ZECH DISTRIBUTION: DERIVATION, PROPERTIES AND APPLICATIONS TO REAL LIFE DATA Текст научной статьи по специальности «Науки о Земле и смежные экологические науки»

ZECH DISTRIBUTION: DERIVATION, PROPERTIES AND APPLICATIONS TO REAL LIFE DATA

Sunday Adeyeye1, Ademola Adewara2, Emmanuela Akarawak3, Adeyinka Ogunsanya4,

Alabbasi Jamal5 •

1 3 Department of Mathematics, University of Lagos, Akoka, Nigeria.

2 Distance Learning Institute, University of Lagos, Akoka, Nigeria.

4 Department of Statistics, University of Ilorin, Ilorin, Nigeria.

5 Al - Nahrain University, Baghdad, Iraq.

E- Mail: 1 [email protected], 2 [email protected], 3 [email protected], 4 [email protected] 5 [email protected]

Abstract

The roles of heavy - tailed distribution in modelling real life events, especially in financial and actuarial sciences, cannot be over - emphasized. In this paper, a new heavy right - tailed, three - parameter continuous distribution with increasing hazard rate called Zech distribution is developed. The proposed model is very suitable for modelling heavy right- tailed data. Zech distribution is the reciprocal of the random variable which follows Gompertz- Inverse - Exponential (GoIE) distribution and it does not involve addition of extra parameter, thereby removing the cumbersomeness in the estimation process posed by other methods involving additional extra parameters, especially where more than three parameters are involved. The statistical properties of the new distribution such as survival function, hazard function, cumulative hazard function, reversed hazard function, quantile function, order statistics, moments, mean, median, variance, skewness, and kurtosis were derived. The Linear representation of the pdf of the newly developed distribution revealed that its probability density function is a weighted exponential distribution. Also, method of maximum likelihood was used in estimating the model's parameters. The simulation results revealed that as the sample sizes increased, the root mean squared errors decreased which showed that the parameters of Zech distribution are stable. The proposed distribution was applied to two real life data sets. The results showed that Zech distribution performs better than Gompertz Inverse Exponential distribution, Weibull Exponential distribution and Gompertz Exponential distribution.

Keywords: Zech distribution, Gompertz Inverse Exponential Distribution, maximum likelihood estimation, simulation studies, moments, linear representation.

1. Introduction

Probability distributions play a crucial role in modelling naturally occurring phenomena. In probability theory and statistics, an inverse distribution is the distribution of the reciprocal of a random variable. To model real life events, there is need for the extension of the classical forms of distributions so as to have a better fit to the real data. Several methods of extending distributions have been proposed in the literature. Among these is 'Inverse Distribution' which does not increase the number of parameter(s) of the parent distribution but provides a better fit. This is a strong motivation for studying inverse distribution as prescribed by the principle of parsimony. Eliwa [12] proposed Inverse Gompertz distribution which was found to out - perform other six competing distributions. The Gompertz Inverse Exponential distribution proposed by Pelumi [5] is good for

modelling right - tailed data. Said [13] introduced Extended Inverse Weibull distribution whose density function can be expressed as a linear combination of the Inverse Weibull densities with increasing and decreasing hazard rates. Ogunsanya [11] developed Weibull Inverse Rayleigh distribution which is an extension of a one - parameter Inverse Rayleigh distribution that incorporated a transformation of the Weibull distribution and Log - logistic distribution as quantile functions. El - Gohari A [2] proposed Generalized Gompertz distribution which is a new generalization of the Exponential, Gompertz and Generalized Exponential distributions. The main advantage of this new distribution is that it has increasing or constant or decreasing or bathtub curve failure rate depending upon the shape parameter. It is this property that makes it suitable for survival analysis. Adewara [3] introduced Gompertz Exponential distribution which is an extension of Exponential distribution by using the Gompertz Generalized family of distributions proposed by Morad [4]. To increase the flexibility of Gompertz Exponential distribution, an extra shape parameter was added to it leading to the introduction of Exponentiated Gompertz Exponential distribution by Adewara [8].

