ON THE CHARACTERIZATION AND APPLICATIONS OF A THREE-PARAMETER IMPROVED WEIBULL-WEIBULL DISTRIBUTION Текст научной статьи по специальности «Математика»

ON THE CHARACTERIZATION AND APPLICATIONS OF A THREE-PARAMETER IMPROVED WEIBULL-WEIBULL DISTRIBUTION

A. S. Mohammed1 , B. Abba2,3*, I. Abdullahi2, Y. Zakari1, A. I. Ishaq1

•

department of Statistics, Ahmadu Bello University, Zaria.

2Department of Mathematics, Yusuf Maitama Sule University, Kano-Nigeria.

3School of Mathematics and Statistics, Central South University, Changsha, Hunan Province, China.

[email protected] [email protected] [email protected] [email protected] [email protected] *Corr esponding author: Email: [email protected]

Abstract

Parametric modeling of complex lifetime data characterized with nonmonotone hazard rate (NMHR) has in recent years attract the interest of many researchers and practitioners. The three-parameter improved Weibull-Weibull distribution introduced in 2022 has demonstrated a better NMHR modeling potential in the analysis of several failure times identified with bathtub hazard rate (BHR). In this study, we present the characterization, properties and two data sets' applications of the distribution. Various properties of the distribution obtained, include moment generating function, moments, skewness, kurtosis, and some types of entropy. Numerical results for mean, variance, skewness, and kurtosis are computed using simulation studies. Estimation of the distribution parameters is performed using the method of maximum likelihood, and the estimation method is assessed by Monte Carlo simulation experiments. The two illustrations further ascertain the capability of the model for modeling lifetime data from different scientific investigation areas.

Keywords: Impr oved Weibull-W eibull distribution, non-monotone failur e rate, characterization, maximum likelihood method, failur e time data.

1. Introduction

Weibull distribution is one of the leading and widely used classical distributions. It has played a vital role in solving many problems in applied areas, such as reliability engineering, renewable energy, weather forecast, and biological studies analysis. For example, the Weibull model is used in modelling the failure time of devices [1], analysis of wind speed data to determine the wind power density [2], further more,[3] applied the distribution to describe soil particle-size and was recently used for fatigue life prediction of mechanical parts [4]. Because of its positiv ely and negativ ely skewed density shapes, the distribution may be the first choice when modeling monotone hazar d rates. One drawback with Weibull is its inability to accommodate non-monotone failur e rates, such as the bathtub and unimodal failur e rates [5]. For instance, the unimodal-shaped failur e rate can be obser ved in the course of a successful sur ger y, wher e the patient is at high risk initially due to infection and other complications. The bathtub-shaped failur e rate can be obser ved

in the course of a population followed from birth or manufactur ed items with early failure due to faulty parts. Different new classes of distributions were developed based on modifications of the Weibull distribution to cope with the non-monotonic failur e rates. Among others, are the exponentiated Weibull by [6], modified Weibull by [7], odd Weibull by [8], beta modified Weibull by [9], Weibull-Weibull by [10], Weibull-exponential by [11], odd generalized exponential-W eibull by [12], new Weibull-W eibull by [13], exponentiated additive Weibull distribution by [14], on monotonic and non-monotonic failur e rates by [15], Four-Parameter Weibull distribution by [16] and flexible additiv e Chen-Gompertz by [17]. These distributions have one or more additional parameter(s) compar ed to Weibull distribution that makes them more flexible for modeling datasets with both monotone and non-monotone failur e rates. The impr oved Weibull-W eibull (IWW3) distribution is a three-parameter model recently established by [18]. The distribution was proposed as an enhanced version of Weibull-W eibull (WW) distribution by [10] introduced in an earlier study to widen its applicability in modeling different complex lifetime data distinguished by various monotone and non-monotone hazar d rate (HR) shapes. The IWW3 distribution is expr essed by the follo wing sur viv al/r eliability function as;

S( x) = exp

{-l 0

h(t)dx| =

exp |— (e? — 1 )'},

t > 0,

(!)

wher e Z = T, V > 0 and p > 0 are the two shape parameters, T > 0 is the scale parameter of the model, and h(x) represents the associated HR function (HRF) of the model, defined as

p— 1

h(x) = Wp—!eZn(eZn — i)i

(2)

The HRF of IWW3 distribution was characterized to have an increasing shape when p, rj > 1, and a decreasing patter n when p, rj < 1. Depending on the chosen values within the ranges of rj > 1 and p rj < 1, the HRF exhibits various bathtub curves. The HRF shapes are displayed in Figure 1, which visually explains the three main forms of the HFR shapes.

