(x) Г

Consider

(в + (1 -e)G(x)y and (в+(1 -e)G(x))

(в + (1 -в)G(x))-2('+1) =5 (-1)

» (-1)'+' Co M= 5 (27+1)! 2

'=0

n

2(' +1) -1

+ '

Л

(1 -в) 'G(x)'7

2(' +1) -1 + j

'

в'1 (1 -в) 'g(x)G(x)

у

Hence the expansion of the pdf is expressed as

2 j

ij-0

Where

V. =

(-1)'

n

(2i +1)! 22

2(i +1)-1 + j j

e'j1 (1 -e)

The cdf can also be expanded as follows:

G(x) '

2 '

fcmo (xe)=1 -

cos

e+(1 -e)G(x)

»(-1) 1 -JV-4

k f 2k

n

G(x)

(2k)! 22k [? + (1 -e)G(x)

tt f — 1 V 2k

■1 - §|k nrG(x>" (e+(1 -G))

' 2(k +1)-1 +/'

(e+(1 -e)G(x)) -J? (-1)1

(1 -?'G(x)'

(7)

fcmo (x;?-1 -J

I \k (-1)

/■o (2k)! 22k Therefore,

tt

fcmo (x;e,£)-1 - j %xxf

Where

2(k +1)-1 + / 1

' / \2k+'

e'(1 -e) g(x)

kj-0

o ■

kj

I \k (-1)

n

(2k)! 22

2(k +1)-1 + / 1

e' (1 -e)'

(8)

Definition 5: Suppose X ~CMO{x\6,£} with cdf and pdf well defined, where 0 > 0 and is a shape parameter and £ is a baseline vector parameter. Then the survival function of X, signified by SFCMO (x?,£j-1 - FCMO (x;?), the survival function for the CMO family of distribution, can be

G(x)

represented by SFCM0 (x;?,£) ■ cos

e+(1 -e)G(x)

Definition 6: Suppose X~CMO(x;0,£} with cdf and pdf well defined, where 0> Oand

e is a shape parameter and £ is a baseline vector parameter. Then the hazard rate function of X, signified by HRFCMO (x?,£)■ fCMO (x;?,£) / SFCMO (x?), the hazard rate function for the CMO family of distribution, can be represented by

hrfcmo (x;e,£) —

?g(x)

(e+(1 -e)G(x))2

-tan

G(x)

e+(1 -?)g{x)

(10)

/■0

71

Definition 7: Suppose X ~ CMO{x\9,E^ with cdf and pdf well defined, where 0 > 0 and

0 is a shape parameter and £ is a baseline vector parameter. Then the Qunatile function of X, signified by

QFCMO (x\O,£) = F~m0 (x;O,£, the Qunatile function for the CMO family of distribution can be obtained as follows:

u = 1 - cos

ф( u) = G-

G(x)

в + (1-в) G(x)

, u e (0,l),

/'cos

1 (1 -u)

(11)

J-(1 -O)Ocos-1 (1 -u)

Definition 8: Suppose X~ CMO{x\9 with cdf and pdf well defined, where 6 > 0 and ^ is a

shape parameter and £ is a baseline vector parameter. Then the rt Moments ofX can be obtained as follow

Mr = j Xf(x)dx

i,J=0

»r = Z ^

i,j=0

where

(12)

* = \ye (x)G(x)*dx

Definition 9: Suppose X ~ CMO(x\9,c} with cdf and pdf well defined, where 9 > 0 and ^ is a

shape parameter and £ is a baseline vector parameter. The rt Moment generating function of X is obtained through

Mx t) = EM = fyxf (x)dx

Thus, the moment generating function of the Cosine Marshall-olkin-G family of distribution is given by:

Mx (t) =ZУ*(*Ш"<**

iJ=0

Mx (t)=ZXV,Y

(13)

iJ=0

where

Y = \y g(x)G(xf dx

Definition 10: Suppose X ~ CMO[x\9,S^ with cdf and pdf well defined, where 9> 0 and ^ is a shape parameter and £ is a baseline vector parameter. The entropy is obtained as given below

rblog i f (x)v

f (х)в={т A*)G{*V

\ij=0

1

в

f W- I ( g(*M4" )

Let c -g(x^G(x)j Therefore,

2"

(

y

f (x) - I "j

V1"

!'(x)-I10

Vi"-0

(

I". \]cedx

V a-00 )