The motivation for this study is to derive a distribution which will be more flexible for modelling heavy right - tailed data and to obtain interesting properties of the new model. Therefore, the inverse of 'Gompertz Inverse Exponential distribution', which will henceforth be called Zech distribution is proposed. The adoption of the name 'Zech distribution' is to avoid the confusion which might arise from using the name: Inverse Gompertz Inverse Exponential Distribution.

Given the cumulative distribution function (cdf) of a random variable Y, the distribution function of a

1

random variable X = - is the reciprocal or inverse of the random variable Y. This implies that the cumulative distribution function G(x) is the inverse function of F(y). This is easier if Y is a continuous random variable and F(y) is strictly on positive supports. The cumulative distribution function of inverse distribution is derived according to the method below: Gx(x) = P(X < x)

= 1-p(x<d

= 1-f(t) (1)

The cumulative distribution function (cdf) and probability density function (pdf) of Gompertz Inverse Exponential distribution are given in equations (2) and (3) respectively.

I<1-

F(y) = 1-e

e --f(y) = a~y2e y

1-e y

;y>O,a>O,fi>O,0>O (2)

er^

l - e y

-fi-1 I

e

1-e y

; y>0,a>0,fi>0,&>0 (3)

II. Zech Distribution

The Zech distribution is the reciprocal of Gompertz Inverse Exponential distribution. The cumulative distribution function of Zech distribution is stated in the following theorem.

Theorem 1: If a non - negative random variable Y follows the Gompertz inverse Exponential distribution expressed as Y~GIE(y; a, 9,p). Assuming a new random variable X = - is defined, then the random variable X follows Zech distribution, written as X~Zech(x; 8, a,p) with the cdf in equation (4).

= eti-[i-

Proof:

G(x) = eP{1[ ] } ; x>0,a>0,p>0,9>0 (4)

e

1-

From equation (1), G(x) = 1 — F

G(x) = 1 —

G(x) = еЛ

■y вхГР\

1 — eP

The result of the first derivative, with respect to x, of equation (4) is the probability density function of Zech distribution given by (5).

g(x) = аве

-вх

[1 — е-вх] "

-ß-l %!-[!■

— I —fz

x>0,a>0,ß>0,9>0

(5)

Figure 1: CDF Plots of Zech distribution.

Figure 2: PDF Plots of Zech distribution.

Figures 1 and 2 illustrate some of possible shapes of the cumulative density function and probability density function respectively of Zech distribution.

III. Estimation of Parameters

The method of Maximum likelihood is used to estimate the parameters of Zech distribution. Assuming each of the random samples Xi , ,' ' ' , Xn

follows the pdf of Zech distribution, the likelihood

function is given by

,xn;a,ß,e) = n|«0e-exi[l — e-8xi]

,■ ■ ■ , Xn; &

Let I denote the log-likelihood function, that is, let I = log L(x1 ,x2 ,■ ■ ■ ,xn; a, ft, 8)

n n

-ß-1ei\1-[1-e 1

I = nloga + nloge - d^Xi - (ft + 1)^ log(l - e-0x) + e-9xi\

i=i i=i i=i The solutions of simultaneous equations obtained from— = 0 , — =0 and — =0 are the maximum likelihood

1 da dp /ff>

estimates of the parameters a, ft and 8. Thus,

(6)

(7)

(8)

d0

a

ß

-вх

n

1 = 1

% — {[1 - - e-^)} - J^U {1 - [1 - e-^Y"} - H=i ln(l -

(10)

% -~e + YH=1xi -(ft + 1)Z?=i(Xie-9xi)(1 - e-9xi)-1 + aZ?=iXie-9xi(1 - e-9xt)-P-1 (11)