-T=1.3, v=1.2 / /

- T=0.4, v=0.7 / /

- x=0.3, v=2.5 / /

- t=0.1, v=3.4 / /

^JJ

—

Figure 1: Curves describing various shapes ofIWW3 hazard rate function at different values.

The probability density function (PDF) of the model is expr essed as

- Wm—iZ ieZ" _

f (x)- —1 ez

P — 1)

q>—1

exp

— — iy

(3)

The distribution was demonstrated to provide a better fit in practice among several other two to five-parameter Weibull and non-Weibull distributions. More specifically , the censored failure and running times of 30 devices by[19] was identified to exhibit bathtub-shaped HR. The findings established the superiority of IWW3 distribution over some well-known Weibull extensions, including the exponentiated Weibull by [6], modified Weibull by [7], exponential Weibull by [20], alongside the WW [10] and other distributions. Motivated by the IWW3 distribution flexibility, and the original study by [18] only proposed the model, discussed its failure rate function and estimation methods. In this study , we present other important aspects of the

distribution, including the model's characterizations and some of its properties. Other real-life data applications of the distributions are also demonstrated.

The rest of the paper is arranged as follows. Section 2 discuss some of the properties and entropies of the distribution. The characterization of the distribution by two truncated moments and based on hazar d function are given in section 3. The estimation of the proposed distribution parameters via the maximum likelihood method is presented in section 4. In sections 5 and 6, we assessed the estimators numerically by simulation studies and applications of the model to two lifetime data, respectiv ely. We conclude the paper in section 7.

2. Distribution properties

Here, we discuss the moment generating function, moments, Renyi entropy and Mathai-Houbold entropy. We obtained an approximation for the values of the mean, variance, skewness and kurtosis of X using Monte Carlo simulation technique.

2.1. Moment generating function and moments

Definition 1: Let X be a random variable with the IWW3 density function (3). Then the moment generating function of X is given by

M (s) — *rr (-')'+'r(*(i + 1)) r T<WT(1 + pi n )

Mx (s)- * r r -T (* (i + 1) - j) r p!0. - * (i + ,))■+ p (4)

The rth moment about origin of X is generated from (4) , and is define as follows.

Definition 2: Let X be a random variable with the IWW3 density function (3). Then the rth (r > 0, real) generalized ordinary moment of X is p.r — f™™ xr f (x) dx. For X ^IWW3 (n, *, t), the rth moment from (4), is given as

^r — E (Xr) — rr (-1)i+;r (* (i +1+))ir (1 + ri n ) Tr * , r £ N (5)

" ( ) ifes jfes i!j!(j - * (i + 1))1+ri nr (* (i + 1) - j)

In particular , the first four moments (for r — 1, ...,4) can be used to calculate the mean (^i), variance (^2), skewness (y^l) and kurtosis (j82) based on some well-known results. The Monte Carlo simulation was performed for N — 1000 samples each of size n — 200 from the IWW3(n, *, t) distribution, with Y — (n0, *0, 1.5 )T - the vector of parameters, where n — 0.7, 0.9, 1.5, 2.0 and 3.0, and * — 0.5, 1.0, 1.5, 2.0 and 3.0. Table 1 listed the numerical results for the mean, variance, skewness and kurtosis with their standar d deviations (SDs) in parenthesis. We can notice From the Table, that the estimates of these properties varies for various combinations of the distribution parameters with a consistent decr ease in the SDs for the mean and variance. The distribution shifted from right to left-ske wed distribution when n* > 2. The skewness and kurtosis plots are displays in Figure 2 as a functions of n and *. A decrease in the values of the skewness and kurtosis are observed from the plots as values of the n and * increases.