(14)

2.5 Cosine Marshall-olkinWeibull Distribution

e

Supposed the baseline distribution is Weibull distribution with cdf and pdf given by:

G(x)■ 1 -e Uj (15)

^■i^CM (16)

Where a is a shape parameter and A is a scale parameter, then the cumulative distribution, probability distribution, hazard and survival function of the Cosine Marshall-olkinWeibull

! \

(CMO-W) distribution is given as: FCM0 (x;0*) -1 -

cos

1 -e

0 + (1-0)

1 -e

and

fCMO (x;0)-

the

n0a\ x i i> 1 e u 2 2(2'

associated

pdf

is

given

(17)

as:

-sin

(

0 + (l-0)

1 -e

1 -e

0 + (l-0)

1-e

\\

a o

— a=0.5 9=0.4 /.= 1.2

- a= 1.2 9=0.6/.= 1.4

■■■■ a= 1.1 9=0.3 /.= 1.2

a=0.3 9=0.5/.= 0.7

/ ' \ '

/ / / \

/ J ! Y \ ' \ ^

/ / / '■' ' \ \\ ■ V

/

— 01=0.5 e=0.4 /.= 1.2 - a=1.2 9=0.6/.= 1.4 ■■■■ oi=1.1 9=0.3 /.= 1.2 oi=0.3 9=0.5/.= 0.7

:[(• \ \ : ' vV 7 '' 'V / ■ \

i.' \ 7 " \ / i \\

;l \ if • ^ X ( '■■ v r V

0 2

— a=1.5 8=2.7 ¡1=1.5

-- <x = 1.7 6=3 1=1.9

a = 2 6=3.2 A=2.4

•-• a=2.3 9=3.5 1=2.7

/A>, / ' V / 1 - A \ ' \

/ ' : > \ i / . - \ \ \\

i : ! ' 1 / ! : ! \ \ \ \ \ ^ V\

\ S ' \ \ ^

/ ' ' \ N

ii ' / ' ■ \ \ ''

j / / / / y // . ■

1 r 0 1 2

I I-1 T

3 4 5 6

/ j / l t _ j > ■_ " . * X / / / / . . ! ■■ ' ' / • / /

/ / / / / i .' / : / i

/ ' / ' i i

/ / : i / / / j j f

/ ' ' / / •

/ / <' '

/ > ' / / / / / / / / / — a = 1.5 6=2.7 1=1.5 -- « = 1.7 6=3 1=1.9 a=2 6=3.2 1=2.4 a=2.3 6=3.5 1=2.7

i i I i 0 12 3 l 1 l 4 5 6

X

Figure 1: Plots of pdf and cdf of CMO-W distribution

The figure 1 above reveals left skewness, right skewness and approximately symmetric pdf shapes. The cdf shape converges to one, validating the CMO-W distribution.

The Hazard and reliability function of the CMO-W distribution is obtained as:

жв

HRFCM0 (x;e,£) = -

( / ча-l (^

a{ x ) Л[ Л'

v

/

в + (1 -в)

1 -e

W

//

-tan

(

1-e

в + (1 -в)

1 -e

(19)

and

sfcmo (*;в,£) =cos

( (

1 -e

\ \

в + (1 -в)

1 -e

(20)

Figure 2: Plots of hazard and reliability function of CMO-W distribution

The figure 2 above reveals the shapes of the hazard and reliability function the hazard shapes obviously shows increasing and decreasing failure rate, and the reliability shapes shows a drop from one to zero with varying values of parameters

III. Results

3.1 Simulation study

In this section, we provide, we provides the simulation of parameters of the CMO-W distribution using Maximum products of spacing estimation method. Random numbers were systematically generated from fixed values of the parameters 0 = 0.5,2 = 2,a = 1 0 = 0.7,2 = 2.2,a = 1 and

0 = 0.6,2 = 2.3,a = 1 and 0 = 0.8,2 = 2.1,a = 1 based on 10,000 replications. The sample sizes (n) considered are 20, 50, 100, 250, 500 and1000. The result is displayed in Table 1 and Table 2

