A novel approach for constructing distributions with an example of the Rayleigh distribution Текст научной статьи по специальности «Математика»

A novel approach for constructing distributions with an example of the Rayleigh distribution

Aijaz Ahmad1, Muzamil jallal2 and Afaq Ahmad3

•

12 Department of Mathematics, Bhagwant University, Ajmer, India 3Department of Mathematical Science, IUST, Awantipora, Kashmir Email: [email protected], Email: [email protected] Email: [email protected]

Abstract

In this paper, we describe a novel technique for creating distributions based on logarithmic functions, which we referred the Log Exponentiated Transformation (LET). The LET technique is then applied to Rayleigh distributions, resulting in a new distribution known as the Log Exponentiated Rayleigh distribution (LERD). Several distributional properties of the formulated distribution have been discussed. The expressions for ageing properties have been derived and discussed explicitly. The behaviour of the pdf, cdf and hazard rate function has been illustrated through different graphs. The parameters are estimated through the technique of MLE. A simulation analysis was conducted to measure the effectiveness of all estimators. Eventually the versatility and the efficacy of the formulated distribution have been examined through real life data set.

Keywords: Log Exponentiated Transformation, Rayleigh distribution, Moments, reliability measures, maximum likelihood function.

Mathematics subject classification: 60-XX, 62-XX, 11-KXX.

I. Introduction

The adoption of an efficient statistical model is critical in a variety of practical analyses. This is especially inconvenient for specific data studies, because the typically employed distributional models are inadequate for producing a plausible fit. Several approaches, such as the generation of families of adaptable distributions, have been presented in recent times. Most of them attempt to increase the effectiveness of a baseline distribution by utilising diverse mathematical expansion approaches. As a result, the related models may incorporate some extra characteristics that provide sufficient flexibility to examine real-life data in many areas of study, such as reliability, survival analysis, computer science, finance, biological research, medicine, and so on. Academics have recently been concerned with developing new techniques for creating new families of distributions so that real data can be adequately analysed and explored. Among them are Marshall and Olkin [9], Eugene et al.[4] , Mudholkar et al. [11], Nadrajah and Kotz [12], Alzaatreh et al. [2], Mahdavi and Kundu [9], Ijaz et al. [8], Anwar Hassan et al.[3]. Based on the argumentation stated above, we suggest a novel family of distributions that adds versatility to the provided family and entitles it Log Exponentiated Transformation (LET). We give a thorough explanation of its fundamental mathematical characteristics, and subsequently employ the Rayleigh distribution as an application.

II. Log Exponentiated Transformation (LET)

This section demonstrates a novel generating family of probability distributions termed as log Exponentiated transformation, abbreviated as LET. If X is a continuous random variable, then the cumulative distribution function (cdf) of the log Exponentiated transformation is described as

F(x;Z,0) = 1 -log(e + ë(g(x;C))û) ;x eÂ,0,Ç > 0 (l)

d (g(xC))

Where G(x;Z) denotes the cdf of baseline distribution and ^ ^ = g(x;Ç). The associated probability density function (pdf) is described as

f (x; zq)=(e-1)q (x; Z)G(x; Pr-1 ; x ÎÂAC > 0 (2)

e + e(G(x; Çf

The survival function s(x;Z,q), hazard rate function h(x;Z,d) and cumulative hazard rate function H (x; Z, d) are stated as respectively

s{x;Z,e) = log(e + e (G(x;ZZ)0)

' (e + eG" (x;Ç))\og(e + e(g(x;Ç))")) H (x; Ç, ") - - log(log(e + e (G(x; Ç))" ))

III. Mixture Form

This section provides an expression for the mixture form of the probability density function. Equation (2) can be written as

f (x; Z, 0) = (e - i)ßg(x; C)Ge-1 (x; + e (g(x; Zfy

= ^dg(x;iPq-1 (x^l + I (GXifJ1 (3)