2.2. Renyi and Mathai-Houbold entropies

Definition 3: Let the random variable X have the density function given by (3). Then the Renyi entr opy of X is given by

It (a) — 1-alog (Mhj (a, Y) ) (6)

„„ / na-1 ^ (-l)i+''Г(*('+l)-(a-l))Г((n-l)(a-l)n-1 + 1 )

wher e, M{. (a, Y) — n-* 0 D>0 - + -( n li+V -11-L, a > 0 and a — 1.

' / Ta 1 Mj,(j-*('+l))(l-1)(a-1)l 1 +1 r(*(i+l)-(a+j-l))

Table 1: Mean, variance, skewness and kurtosis with standard deviations between parentheses; t = 1.5 and some values of n and p.

V P Mean Variance (a2) Skewness^Kurtosis(32)

0.5 1.195 (0.128) 3.021 (0.635) 2.045 (0.408) 7.729 (3.268)

1.0 0.830 (0.057) 0.609 (0.091) 1.277 (0.273) 4.563 (1.598)

0.7 1.5 0.778 (0.038) 0.284 (0.033) 0.785 (0.197) 3.264 (0.833)

2.0 0.772 (0.030) 0.173 (0.018) 0.466 (0.163) 2.784 (0.510)

3.0 0.785 (0.021) 0.089 (0.008) 0.074 (0.143) 2.617 (0.275)

0.5 1.066 (0.094) 1.660 (0.263) 1.535 (0.257) 5.099 (1.506)

1.0 0.875 (0.048) 0.450 (0.053) 0.870 (0.186) 3.295 (0.774)

0.9 1.5 0.862 (0.034) 0.229 (0.022) 0.451 (0.151) 2.690 (0.439)

2.0 0.871 (0.027) 0.144 (0.013) 0.177 (0.139) 2.560 (0.293)

3.0 0.894 (0.019) 0.075 (0.007) -0.160 (0.142) 2.711 (0.227)

0.5 1.007 (0.061) 0.710 (0.071) 0.780 (0.139) 2.736 (0.400)

1.0 1.001 (0.036) 0.259 (0.022) 0.231 (0.122) 2.376 (0.231)

1.5 1.5 1.030 (0.027) 0.140 (0.013) -0.096 (0.127) 2.524 (0.190)

2.0 1.054 (0.021) 0.088 (0.009) -0.304 (0.139) 2.782 (0.234)

3.0 1.086 (0.015) 0.045 (0.005) -0.546 (0.164) 3.246 (0.372)

0.5 1.032 (0.050) 0.492 (0.041) 0.448 (0.116) 2.238 (0.217)

1.0 1.077 (0.031) 0.188 (0.016) -0.062 (0.116) 2.376 (0.162)

2.0 1.5 1.116 (0.022) 0.101 (0.010) -0.349 (0.135) 2.766 (0.232)

2.0 1.142 (0.018) 0.063 (0.007) -0.522 (0.154) 3.126 (0.334)

3.0 1.173 (0.012) 0.031 (0.004) -0.711 (0.182) 3.624 (0.500)

0.5 1.098 (0.039) 0.305 (0.023) 0.041 (0.104) 2.064 (0.117)

1.0 1.178 (0.024) 0.115 (0.011) -0.416 (0.127) 2.728 (0.231)

3.0 1.5 1.219 (0.017) 0.059 (0.007) -0.646 (0.158) 3.306 (0.392)

2.0 1.243 (0.013) 0.036 (0.004) -0.772 (0.180) 3.702 (0.521)

3.0 1.269 (0.009) 0.017 (0.002) -0.895 (0.206) 4.140 (0.683)

Figure 2: Plots of the Skewness and Kurtosis of the IWW3 as a function of n and p, respectively.

Definition 4: Let X have the PDF given by (3). Then the Mathai-Houbold (M-H) entropy of X is given by

(?) l~6p2-6M*j (6, T) - 1

jmh (6) =

(7)

wher e, (5, Y) = £>o j (—^ 5))Г((2—5)—(1—5)п—1), 5 = land 5 < 2.