Table 1: The MPSs parameter estimates (Est. value), Biases and RMSEs of various parameters values

n Parameters Est. value Bias RMSE Est. value Bias MSE

20 e 0.6195 0.1195 0.4675 0.8346 0.1346 0.5855

X 2.1325 0.1325 0.8886 2.3199 0.1199 0.9108

a 0.9536 -0.0464 0.2357 0.9563 -0.0437 0.2327

50 e 0.5984 0.0984 0.3570 0.8354 0.1354 0.5037

X 2.0474 0.0474 0.7208 2.2352 0.0352 0.7881

a 0.9581 -0.0419 0.1652 0.9608 -0.0392 0.1722

100 e 0.5719 0.0719 0.2906 0.7981 0.0981 0.3977

X 2.0375 0.0375 0.6079 2.2232 0.0232 0.6503

a 0.9703 -0.0297 0.1247 0.9712 -0.0288 0.1281

250 e 0.5572 0.0572 0.2226 0.7788 0.0788 0.3004

X 2.0014 0.0014 0.4941 2.2114 0.0114 0.5370

a 0.9786 -0.0214 0.0970 0.9787 -0.0213 0.1010

500 e 0.5318 0.0318 0.1674 0.7592 0.0592 0.2422

X 2.0092 0.0092 0.3890 2.2032 0.0032 0.4317

a 0.9873 -0.0127 0.0745 0.9832 -0.0168 0.0801

1000 e 0.5197 0.0197 0.1254 0.7326 0.0326 0.1810

X 2.0041 0.0041 0.2914 2.2011 0.0010 0.3362

a 0.9918 -0.0082 0.0547 0.9904 -0.0096 0.0608

Table 2: The MPSs parameter estimates (Est. value), Biases and RMSEs of various parameters values

n Parameters Est. value Bias RMSE Est. value Bias RMSE

20 e 0.7259 0.1259 0.5302 0.9459 0.1459 0.6535

X 2.4416 0.1416 0.9796 2.2125 0.1125 0.8606

a 0.9567 -0.0433 0.2366 0.9607 -0.0393 0.2402

50 e 0.7174 0.1174 0.4218 0.9525 0.1525 0.5519

X 2.3376 0.0376 0.8233 2.1128 0.0128 0.7333

a 0.9593 -0.0407 0.1690 0.9588 -0.0412 0.1711

100 e 0.6863 0.0863 0.3426 0.9240 0.0981 0.1240

X 2.3231 0.0231 0.6725 2.1063 0.0232 0.0063

a 0.9703 -0.0297 0.1253 0.9698 -0.0288 -0.0302

250 e 0.0722 0.0722 0.0722 0.9008 0.1008 0.3486

X -0.0181 -0.0181 -0.0181 2.0640 -0.0360 - 0.4893

a -0.0226 -0.0226 -0.0226 0.9752 0.0248 0.1006

500 e 0.6444 0.0444 0.2047 0.8628 0.0628 0.2716

X 2.2928 -0.0072 0.4473 2.0824 -0.0176 0.4125

a 0.9851 -0.0149 0.0772 0.9840 -0.0160 0.0808

1000 e 0.6300 0.0300 0.1558 0.8433 0.0433 0.2073

X 2.2845 -0.0155 0.3418 2.0783 -0.0217 0.3206

a 0.9897 -0.0103 0.0585 0.9886 -0.0114 0.0624

3.2 Applications

Application of the CMO-W distribution to two real life data sets are provided and revealing it applicability in practice along with comparison with its comparators. The proposed Cosine Marshall-olkin-Weibull distribution (CMO-W) is compared with four other Cosine extended Weibull distributions, namely: Cosine Topp-Leone Weibull (CTL-W) distribution [36], Extended Cosine Weibull (ECS-W) distribution [37], New Alpha Power Cosine-Weibull (NACos-W) distribution [38] and Cosine Weibull (C-W) distribution [39].

The information criteria explored to investigate the goodness-of- fit of the distribution appropriate for the data are Akaike's Information Criterion (AIC), Consistent Akaike's Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn Information Criterion (HQIC). The computation can be seen as follows A\C = -2£ + 2p

CAIC = —21 + 2UP n-p-1

BIC = -2£ + plog(fl)

HQIC = -2 £ + 2plog(logO)),

where t is the maximized log likelihood of the parameter vector Q = [0,X,a), p is the number of estimated parameters and n is the number of observations. The best fitted model is selected based on minimum value obtained through the information criteria measures.