¥

We know that (l + z)-1 = ^ (- l)p zp ; |z| < 1, using it in equation (3), we have

p=0

Np

f(x;cq) = ^q(x-zXGixzT11 (-i)pI e I (G(x;C)y

p=0

After simplification, we obtain the mixture form of pdf as

Np+i

¥ / f 1 *

f(xz,e)=£(-W -1 (g(x-,cMx;c)ip+q (4)

p=0 ^ e 0

IV. Log Exponentiated Rayleigh Distribution with properties

The Rayleigh distribution, named after the Lord Rayleigh, is a continuous probability distribution. Due to its wide range of applications, researchers have extended Rayleigh distribution for instance Exponentiated Rayleigh distribution by Voda [13], Weibull-Rayleigh distribution by Faton Merovci et al.[5], transmuted generalized Rayleigh distribution by Faton Merovci [6], Topp-Leone Rayleigh distribution with application by Fatoki O [7] and inverse Weibull Rayleigh distribution by Aijaz et al. [1]. The probability density function (pdf) of Rayleigh distribution with scale parameter« is defined by

a 2

g(x;a) = ae 2 ;x>0,a>0 (5)

The related cumulative distribution function (cdf) is given by

a 2

G(x;a) = 1 -e 2X ;x>0,a> 0 (6)

The cumulative distribution function (cdf) of the formulated distribution can be obtained by substituting the value of equation (6) in equation (1), which follows

F(x; a, 0) = 1 - log

qö

e + e

--x

1 - e 2

; x > 0,a,ö> 0

(7)

The related probability density function is stated as

a

q(e-1)

f \0-i a 2i

xe

f (x,a,0) = -

1 - e 2

; x > 0,a,ö > 0

(8)

e + e

--x

1 - e 2

Equation (8) may be stated in mixture form by substituting equations (5) and (6) in equation (4).

.P+1 a , ( a , VP^-

f (x; a, 0) = £(-1У

p=0

aQxe 2

1 - e 2

(9)

¥ f b - Л

Since (l - z)b-1 = ^ (-1 )q zq ; |z| < 1, using it in equation (9), we have V q 0

q=0 ¥ ¥

f(x;aq) = ££ (- l)p+q f(p +l)q- ^f P+\(ke 2 p=0q=0 V ^

■lid

V J /

(q+i)..

, oxe

2

-x

(10)

p=0q=0

Where

SP,q =(-1)

p+q

f (p+i)q- i

q

v e ö P+1

e) q

Figures (1.1), (1.2), (1.3), and (1.4) depict several probable pdf and cdf layouts of LERD for distinct parameter selections.

a2 --x

2

a

e

2

alpha=0.6,theta=1.20 alpha=0.70,theta=2.30 alpha=0.80,theta=2.50 alpha=0.90,theta=2.90 alphas 1.50,theta=2.0

Figure 1.1 :pdf of LERD under different values to parameters

□ alpha=1 30,theta=0.8

□ alpha=0.70,theta=0.90 ■ alpha=1.80,theta=0.90

□ alpha=0.80,theta=1.E s alpha=0.50,theta=1.60

Figure 1.2:pdf of LERD under different values to parameters

Figure 1,3:cdf of LERD under different values to parameters Figure 1,4:cdf of LERD under different values to parameters

V. Mathematical Properties of LER Distribution I. Moments of LER Distribution

Let suppose X denotes random variable follows LERD. Then rth moment denoted by ¡j.r is given as

¥

ni = e(xv )= J xrf (x;a,0)dx

0

Using equation (10), we have

¥ ¥

Vr = J ^

0 p=0 q=0

r= J-r EI

0 p=0 q=0 ¥ ¥

p=0 q=0 0

a ,

a(q+1) x 2 Sp q axe 2 dx

a(.q+1) x

xr+le 2

dx

Making substitution — (q + l)x2 = z , so that 0 < z < ¥, we have

p=0 q=0

2 ) 2 1

a(q +1) J (q +1)

\z 2 e 2dz

After solving the integral, we get

^ ¥ ¥

p=0 q=0

\

a(q +1)