4Y , ^>0Ч>0 iljl(j-v(2-5+i))(2-5)-(1-5)4 1 r(q(2—5+i) —(1 —5+j))

We presented some numerical values for the Renyi and Mathai-Houbold entropies in Table 2. We note a decrease in the entropy as the values of either ц or q increases for the Renyi. While for the M-H, we observed that the entropy increases with an increase in ц and decreases with an increase in q. It can further be seen that entropy can take negativ e values, which may be understood as a loss of infor mation in physical systems [21].

Table 2: Renyi and Mathai-Houbold entropies for some values of the parameters

n i q T a Renyi q i V T a Renyi

0.1 2.9245 0.7 1.1692

0.2 1.2632 0.9 0.9486

0.4 0.8462 1.2 0.7302

0.6 1.1 1.2 0.6 0.7562 1.4 0.5 1.2 0.6 0.6229

0.7 0.7243 1.6 0.5330

0.8 0.6940 1.8 0.4549

0.85 0.6790 2.0 0.3853

q T 5 M-H rj T 5 M-H

0.58 0.60 0.62 0.64 0.66 0.68 0.70

0.7 1.2

0.6

0.3928 0.6530 0.8740 1.0635 1.2278 1.3716 1.4986

0.68 0.70 0.72 0.74 0.76 0.78 0.80

0.5 1.2

1.6

2.0974

1.4760

0.9609

0.5311

0.1700

-0.1352

-0.3946

3. Characterization

Characterization guides an investigator in designing a stochastic model for a particular modelling problem to know if the model fits the conditions of a specific underlying probability distribution. The investigator will depend on the characterization of the chosen distribution. The technique characterizes distribution and its random variable when the distribution conditions are similar to those of the random variable. This section presents the IWW3 distribution characterization in two directions (i) in terms of the simple relationship for the ratio of two truncated moments and (ii) based on the hazard function. We employed a theorem due to [22], for the first characterization (see Theorem 1). Note that the first characterization can be utilized even when the CDF's closed-for m does not exist.

3.1. Characterizations based on two truncated moments

Theorem 1. Let (Q, E, P) be a given probability space and let H — [a1, a2], - be an interval for some a1 < a2 (a1 — - <x>, a2 — ™ might as well be allowed). Let X : Q —>• H be a continuous random variable with the distribution function F and let q1 and q2 be two real functions defined on H such that

E [q2(X)|X > x] — E [q1 (X)|X > x] n(x),x £ H

is defined with some real function n. Assume that q1, q2 £ C1 (H), n £ C2 (H) and F is twice continuously differentiable and strictly monotone function on the set . Finally, assume that the equation nq1 — q2 has no real solution in the interior of H. Then F is uniquely deter mined by the functions q1, q2 and n , particularly

n' (u)

F(x) = i;c

n(u)q1 (u) - q2(u)

exp [—s(u)] du

wher e the function s is a solution of the differential equation sf = n q' and C is a constant

m n qi -q2

chosen to make JHdF = 1 .

Proposition 1: Let X : Q —>• (0, œ) be a continuous random variable and let q1 (x) = 1 and q2 (x) = exp [—k? ], wher e k = e(x/ t) — 1 and x > 0. The random variable X has PDF (3), if and only if the function n(x) defined in Theorem 1 has the form

n(x) = 1 exp [—k?], x > 0.

Proof. Let the random variable X has the PDF (3), then

(1 — F(x))E [q1 (X)|X > x] = exp [—k?], x > 0,

and

(1 — F(x))E [q2(X)|X > x] = 1 exp [—k?], x > 0.

Further ,

n(x)q1 (x) — q2(x) = —-exp [—k?] < 0, for x > 0

Conversely, if n is given as above, then

S(x) = f n(xx)q1 (x\, = n?(x/ T)n—1 (K + 1)K?—1 = ^, for x > 0

n(x)q1 (x) — q2 (x) rv F(x)

Therefore, according to Theorem 1, X has PDF (3).