Equating — — 0, — — 0 and — — 0, we have

1 "da dp d8

™_L IV" U- h _ „-dxil-!3

«a + 1Z?=1{l-[l-e-exq-ß} = 0 (l2)

fö=i{[l - e-9xi]~ß ln(l - e-^)} - j-2Zf=1[l -[l- e-^]-} - Y?=1ln(l - e-Sxi)

ß

j-П,i{l-[l-e-exi]-ß} = 0 (l3)

% + I.?=1Xi-(P + 1)I,?=1(xie-(>xi)(1-e-exi) 1 + aZ?=1Xie-exi(1-e-exi) P 1 — 0 (14)

The Maximum likelihood estimate for parameter a can be obtained from (12) in the form below for a given ft and 6

a —--m (15)

In=i{1-[1-e-0Xl]-^)}

To obtain the MLE of ft and 6, equation (15) can be substituted into equations (13) and (14). The resulting system of non - linear equations can be solved numerically.

IV. Linear Representation

Theorem 2: The pdf of Zech distribution is a weighted function of an exponential distribution with rate

parameter 6(1 + ¿).

Proof.

g(x) — a6e-Sx[1 - e^^ell1-1--6^} Using the exponential expansion

xk

ex — (16)

g(x) = e-*x[l - e-9x]~ß~1 £{l-[l- ^T^gf W)

k=0

k! \ßJ

From the mixture representation, °>(k + j) F(k)j

, v^rrk + j) .

j=0

{1 - [1 - e-"Y»f—^-y^b -(19)

j = 0

Inserting equation (18) into (16), we have

Simplifying, [1 - e-ex]-Pi [1 - e-ex]--1 — [1- e-9x]-[M+1)+1 (21)

Inserting equation (20) into equation (19), we have

f , V V r(k +j) , ^iad(a\k -exil -gx-\-[P(i+1)+1] ^

k=0 j=0

By using mixture representation,

ro

(1-z)-k = IlxmzL (23)

ro

e ^ =1 T[ß(J + 1) + 11il e (24)

i=0

ro ro ro

9(x) = LLL-mß(-1)}j!{ß) r[ß(j + 1) + 1iv. e e (25)

k=0j=0 ¿=0

rororo

g(x) 111 r(k)j\ ( 1 k\\ß) r[ß(j + 1) + 1H\ e ( )

k=0j=0 i=0

; r(k+j) r[(ff(7 + 1) + 1) + t1 _ T(k)j! r[ß(j + 1) + 1]i! w4,k

Let(-1)y ^^^тт-ттттттг = wiJ,k (27)

rororo

°(X)=avQ 111W^e-d(1+l)X (28)

k=0j=0 i=0 вак+1_111 в(1+0х

9(x) = ~ß4^111w^e-ti(1+l (29)

k=0j=0 i=0

sW=JLäTb^1+i)e-"1+'K] (31)

k=0j=0 ¿=0

rororo

ß«k!(1 + i)111Wij,k[

k=0j=0 i=0

V. Reliability Properties

The reliability function can be obtained from

S(x) = 1 — G(x) (32)

Therefore, the survival function of Zech distribution is given as

«{1-[l-e-e*\-P}

S(x) = 1-ert [ \ } ; x > 0,a> > 0,9 > 0 (33)

The hazard function of Zech distribution is obtained from

h(x)=3M (34)

The hazard function of Zech distribution is given by

aee-°*[l-e-°*YP-1e%1-[1-e-eX]-il} h(x)=---^-^-; x> 0,a> 0,B > 0,6 > 0 (35)

Figure 3: Survival Plots of Zech distribution.

Figure 4: Hazard plots of Zech distribution.

Figures 3 and 4 illustrate some of possible shapes of the Survival function and Hazard function respectively of Zech distribution.