2 G' k +1

II. Moment Generating Function of LER Distribution

Let X be a random variable follows LERD. Then the moment generating function of the distribution denoted by MX (t) is given

¥

MX (t) = E (etx )= J etxf (x;a,d)dx

r

¥ r

2

Using Taylor's series

гГ (tx)2 (tx)3 V (

=Jl1+*+Чт+Чт+...f(x;i

¥ tr r = ^ — I xrf (x; a, d)dx

r=0 r! 0 ¥ tr 1 \ = I )

r=0 '

¥ ¥ ¥ Г

(t)=§5Ii^ir+i),12 +1

-+i

III. Quantile Function of LER Distribution

The quantile function of any distribution may be described as follows:

Q(u ) = Xq = F(u) Where Q(u) denotes the quantile function of F(x) for u e (0,l) Let us suppose

F(x) = 1 - log

f f

e + e

V v

a 2

--x

1 - e 2

Л0

= u

yj

I)

After simplifying equation (ll), we obtain quantile function of LER distribution as

Q(u )=x. =

-flog

f 1 ö (e1-" -e

1 -

e

v /

v 0

VI. Mean Deviation From Mean and Median of LE R Distribution

The entirety of deviations is apparently a measure of amount of dispersion in a population. Let X be a random variable from LER distribution with mean ¡i . Then the mean deviation from mean is defined

as.

D(j) = д(X - j)

¥

= J| X - jf (x)d: 0

M

= 2jF (m) - 2 J xf (x)dx

(12)

Now

ß ¥ ¥ ß a(q+\)

j xf^x = ZTSp,qaj"2

0 p=0 q=0

S„qa\ x e 2 dx

Р1Ч

Making substitution a(q+ 1 = z so that 0 < z < ^ ■ -2

ц , we have

¥

H- ¥ ¥

J xf(x)dx = ^^

2 12I 1

0 p=09=0

After solving the integral, we have

p'9 a(q +1)J ^ q +1

9 +1) J

x(9+1) 2 2 1

z 2 e Zdz

A* ¥ ¥

J xf (x)dx = ^^

d

( 2 ^ 2 ( 1 ( 3 a(q +1) 2

(q +1) J ^ q +1

g

{2 2

0 p=0 q=0

Substituting value of equation (7) and (13) in equation (12), we get

( ( (

(13)

D(v) = 2v

1 - log

e + e

1-e

2

V v

/0

- 2

p=0 q=0

( 2 \ 2 ( 1 \ (3 a(q +1)

a

(q+1)

q + 1

g T

2 2

M

LetX be a random variable from LER distribution with medianM . Then the mean deviation from median is defined as.

M

D(M) = e(x - M\) = J| X - M\f (x )dx = /U - 2 J xf (x)rfx

(14)

Now

M

J xf (x)dx = YjYjSp,qa\

M a(q+1)

dp qa | x2e 2 dx

a(q +1)

Making substitution —^—- = z so that 0 < z <

P=0 q=0 0

M2, we have

a(q+0» * 2

a(q+\)

1V1 ¥ ¥ J xf (x)dx = ^^

d

2 12 i 1

0 p=0 q=0

After solving the integral, we have

a

(q +1) J ^ q +

q +1) J

M2

2 1

z 2 e - 2dz

M ¥ ¥ J Xf (X)dx = ^^

d

( 2 \2 ( 1 \ (3 a(q +1)M2

(q +1)0 ^ q +1

g

22

0 p=0 q=0

Substituting value of equation (15) in equation (14), we get

D(M )=v-

p=0 q=0

2 12 ( 1

p,q a(q +1)0 Vq +10 V2 2

g 3, ak+) M 2

(15)

1

¥

VII. Ageing Properties of LER Distribution

Suppose X be a continuous random variable with cdf F(x), x > 0. Then its reliability function which

is also known survival function is stated as

¥

x) = f (x)dx = 1 - F(x)

S (x) = pr (X > x) = J f (x)dx = 1 - F (x)