Corollary 1. Let X : Q —>• (0, œ) be a continuous random variable and let q1 (x) be as in Proposition 1. The PDF of X is (3) if and only if there exist functions q2(x) and n(x) defined in Theorem 1 satisfying the differential equation

n' Wq1 (x) = n? (x/ r)n—1 (k + 1)k?—1, where k = e(x/ t)n — 1 and x > 0

n(x)q1 (x) — q2 (x) rv

Remark 1. The general solution of the differential equation in Corollary 1 is

n(x) = exp [—k?]

■ ç (x/ t)n—1 (k + 1)k?—1 exp [—k?] (q1 ) — 1 q2dx + D

where D is a constant. Note that one set of functions satisfying the differential equation is given in Proposition 1 with D = 0.

3.2. Characterization based on hazar d function

The hazar d function, h(x) of a twice differentiable distribution function, F(x), satisfy the following first order differential equation

f'(x) h'(x) .. .

= - h(x) (8)

f(x) h(x)

Proposition 2: Let X : Q —>• (0, to) be a continuous random variable. The random variable X has PDF (3), if and only if its hazar d function h(x) satisfy the following differential equation

h'(x) - | T (x/ T)(n-1) + ^^ (x/ t)(-1)} h(x) = ^^^ (x/ T)2(n-% + 1)2

x k?-2

under the boundar y conditions h(0) > 0n and k = e(x/ t) - 1. Proof. If random variable X has the hazar d function given in (2), then

h'(x) = n (l) (x/ T)2(n-1)(k + 1)2k?-2 + ^ (x/ r)2(n-1)(k + 1)k?-1

+ (x/ t)(n-2)(K + 1)K?-1

and hence,

hM - h(x) — (xi T)(n-1)(K + l)K-l + n-(xi T)(n-1) + (xi T)(-1)

h( x) T T T

- T(xi t)(--1)(k + 1)K*-1

Similarly ,

f' (x) — f (x) { (xi T)(n-1)(K + l)K-l + T (xxi T)(n-1) + (xi T)(-1)}

- (xi t)(--1)(k + 1)K*-1 f (x)

(9)

and thus,

f (x)

— n(*~ l) (xi T)(n-1)(K + l)K-l + T(xi T)(n-1) + ^(xi T)(-1) - -T*(xi t)(--1)(k + 1)K*-1

(10)

Equations (9) and (10) satisfied the differential equation (8), and hence, the IWW3 random variable X has the hazar d function (2).

4. Parameter estimation

In this section, we use the method of maximum likelihood to estimate the unknown parameters of the distribution for complete dataset. Let x1, x2, . . . , xn be a random sample of size n from the IWW3 model with the vector of parameters Y — (n, *, t). Then the log-likelihood function of Y from the PDF (3) is

/ n* \ n n n .

£(Y) — nlog (nT) +(n - 1) r log (zi) + * r z'- + (* - 1) r log (l - e-Zin)

i—1 i— 1 i—1

n

- r {eZiii - ^ *

i—1

wher e, z' — T. The estimate Y — (-, *, t)T of Y — (-, *, t)T is deter mined by maximizing the log-likelihood function £ (Y) with respect to each of the IWW3 parameters. Thus, we have following score functions.

d£dp — n + r logz' - * r (f - l\*-1 e^ z-log ^) + * ITz'1 log (tz{)

+ (* - 1) f^z-log (tz') (e^ - l)

'—1 '—1 '—1

n \ -1

(11)

'—1

^ * - ^ P- l\*log if'1" - l\ + r V + Dog (l - 'e-'1''\ <12)

T T '—1 '—1 '—1

and

3£ (Y) _ -

dT

(13)

+ * rr z'1 (l - ezi- (ezi1 - l\ - (* - 1) r z'1 (ez'- - l\ -1 '—l V / '—1

Solving equations (11)-(13) analytically may be intractable. Thus, a numerical appr oach is adopted to obtain the maximum likelihood estimates (MLEs) of the parameters Y — (-, *, t)T with a good set of initial values using R statistical package. To obtain the asymptotic interval estimation of