The cumulative hazard function, H(x) of a continuous random variable X from Zech distribution is derived from

H(x) = -\og(S(x)) (36)

Substituting equation (33) into equation (36)

H(x)zech = - log (l - ell1-[1-e-9X] (37)

The reversed hazard function of a random variable x of Zech distribution is obtained from

r(x)—wi)

Therefore,

r(x)zech = <xQe~6x\l — е~вх]

-вх!--1

(38)

(39)

VI. Quantile Function and Median of Zech distribution.

Quantile function is very important for generating random numbers which can be used for simulation studies. Aside that, it can also be used for finding quantiles i.e. quartiles, octiles, deciles and percentiles of a distribution which are necessary for deriving the measures of skewness and kurtosis. The quantile function is derived by inverting the cdf

Q(u) = G-1(u)

Q(u) = -i{ln

1 — (1 — llnu

Where u ~ Uniform (0,1).

To generate random numbers from Zech distribution, it is sufficient that

X=-e{ln

l-(l-llnu

(40)

(41)

(42)

1

1-

The median of Zech distribution can be obtained by substituting u — 0.5 in equation (41) as follows:

i

Median = —{In

в

-(1-Zln0.5)-

(43)

1

Other quantiles can also be derived from (41) by substituting appropriate values of "un.

VII. Quantile - based measures of skewness and kurtosis

The measure of Skewness(S) and Kurtosis (K) of Zech distribution using quantile function, defined by Galton [6] and Moors [7] are given by equations (44) and (45) respectively.

K =

(44)

(45)

©'G ©'G (4)'G (8) ' G (8)' and Q © can be obtained by suteütatag 8 ,8 1,8 *<,and 8 for u respectively in equation (41). Therefore,

Q(8) = -ß

QQ) = -1e =

Q(8)=-l■ Q(8)=-l■

ln

ln

In

ln

ln

ln

ln

1-(1-H8)) 1-(1-H8)) Hi-'«»®) i-(i-H8)) i-(i-H6))

(46)

(47)

(48)

(49)

(50)

(51)

(52)

The skewness of Zech distribution is derived by substituting the values of Q (6), Q (4) and Q (2) into equation

(44).

Therefore, SZech =

>

Hi-lHi))'"

— —

i-lHi))

6W P

->

In

— —

In

— —

(53)

Simplifying (53) by factoring out (1), the result shows that symmetry or asymmetry of Zech distribution is

independent of parameter 1.

SZech = '

2 {In

In

— --

l-K,

In

i-(i-EaH(l

))

In

In

(54)

The kurtosis of Zech distribution is derived by substituting the values of

Q(8),Q © ,Q(8),Q(1),Q (8) and Q ^ into equation (45)

1

1

1

1

1

1

1

1

1

1

e

e

1

1

P

P

K,

?jln

Zech

-?jln

?jln

HHln

HHln

-?jln

— —

(55)

Simplifying equation (55) by factoring out (1), the result shows that the peakedness or otherwise of Zech distribution is independent of parameter &■

jln

K.

Zech

Hi-148,

))-

jln

jln

))-

Hi-Hi

))-

jln

-jln

Hi-143

))-

))-

-jln

Hi-K

))-

(56)

VIII. Distribution of Order Statistics

Let x1,x2, ...,xn be a random sample from a cdf and pdf of Zech distribution as defined in (4) and (5) respectively. The pdf of jth order statistics of any random variable X is given by:

n\

fr-n(x) = Q-1)](n-jy9(x)G(x)i-1[1 — G(x)]n-i

(57)

From (56), putting the pdf of the jth order statistics of Zech distribution,

fy.n(x) =

(j-1)\(n-])\

-аде

4-ß-1e%1-[l-e-6Xn {ет{1-[1-е-вХГ}

~вх[1 — е-вх] ^ ieß{

ß)V-1

1—

Simplifying equation (58),

(58)

(j-l)\(n-j)

-вхи _ „-вхЛ-Р-1 J „f(1-[l-H

аве-вх[1 — е-вх]

-m7

1 — (eß

iU-[i-e-exrß}

П-J

fj\n(x) = (59)

Therefore, the distribution of minimum and maximum order statistics for the Zech distribution is given by f1vn(x) i■ e when j = 1 and fn:n(x) respectively in equations (60) and (61) respectively.