Dataset 1:

"The data set shown below represents the civil engineering data with 85 hailing times, previously used by Kotz and Dorp (2004):"

4.79, 4.75, 5.40, 4.70, 6.50, 5.30, 6.00, 5.90, 4.80, 6.70, 6.00, 4.95, 7.90, 5.40, 3.50, 4.54, 6.90, 5.80, 5.40, 5.70, 8.00, 5.40, 5.60, 7.50, 7.00, 4.60, 3.20, 3.90, 5.90, 3.40, 5.20, 5.90, 4.40, 5.20, 7.40, 5.70, 6.00, 3.60, 6.20, 5.70, 5.80, 5.90, 6.00, 5.15, 6.00, 4.82, 5.90, 6.00, 7.30, 7.10, 4.73, 5.90, 3.60, 6.30, 7.00, 5.10, 6.00, 6.60, 4.40, 6.80, 5.60, 5.90, 5.90, 8.60, 6.00, 5.80, 5.40, 6.50, 4.80, 6.40, 4.15, 4.90, 6.50, 8.20, 7.00, 8.50, 5.90, 4.40, 5.80, 4.30, 5.10, 5.90, 4.70, 3.50,

6.80.

Figure 3: The boxplot and kernel density of the data set 1

Dataset 2:

"The data set shown below represents the strength of carbon fibers tested under tension at gauge lengths of 10mm, previously used Bi and Gui (2017):"

1.901, 2.132, 2.203, 2.228, 2.257, 2.350, 2.361, 2.396, 2.397, 2.445, 2.454, 2.474, 2.518, 2.522, 2.525, 2.532, 2.575, 2.614, 2.616, 2.618, 2.624, 2.659, 2.675, 2.738, 2.740, 2.856, 2.917, 2.928, 2.937, 2.937, 2.977, 2.996, 3.030, 3.125, 3.139, 3.145, 3.220, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332, 3.346, 3.377, 3.408, 3.435, 3.493, 3.501, 3.537, 3.554, 3.562, 3.628, 3.852, 3.871, 3.886, 3.971, 4.024, 4.027, 4.225, 4.395, 5.020.

o

1 2 3 4 5 N = 63 Bandwidth = 0.244 Figure 4: The boxplot and kernel density of the data set 2

i/n i/n

Figure 5: The TTT plot of data set 1 and 2

Table 3: MPSs, Log-likelihoods and Goodness of Fits Statistics for the Data Set 1

Distributions 2 a e P LL AIC CAIC BIC HQIC

CMO-W 3.9229 2.2488 2.8840 -110.2788 226.5576 226.8539 233.8856 229.5051

CTL-W 0.0026 1.3745 3.2049 -132.9372 271.8744 272.1707 279.2024 274.8219

ECS-W 3.8988 0.0382 0.7231 -258.3757 522.7514 523.0477 530.0794 525.6989

NACos-W 4.8815 3.1622 0.0026 -193.4512 392.9024 393.1987 400.2304 395.8499

C-W 2.8953 0.0094 0.1471 -138.7963 283.5926 283.8889 290.9206 286.5401

Table 4: MPSs, Log-likelihoods and Goodness of Fits Statistics for the Data Set 2

Distributions 2 a e P LL AIC CAIC BIC HQIC

MO-W 4.4779 6.1830 0.0415 -82.1587 170.3174 170.7242 176.7468 172.8461

CTL-W 0.3398 12.5851 1.4887 -86.6096 179.2192 179.6260 185.6486 181.7479

ECS-W 0.0027 0.9870 4.4644 -84.5562 175.1124 175.5192 181.5418 177.6411

NACos-W 8.1270 0.0128 2.6986 -85.2837 176.5674 176.9742 182.9968 179.0961

C-W 0.5119 6.9721 0.0020 -86.0652 178.1304 178.5372 184.5598 180.6591

IV. Discussion

We introduce a novel Cosine Marshall-Olkin family of distribution and its properties, therein, we extended the Weibull distribution to form a new sub-model known as Cosine Marshall-Olkin Weibull distribution. We conducted a comprehensive study of the new Cosine Marshall-Olkin Weibull distribution properties. Furthermore, we investigate the consistency and efficiency of the estimates obtained from the parameters of the novel distribution. We employ the maximum products of spacing estimation technique, which enabled us to access the values of the parameters effectively. To demonstrate the applicability of the proposed distribution, we provide insights on its performance using two real-life datasets. The analysis reveals that the new model outperforms other trigonometric family of distribution with the same baseline.

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