Y — (-, *, T),we determine the observed Fisher information matrix. The 3 x 3 Fisher information matrix is

fJ--(Y) J- * (Y) J-t (Y)\ J (Y) — - ( J*- (Y) J** (Y) J* (Y) I \Jt-(Y) Jt*(y) Jtt(Y)J

where the expressions of Jfi,Yj — dY^f1-, ',' — 1,2,3. Thus, the approximate variance of Y — ( , *, T) can be obtain as

(var(-j) cov(-, *) cov(-, T)\

cov(*,') var(*) cov(*, T) I

cov(T,if) cov(T, *) var(T) )

Hence, the 100(l - a)% asymptotic confidence intervals of Y — (-, *, t) are given by - ± Z var(ij), * ± Z var( *), and -f ± Z a var(T).

wher e Za is the upper ath percentile of the standar d normal distribution.

5. Simulation results

The main objectiv e in this section is to evaluate the perfor mance of the maximum likelihood method for estimating the IWW3 distribution parameters for a complete dataset via Monte Carlo simulation. For this purpose, we used six different combinations of the distribution parameters, including (1.8, 0.5, 0.5), (1.8, 0.7, 0.5), (1.8, 0.9, 0.5), (2, 0.5, 0.5), (2, 0.7, 0.5), and (2, 0.9, 0.5). The process is repeated 1000 times for four sample sizes n = 100, 150, 200, and 300. Table 3 presents the MLEs, Biases, and Mean Square Errors (MSEs) of the parameters. Based on the results, we observe that the ML method performs well for estimating the distribution parameters. Also, as the sample size increases, the biases and the MSEs of the MLEs decreases as expected.

6. A pplications

In this section, we analyze two different datasets to assess the potentiality of the IWW3 distribution in practice. The datasets are Aarset data [23] and Meeka and Escoba data [19]. Both the two datasets have a bathtub-shaped hazar d rate. We compar ed the results of the IWW3(Y) with Weibull and other Weibull extended models, including the exponentiated Weibull (EW) by [6], Weibull-W eibull(WW) by [10], Weibull-exponential (WE) by [11] and new Weibull-W eibull (NW-W) by [13]. To accomplish the purpose, We manage the maximum £(Y),Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC) and Bayesian Information Criterion (BIC). The Kolmogor ov-Smir nov (K-S) test is used to measur e the closeness between the empirical and the fitted distribution.Generally , the smaller the value of these statistics, the better the model fit the dataset. All computations were done using RStudio 1.2.5042 software.

6.1. Aarset data

Here, we employed Aarset data [23], which is considered by many authors, such as [6], as standar d data for assessing distributions with bathtub-shaped FR. It repr esents the failur e times of fifty components placed on life test at time zero. The data revealed a bathtub-shaped FR, as shown by the TTT-plot in Figure 3. Table 4 presents the MLEs of the parameters of IWW3(Y) together with that of EW, WE, W, WW, and NW-W for the Aarset data. From Table 5, the IWW3(Y) has the smallest -£(Y), AIC, CAIC, and BIC values, thus, the IWW3 model provide a best fit for the Aarset data. For the non-parametric goodness-of-fit statistics, the IWW3 model has the smallest K-S value with the highest p-value, which suggests that the IWW3 model has a better fit for the data set than the other competing models.

Table 3: MLEs, Biases, and MSEs of the distribution parameters.

Estimates

Biases

MSEs

n V

V

T

V

T

V

T

100

150

200

300

1.8 1.8 1.8 2.0 2.0 2.0 1.8 1.8 1.8 2.0 2.0 2.0 1.8 1.8 1.8 2.0 2.0 2.0 1.8 1.8 1.8 2.0 2.0 2.0

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

1.9714 2.0089 2.0485 2.1910 2.2326 2.2756 1.9050 1.9312 1.9570 2.1184 2.1458 2.1742 1.8735 1.8896 1.9039 2.0825 2.1005 2.1154 1.8509 1.8621 1.8712 2.0558 2.0688 2.0790