Therefore, the survival function for LER distribution is given as S (x,a,0) = 1 - F (x,a,q)

( ( a 2 ^

= log

e + e

1 - e

//

V v

The hazard rate function of a random variable x is given as

h(x,aq)= «a)

S (x,a,&)

(16)

2

Using equation (8) and (16) in equation (17), we have h(x,a,d) = y

ав(в - l)xe 2

/

I a 2 \ --x

1 - e 2

e + e

a 2 —x

1 - e 2

леУ ( î

log

e + e

a 2 --x

1 - e 2

0V V V

Figures (1.5) and (1.6) depict several probable hazard rate function layouts of LERD for distinct parameter selections.

□ alpha=0.6,theta-0.20

□ alpha-0.70,theta-0 30

■ alpha=0-80,theta=0 50

□ alpha-0.10,theta=0.20

■ alpha-0.20,theta-0.30

□ alpha=0.30,theta=0.10

□ alpha=0.40.theta=0 20

□ alpha=0.50,theta=0.30 n alpha=0.60,theta=0.40 ■ alpha=0.70.theta=0 50

Figure 1.5:hrf of LERD under different values to parameters

Figure 1,6:hrf of LERD under different values to parameters

The cumulative hazard rate function of a continuous random variable x is defined as

H (x, a, q) = - log[F (x; a, 0)\ (l8)

Using equation (16) in equation (18), we obtain the cumulative hazard rate function of LER distribution

H(x,a,e) = - log

log

œ œ

e + e

a 2 --x

1 - e 2

è v

0/

VIII. Renyi Entropy of LER Distribution

If X denotes a continuous random variable having probability density function f (x). Then Renyi entropy is stated as

T

(d) = p7log| J f8 (x)dx ü, where S > 0

and d -Ф-1

Thus, the Renyi entropy of LER distribution is given as

(e -1)9 g (x;Z)(G(x;Z))'

TR (^) = Jidl0g

№ i

9-1 ö

dx

e + e (G(x;Z))9

¥

^log^e-l)se-ses\(g (x;Z))d(G(x;Z))(9-1Hl + | (g(x;Z))9] dxl

(19)

d

Since (l + z )-b = £ (-1)

(b + p-lö

p=0

zp ; Izl < 1, using it in equation (19), we have

v f /

^ (d)- - l)deq (g (X;C)f(G(X;C)ri)S± (-)P [ p '](ee ] " (G(xz))P& dx

{p=o

^log^ ("l)p+d[ P S (g X z)f (G(x; c))q P+S'hS dx I

p+S

Using equation (5) and (6), we have

Tr (d)--1-log-

1 — S

(b ö

E (-1)

p+S

f c i\/ — \P+S <v( a 2

l p + S — 1 ir e ö _°rl ~x2

p=0

\ f /

-I 9° 11 axe 2

sd( p+S°—S

1-e 2

dx

(20)

Since (1 - z)b = £ (-1)q zq ; |z| < 1, using it in equation (20), we have

q=0 Vq0

.. I ¥ ¥

w-i-Hn (-1)

,p+q+S

p+8-iöfö(p+d)-döf e ö p+d

p-0q-0

¥ ¥ ¥ a(q+S)x2

xde 2 dx

¥ qjq+S) x2 1 lsas\ x8e 2 dx)

^^¿¿Wq j

p-0q-0 0

Where

w

p,q

-(-1)

,p+q+S

f p+d-iöfe(p+s) s)( e ö p+s0sas

v p A q

Making substitution + d x2 - z so that 0 < z < ¥, we have

d-i

¥ ¥ ^ 2 ¿¿"p*—-rd+i ,

p-0 q-0 (a(q + S)) 2 0

¥ d+i

2 e z dz

After solving the integral, we get

d-i

2 2 (s+1 wp,q—-^TT1 I ~Y~

1Z.. ^

p-0 q-0 (a(q + d)) 2

IX. Maximum Likelihood Estimation of LER Distribution

Let X1, X2,.., Xn be a random sample of size n from LERD then its likelihood function is given by

n

l = n f (yi

ad(e - l)xe 2

n

--x

1 - e 2

e + e

1 - e 2 v 0

The log likelihood function is given as

¥

S

R

q

e

NÖ-1

a

a 2i

( a

log l = n loga+ n logq + n log(e -1)- —' x, + ' logfa )