fl:n(x) = i;

(j — 1)\(n — 1)\ After some simplifications,

f1. n(x) = паве-вх[1 — e-ex]-ß-\ е$1-1-е-вХП {1 — (ell1-[l-e~

.аве-вх[1 — е-е^-\е%1-[1-е-вхП{1 — (е^-^П)}

i-ß]

60)

Also,

fn.n(x)

(п — 1)\(п — п)\

аве-вх[1 — е-вх]

■-I-P]

1 — \еР

After some simplifications, fmn(x) = naee-9x[l-e-9x]~

-ß-1 (е^1-[1-е-оХ]-^}

(61)

i

1

1

1

ß

1

1

1

1

1

1

ß

ß

ß

ß

1

1

ß

ß

n\

П

n-1

n

n-n

а

вх

П

1-11-e

e

n

IX. Moments of Zech distribution

The moment about the Origin of Zech distribution is derived as follows: recall from the linear expansion of the pdf of Zech distribution,

k+1

= + i)e-e(1+0x]

The rth moment about the origin of a random variable X is given by

E(xn = j*rmdx m

, œ œ œ

vk+1

E(Xr) = S Xr B*k\(l + iyLYLWijk[d{1 + i)e-9(1+i)x]dx (63)

0 k=0j=0 i=0

œ œ œ œ

k+1 œ œ œ

E(Xr) = jk^YL Z Wij'k S xre-e(1+i)X dx (64)

0a

W; ik | x'e k=0 =0 =0 0 From gamma expansion,

œ

Ta i ,

xa-1e-bx dx

Fa f . ,

— = I xa-1e-bxdx (65)

0

a — 1=r,a = r + 1,b = 0(1 + i)

œ

S xre-9(1+i)xdx = J}r +1 , (66)

J [0(1 + i)]r+1

0

œœœ k+1 œ œ œ

ûak+i r(r + 1)

E(xr) = l^LLLw^ [9(1+-^ (67)

k=0j=0 i=0

The first moment about the origin represents the mean of Zech distribution. This can be done by setting r = 1,

. „ œ œ œ

9ak+1V'VV r(2)

E(x) = lakkÂLLLw^-k[9^)yr2 rn

k=0j=0 i=0

The second moment about the origin of Zech distribution is

œœœ

, „ r(3)

E(x2) = ia^ZZZw^mr+i>F (69)

k=0 =0 =0

The variance of Zech distribution is obtained from

Var(X) = E(X2) — [E(X)]2 (70)

2

, , 0ak+^ W r(3) 0akl1sr VV

Var(x) = iküLLL Wi^k [S(i + i)]3 —iküLLL Wi■>-

k=0 =0 =0

ßkk\ Wi,i,k [0(1 + o]3

ßkk\ ZZZWi'i'k[0(i + i)]2

k=0 =0 =0

(71)

The Moment about the Mean of Zech distribution is thus derived.

The rth central moment of a random variable X having Zech distribution is given by

œ

E«x—»r) = S(x — <'rHx)dx

(72)

0

œ œ œ œ

a k+1

E((x — ß)r) = S (x — + 0 ZZZ Wi-j-k [0(1 + i)e-9(1+i)x] dx (73)

0 k=0 =0 =0 œ œ œ œ

E((x — ß)r) = Sk+YY Z Wi'jk S(x — V)r [e-e(1+i)x] dx (74)

k=0 =0 =0 0

By setting y = x — ß, 1, dx = dy, x = y + ß

œ œ œ œ

0 a k+1

E((x — ß)r)=-^ZZzWi'i'k J yre-9(1+i)(y+ï>dy (75)

k=0 =0 =0 0

œœœ

œ

E(Xr) = IkwYY Z Wi']* J yr e-e(1+0y. e-S(1+^dy (76)