0.5414 0.9619 1.5201 0.5405 0.9666 1.5279 0.5146 0.7964 1.1864 0.5147 0.7968 1.1878 0.5085 0.7463 1.0521 0.5083 0.7456 1.0523 0.5050 0.7222 0.9718 0.5053 0.7222 0.9729

0.5263 0.6985 0.8309 0.5189 0.6548 0.7601 0.5039 0.5378 0.6115 0.5037 0.5304 0.5875 0.5027 0.5112 0.5388 0.5024 0.5090 0.5315 0.5018 0.5037 0.5136 0.5016 0.5033 0.5126

0.1714 0.2089 0.2485 0.1910 0.2326 0.2756 0.1050 0.1312 0.1570 0.1184 0.1458 0.1742 0.0735 0.0896 0.1039 0.0825 0.1005 0.1154 0.0509 0.0621 0.0712 0.0558 0.0688 0.0790

0.0414 0.2619 0.6201 0.0405 0.2666 0.6279 0.0146 0.0964 0.2864 0.0147 0.0968 0.2878 0.0085 0.0463 0.1521 0.0083 0.0456 0.1523 0.0050 0.0222 0.0718 0.0053 0.0222 0.0729

0.0263 0.1985 0.3309 0.0189 0.1548 0.2601 0.0039 0.0378 0.1115 0.0036 0.0304 0.0875 0.0027 0.0112 0.0388 0.0024 0.0090 0.0315 0.0018 0.0037 0.0136 0.0016 0.0033 0.0126

0.4260 0.7313 1.1138 0.5269 0.9076 1.3719 0.2569 0.4453 0.6846 0.3217 0.5492 0.8443 0.1610 0.2781 0.4310 0.1995 0.3430 0.5321 0.1077 0.1887 0.2973 0.1321 0.2327 0.3669

0.1539 2.2978 3.2315 0.0671 1.3555 2.1311 0.0013 0.2475 0.7578 0.0012 0.1516 0.4836 0.0007 0.0263 0.1530 0.0006 0.0145 0.1015 0.0004 0.0011 0.0362 0.0003 0.0010 0.0366

0.1529 2.2686 2.9566 0.0658 1.3084 1.8045 0.0011 0.2397 0.6882 0.0010 0.1431 0.4085 0.0006 0.0243 0.1314 0.0005 0.0125 0.0793 0.0004 0.0006 0.0313 0.0003 0.0005 0.0315

n

Figur e 4 presents the plots of the fitted PDFs (see Figur e 4, Fitted PDFs) and the estimated CDFs (see Figur e 4, estimated CDFs), which equally illustrate that the IWW3 model has fitted the data well compare to the other competing models. Moreover, Figure 4 (estimated hazard rate function) has indicated that the hazar d rate function is bathtub shaped, and hence, has ascertained the actual behavior of the data.

Table 4: MLEs and their standard errors (in parentheses) for the models fitted to the Aarset data.

Models n <P T e a

EW 0.0109 4.6713 0.1450

(0.0009) (0.0246) (0.0217)

WE 0.1742 0.3851 0.0778

(0.0621) (0.1029) (0.0257)

W 0.9488 44.847

(0.1196) (6.9313)

WW 0.2705 0.2558 0.0066

(0.0700) (0.0584) (0.0059) 1.6026

NW-W 0.4929 0.0021 1.4560 (0.2263)

(0.0893) (0.0004) (0.0690)

IWW3 5.4238 0.1363 61.067

(0.0040) (0.0230) (3.6212)

Table 5: The values of £(Y), AIC, CAIC, BIC, K-S(w'th 'ts p-value) statistics for the models fitted to the Aarset data.

Models -£(Y ) AIC CAIC BIC K-S p- value

EW 229.136 464.272 464.7937 470.008 0.2057 0.0291

WE 225.6185 457.2371 457.7588 462.9731 0.1289 0.3774

W 241.0019 486.0037 486.259 489.8278 0.1933 0.0477

WW 220.902 449.804 450.6929 457.4521 0.1308 0.3598

NW-W 230.2974 466.5948 467.1166 472.3309 0.2040 0.0312

IWW3 218.3491 442.6982 443.2199 448.4342 0.1192 0.4762

Figure 4: Fitted PDFs (left panel), estimated CDFs (center panel) and estimated hazard rate function (right panel) for some of the fitted models to Aarset data.