+(q-1)' log

a i --xi

1 - e 2

i=1

œ

i=1

Slog

e + e

a i

--x,

1 - e 2

Differentiate equation (21), partially with respect parameters, we have

д log l n 1

-— =---> x, +

дa a i

> x, +a(e-l)>

xe

-ae>

ai --xi

1 - e i

,e-l

ai

--x,

ii

1-e

e + e

ai

--x¡

1 - e i i

д log l n

дa ~ в

S log

i=1

a2

--x,

1 - e i

+e

S

i =1

--x,

1 - e i

log

v

a2

--x¡

1 - e i

e + e

1-e

--x,

2i

(21)

(22)

(23)

The equations (22) and (23) are non-linear equations and hence cannot be expressed in compact form. Therefore to solve these equations explicitly for a and 0 is difficult. So we can apply iterative methods such as Newton-Raphson method, secant method, Regula-falsi method etc. The MLE of the parameters denoted as V(a, q)of g(a,d)can be obtained by using the above methods.

For interval estimation and hypothesis tests on the model parameters, an information matrix is required. The 2 by 2 observed matrix is

Z (v) =

The elements of above information matrix can obtain by differentiating equations (22) and (23) again partially. Under standard regularity conditions when n the distribution of ç can be approximated by a multivariate normal N(o, I(ç) _1) distribution to construct approximate confidence interval for the parameters.

Hence the approximate 100(l - y)% confidence interval for a, 9 and l are respectively given by

a ±and 0 ±Z^JqV) 2 2 Where Zy denotes the çth percentile of the standard normal distribution.

X. Simulation Analyses

y 2

In this segment, a Monte Carlo simulation analysis was performed using R software to evaluate the consistency of the MLE's. This analysis was performed 500 times using sample sizes of n=30, 50, 150, 250,350 and 450 and various parameter combinations (0.5, 0.7) and (0.7, 0.5) created from LERD. In each case, the bias, variance, and mean square errors (MSEs) were calculated. Table 10.1 shows the simulation findings. In particular, we see that, pursuant to the theory, the MSEs and bias decrease as sample size increases.

i=1

i=1

a

xe

x

2

a --x

2

\в

a

Table 1: Average bias, variance and MSEs of500 simulations of LERD ifor different parameter combinations.

Sample parameters a = 0.5, 0 = 0.7 a = 0.7 , 0 = 0.5

Size n

Bias Variance MSE Bias Variance MSE

30 a 0.04106 0.01906 0.02075 0.06495 0.04404 0.04826

e 0.08050 0.05995 0.06643 0.04658 0.02140 0.02357

50 a 0.01797 0.00826 0.00858 0.03166 0.02274 0.02375

e 0.03925 0.02402 0.02556 0.02152 0.01128 0.01175

150 a 0.01085 0.00321 0.00333 0.01693 0.00608 0.00637

e 0.01481 0.00729 0.00751 0.01100 0.00305 0.00317

250 a 0.00280 0.00186 0.00187 0.00456 0.00366 0.00368

e 0.00702 0.00401 0.00406 0.00280 0.00169 0.00170

350 a 0.00219 0.00102 0.00102 0.00461 0.00271 0.00273

e 0.00175 0.00232 0.00232 0.00296 0.00123 0.00124

450 a 0.00309 0.00088 0.00089 -0.0002 0.00200 0.00200

e 0.00311 0.00222 0.00223 0.00188 0.00102 0.00103

XI. Data Analysis

This section assesses the effectiveness of the stated distribution using real-world data. We fitted the LER distribution to many other models for comparative purposes, including Weibull distribution (WD), Exponentiated exponential distribution (EED), Frechet distribution (FD), inverse Burr distribution (IBD), Rayleigh distribution (RD) and exponential distribution (EXD).