œ

œ œ œ œ

E((x — ß)r) = ^Г+Т^^ £ Щ.ьк e-8(1+i)ß | Уг e-e(1+i)ydy

k=0j=0i=0

Using the Gamma function expansion,

Га Г

— = I xa' ba J

xa-1e-bx dx

a — 1 = r, a = r + 1, b = e(1 + i)

ÎC.Ûry^ + 1

ouu ^œ гю гю Bkk\ bk=0bj=0bi=0 Wi,j,l

(77)

E((x—»)r) = ^e-e(1+l)" m + wr+1) (78)

X. Moment - based measures of Skewness and Kurtosis

The skewness of Zech distribution based on central moment is given as

Skewness= (79)

Where ß3 = Third central momen ß2 = Second central moment

. „ m m m ..

6Qak+1VW e-e(1+l)^

k=0j=0 i=0 , „ m m m ^ ,„

29ak+^W e-e(1+L^

^ ^s^kTLLLm+w (81)

k=0j=0 i=0

yk+1 p-S(i+i)ß-\2

i ^— ^m rim e 1

, „ - =owi,i,k [g(1+i)]4 \

Skewness =-? (82)

-gu учю yiœ yœ e

вкк\ Lk=0^i=0^i=0Wi,i,k [0(1+j)]3j

Kurtosis=

(83)

té

But ^ = 2-4kr nœ=o z!=o zœ=o wlJJC (84)

The Kurtosis, based on central moment of Zech distribution is given by

IAQryk+1 0-Q(1 + ï)u

Y'œ y œ yœ c

Bkk] Lk=0Lj=0Li=0Wi,j,k .)]5

Kurtosis =-7 (85)

-gu y,œ yiœ yœ e

Bkk\ Lk=0^i=0^i=0Wi,i,k [ea+0pJ

0

œ

0

XI. Results

I. Simulation Studies

The behavior of the parameters of Zech distribution was investigated through simulation studies using R statistical software. Data were replicated 1000 times. A random sample of sizes 50, 100, 150 and 200 were selected. The parameters were varied as follows: a = 0^5, 8 = 0^5, and ft = 0^5; and a = 1, 9 = 1, and ft = 1; and a = L5, 8 = L5, and ft = L5 respectively. The maximum likelihood estimates of the true parameters, the bias, standard error and Root Mean Square Error were obtained from the simulation. The results are shown in Tables 1, 2 and 3.

Results of Simulation Studies

Table 1: Simulation study at a = 0.5, 9 = 0.5, and p = 0.5

N Parameters Means Bias Std. Error RMSE

50 a = 0.5 9 = 0.5 P = 0.5 0.0339 0.6382 0.3763 0.4661 - 0.1382 0.1237 0.0152 0.1931 0.2300 0.0174 0.0621 0.0678

100 a = 0.5 9 = 0.5 P = 0.5 0.0466 0.5295 0.5720 0.4534 - 0.0295 -0.0720 0.0170 0.1348 0.1457 0.0130 0.0367 0.0382

150 a = 0.5 9 = 0.5 P = 0.5 0.0667 0.4228 0.5170 0.4333 0.0772 - 0.0170 0.0185 0.0849 0.1100 0.0111 0.0238 0.0271

200 a = 0.5 9 = 0.5 P = 0.5 0.0818 0.4815 0.6211 0.4182 0.0185 - 0.1211 0.0221 0.0927 0.0987 0.0105 0.0215 0.0222

Table 2: Simulation study at a = 1.0, 9 = 1.0, and p = 1.0

N Parameters Means Bias Std. Error RMSE

50 a = 1.0 9 = 1.0 p = 1.0 0.0627 0.2452 0.1369 0.9373 0.7548 0.8631 0.0446 0.4291 0.4911 0.0299 0.0926 0.0991

100 a = 1.0 9 = 1.0 p = 1.0 0.0834 0.8306 0.9365 0.9166 0.1694 0.0635 0.0363 0.1951 0.2667 0.0191 0.0442 0.0516