6.2. Meeker and Escobar data

The second data used is [19] data, which represents the failure and running times of n — 30 devices. It has been analyzed by many authors, including [24]. The data revealed a bathtub-shaped FR, as shown by the TTT-plot in Figure 5. Table 6 lists the MLEs of the parameters of IWW3( Y) together with that of EW, WE, W, WW, and NW-W for the data. From Table 7, it is note that the IWW3(Y) has the smallest -£(T), AIC, CAIC, and BIC values, thus, the IWW3 model provide a better fit for the data. For the formal non-parametric goodness-of-fit statistic, the IWW3 model has the smallest value for K-S, with the highest p-value, which also ascertains the IWW3 model well fits the Meeker and Escoba data.

Figure 6 presents the plots of the fitted PDFs (see Figure 6, Fitted PDFs) and the estimated CDFs (see Figur e 6, estimated CDFs), which illustrate that the IWW3 model has fitted the data well

compar e to the other competing models. Additionally , Figur e 6 (estimated hazar d rate function) has indicated that the hazar d rate function is bathtub shaped, and hence, has ascertained the actual behavior of the data.

Table 6: MLEs and their standard errors (in parentheses) for the models fitted to the Meeker and Escobar data.

Models n <P t 0 a

EW 0.0030 5.5320 0.1620 0.5886

(0.0002) (0.2077) (0.0319) (0.1375)

WE 0.1258 0.4976 0.0187

(0.0669) (0.2022) (0.0093)

W 1.2550 180.652

(0.2043) (26.8740)

WW 0.0860 0.4936 0.0539

(0.0409) (0.1760) (0.0304) 0.8357

NW-W 0.8785 0.0020 1.0669 (0.1147)

(0.1653) (0.0003) (0.0371)

IWW3 6.9601 0.1479 239.38

(0.0085) (0.0297) (14.149)

Table 7: The values of £(Y), AlC, CAIC, BIC, K-S(with its p-value) statistics for the models fitted to the Meeker and Escobar data.

Models ) AIC CAIC BIC K-S P- value

EW 177.9146 361.8293 362.7524 366.0329 0.2159 0.1219

WE 177.1573 360.3145 361.2376 364.5181 0.1724 0.3347

W 184.35 372.6999 373.1443 375.5023 0.2358 0.0712

WW 178.0157 364.0313 365.6313 369.6361 0.1650 0.3873

NW-W 180.3075 366.6149 367.538 370.8185 0.2234 0.1001

IWW3 170.804 347.608 348.5311 351.8116 0.1425 0.576

^-\-\-1-1-\

O.O 0.2 0.4 0.6 0.8 1 O

i/n

Figure 5: TTT plot for Meeker and Escoba data.

Figure 6: Fitted PDFs (left panel), estimated CDFs (center panel) and estimated hazard rate function (right panel) for some of the fitted models to Meeker and Escoba data.

7. C onclusion

This paper defined and studied a new generalized Weibull distribution called the Improved Weibull-W eibull (IWW3) distribution. It is a three-parameter flexible distribution with the ability to accommodates monotone and non-monotone failur e rates lifetime data. We obtain explicit expressions for the moment generating function, moments, quantile function, Renyi entropy, and Mathai-Houbold entropy. Numerical results for median, Renyi entropy, Mathai-Houbold entropy and conduct a Monte Carlo simulation study to obtain some numerical results for the mean, variance , ske wness, and kurtosis of the distribution. We also characterize the IWW3 model based on tw o truncated moments and in ter ms of the hazar d function. Estimation of the distribution parameters is perfor med using the method of maximum likelihood, and the estimation method is assessed by Monte Carlo simul ation experiments which yield consistent estimates in the samples consider ed. Two failur e time data having non-monotone failur e rate functions are analyzed to demonstrate the potentiality of the distribution.

Disclosur e statement

On behalf of The authors, I declare that no potential conflict of interest was reported.

Funding

No funding was provided for the work.

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