We will use certain measures to evaluate which of the competitive models is the strongest, including AIC (Akaike Information Criterion), CAIC (Consistent Akaike Information Criterion), BIC (Bayesian Information Criterion) and HQIC (Hannan-Quinn Information Criterion). Such criteria can be represented mathematically by

2kn

AIC = 2k - 2 In l CAIC --2ln l

n - k-1

BIC = k Inn - 2Inl and HQIC = 2kln(ln(«))-21nl

We compute Anderson-Darling (A*), Cramer-Von Misses (W*), Kolmogorov-Smirnov Statistic, and P-value in addition to the aforementioned goodness of measures. The model with the lowest value of these indicators and the greatest p-value is considered the best among the competing models.

Data Set: The data set was originally reported by Bader and Priest (1982), on failure stresses (in GPa) of 65 single carbon fibres of lengths 50 mm, respectively. The data set is given as follows

1.339,1.434,1.549,1.574,1.589,1.613,1.746,1.753,1.764,1.807,1.812,1.84,1.852,1.852,1.862,1.864,1.931,1.952,1

.974,2.019,2.051,2.055,2.058,2.088,2.125,2.162,2.171,2.172,2.18,2.194,2.211,2.27,2.272,2.28,2.299,2.308,2.33

5,2.349,2.356,2.386,2.39,2.41,2.43,2.458,2.471,2.497,2.514,2.558,2.577,2.593,2.601,2.604,2.62,2.633,2.67,2.68

2,2.699,2.705,2.735,2.785,3.02,3.042, 3.116, 3.174.

Tabie 2: The -Mn q Med Mean q Kurt Skew m~~ descr¥ive statistics

for data set

Min Qi Med. Mean Q3 Kurt. Skew. Max

1.339 1.914 2.271 2.241 2.563 2.5270 0.0419 3.174

Table 3: The ML Estimates for data set

Model ML Estimates

Standard Error

a в a в

LERD 1.3473 12.660 0.1356 3.7652

WD 0.0059 5.8363 0.0022 0.4026

EED 2.3310 115.52 0.2045 46.011

FD 1.9940 4.9923 0.0530 0.4439

RD 0.3849 0.0481

IBD 5.0822 34.299 0.4311 9.5300

EXD 0.4462 0.0557

(standard error in parenthesis)

Table 4: Comparison criterion and goodness of fit statistics for data set

Model -2 log l AIC CAIC BIC HQIC

LERD 69.712 73.712 73.909 78.030 75.413

WD 70.756 74.756 74.952 79.073 76.457

EED 76.657 80.657 80.853 84.974 82.358

FD 86.443 90.443 90.642 94.761 92.144

RD 149.168 151.16 151.23 153.32 152.01

IBD 85.506 89.506 89.702 93.824 91.207

EXD 231.29 233.29 233.35 235.45 234.14

Table 5: Other goodness of fit statistics criterion for data set

Model W* A* K-S value p-value

LERD 0.04714 0.2987 0.0670 0.9357

WD 0.0590 0.3836 0.0787 0.9181

EED 0.1173 0.7114 0.1006 0.5363

FD 0.2547 1.5484 0.1221 0.2949

RD 0.0834 0.3266 0.3501 3.054e-07

IBD 0.2428 1.4748 0.1186 0.3288

EXD 0.04735 0.3986 0.4677 1.374e-12

XII. Conclusions

In this study, a novel technique known as log exponentiated transformation (LET) is suggested. As an illustration, the Rayleigh distribution is employed as the baseline distribution, and a novel two-parameter log exponentiated Rayleigh distribution (LERD) which proved more flexible has been studied. Several mathematical aspects of the newly developed distribution are deduced and analysed. The MLE approach is used to acquire the parameters. From table 8.3 and 8.6 it is evident that the formulated distribution outranks than compared ones.

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