150 a = 1.0 9 = 1.0 p = 1.0 0.1469 1.0107 0.9909 0.8531 - 0. 0107 0.0091 0.0536 0.1898 0.2452 0.0189 0.0356 0.0404

200 a = 1.0 9 = 1.0 p = 1.0 0.1609 0.8606 0.9995 0.8391 0.1394 0.0005 0.0515 0.1484 0.1901 0.0160 0.0272 0.0308

Table 3: Simulation study at a = 1.5, 9 = 1.5, and p = 1.5

N Parameters Means Bias Std. Error RMSE

50 a = 1.5 9 = 1.5 p = 1.5 0.0636 1.5227 1.9951 1.4364 - 0.0227 - 0.4951 0.5580 0.5782 0.5450 0.0334 0.1075 0.1044

100 a = 1.5 9 = 1.5 p = 1.5 0.0821 1.2049 1.9077 1.4179 0.2951 -0.4074 0.0517 0.3506 0.3208 0.0227 0.0592 0.0566

150 a = 1.5 9 = 1.5 p = 1.5 0.1249 1.0912 0.1713 1.3751 0.4088 1.3287 0.0621 0.2543 0.2802 0.02035 0.04120 0.04320

200 a = 1.5 9 = 1.5 p = 1.5 0.2188 1.3248 1.6958 1.2812 0.1752 - 0.1958 0.0825 0.2275 0.2539 0.02031 0.03370 0.03560

The tables 1, 2 and 3, for each of the selected true parameter values show that as the sample sizes increase, the root mean square errors decrease. This implies that the parameters of Zech distribution are stable.

II. Applications to Real Life Data Sets

The performance of Zech distribution, as well as goodness of fit tests when fitted to the real life data sets is hereby compared with other three - parameter distributions such as Gompertz Inverse Exponential (GIE) distribution, Weibull Exponential (WE) distribution and Gompertz Exponential (GE) distribution are provided in tables 6 and 7 respectively.

To select the best among the competing distributions, the following statistical criteria are used: Negative Log-likelihood, Akaike Information criterion (AIC) and Bayesian Information criterion (BIC). The distribution having the least value of the criteria above is adjudged to be the best. Also, the goodness of fit tests like Kolmogorov-Smirnov Statistic (KS) and Anderson-Darling Statistic (ADS) are also computed to select the best fit. The best distribution has the least value of the statistics above.

Data 1: The first data set represents the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli observed and reported by Bjerkedal [9] and used by Adewara [1].

10, 33, 44, 56, 59, 72, 74, 77, 92, 93, 96, 100,100,102, 105, 107, 107, 108, 108, 108, 109, 112, 113, 115,116, 120, 121, 122, 122, 124, 130, 134, 136, 139, 144, 146,153, 159, 160, 163, 163, 168, 171, 172, 176, 183, 195, 196,197, 202, 213, 215, 216, 222, 230, 231, 240, 245, 251, 253,254, 254, 278, 293, 327, 342, 347, 361, 402, 432, 458, 555

Table 4: Descriptive Statistics for data 1.

Min 1st Quartile Median Mean 3rd Quartile Max. Standard Deviation Skewness Kurtosis

10.0 108.0 149.5 176.8 224.0 555.0 103.4549 1.341869 4.991056

Table 5: Performance rating for the fitted models using data 1.

Distributions Estimates -LL AIC BIC KS ADS

a = 5.1884384

Zech ¡3 = -0.6383047 S = 0.0128380 424.8790 855.7579 862.5879 0.0835 0.4925

a = 0.02683774

GIE ¡3 = 1.88823061 S = 22.03993914 427.6661 861.3322 868.1622 0.1095 1.0777

a = 1.157801294

WE ¡3 = 1.340608545 S = 0.002981762 431.3887 868.7774 875.6074 0.1195 1.9894

GE a = 0.004089507

¡3 = 1.08290306 S = 0.002751085 434.3901 874.7802 881.6102 0.1759 2.6168

The distributions tested showed the performances of each. The results revealed that the distribution with the lowest value of -LL, AIC, BIC, KS and ADS is considered to be the best. From Table 5, Zech distribution had the least value of 424.8790 for -LL, AIC= 855.7579, BIC = 862.5879, KS = 0.0835 and ADS = 0.4925 hence, it was considered the best fitted distribution among other distributions.

Figures 5 and 6 respectively depict the performance of the new distribution with survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli observed, this is compared to other distributions mentioned in the research.

Data 2: The second dataset represents the gauge of length of 10mm observed by Mohammed [10] 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

Table 6: Descriptive Statistics for data 2.

Min 1st Quartile Median Mean 3rd Quartile Max. Standard Deviation Skewness Kurtosis

1.901 2.554 2.996 3.059 3.422 5.020 0.6209216 0.6178407 3.286345

Table 7: Performance rating for the fitted models using data 2.

Distributions Estimates -LL AIC BIC KS ADS

a = 238.223663

Zech ß = -5.177056 9 = 1.971310 56.5097 119.0194 125.4488 0.0885 0.3571

a = 330.740054

GIE ß = -41.32409 9 = 18.50223 57.28159 120.5632 126.9926 0.0903 0.3846

a = 1.8379671

WE ß = 3.7232674 9 = 0.1842052 63.6584 133.3168 139.7462 0.0986 1.1866

GE a = 0.944957015

ß = 1.480487938 9 = 0.008481201 69.1480 144.2960 150.7254 0.1389 1.9840

The distributions tested showed the performances of each. The results revealed that Zech distribution had the lowest value of -LL, AIC, BIC, KS and ADS and is considered to be the best. From Table 7, Zech distribution had the least value of = 56.5097 for -LL, AIC=119.0194, BIC =125.4488, KS = 0.0885 and ADS= 0.3571 hence, it was considered the best fitted distribution among other distributions.

Histogram and theoretical densities

P-P plot

Theoretical probabilities

Figure 7: Histogram and theoretical densities for data 2

Figure 8: P - P plot for data 2

Figures 7 and 8 respectively depict the performance of the new distribution with gauge of length of 10mm data, and compared to other distributions mentioned in the research

The histogram and theoretical densities plot shows that Zech distribution fits data 1 and 2 best. Also, the probability plot i.e. (the PP plot) which compares the empirical cdf of data sample with specified theoretical cumulative distribution, reveals that Zech distribution is closer to the line than the remaining three fitted models. Tables 1, 2 and 3, for each of the selected true parameter values show that as the sample sizes increase, the Root Mean Square errors decrease which shows that the parameters of Zech distribution are stable.

Tables 4 and 6 show that the data is skewed to the right. Interestingly, the shape of the pdf graph of Zech distribution is also positively skewed. Also, the Kurtosis values of 4.991056 and 3.2866345 suggest that the data is leptokurtic. Data 1 and 2 have a kurtosis of 1.991056 and 0.2866345 respectively above that of normal distribution which is 3.0

Tables 5 and 7, show that Zech distribution has the lowest value of -LL, AIC, BIC, KS and ADS and is considered to be the best when compared with the competing distributions such as Gompertz Inverse Exponential distribution, Weibull Exponential distribution and Gompertz Exponential distribution.

In this paper, a new three - parameter continuous distribution named Zech distribution was proposed from Gompertz Inverse Exponential distribution. Its probability density function was plotted and the result revealed a heavy positively skewed distribution which is suitable for modelling heavily right-tailed data. Several statistical and mathematical properties of the new distribution were derived. The results of the simulation studies revealed that the parameters of the new distribution are stabled and as the sample sizes increased, the Root Mean Square (RMS) errors decreased. The applications to two real life data sets showed that Zech distribution has the lowest -LL, AIC, BIC, KS and ADS when compared with other competing distributions such as Gompertz Inverse Exponential distribution, Weibull Exponential distribution and Gompertz Exponential distribution used in this research paper.

XII. Discussion

XIII. Conclusion

References

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