The Marshall-Olkin Right Truncated Fréchet-Inverted Weibull Distribution: Its Properties and Applications ()
1. Introduction
In the past, researchers highlighted on the inversion of univariate probability models. They have applied the inverse technique for many distributions. For example, there are many examples such as inverted beta [1], inverse Rayleigh [2], inverse Gaussian [3], inverse Weibull [4], inverted Burr type XII also called Burr type III [5], inverted exponential [6] and many other distributions. The Weibull distribution is a continuous probability distribution which identified by Fréchet [7]. Inverse Weibull distribution has two parameters
and
with probability density function of a random variable X is denoted by
(1)
The basic objective in this paper is the study how the Marshall and Olkin right truncated Fréchet-Inverted Weibull distribution applied. Later, Haq [8] used this conversion to improve Marshall-Olkin length biased moment exponential distribution. Marshall and Olkin [9] proposed an ingenious approach for adding an additional shape parameter to the existing distribution. So, we find the cumulative distribution function
of right truncated Fréchet-inverted Weibull distribution (RTFIWD)
(2)
The pdf
of (RTFIWD) is
(3)
the reliability function
(the survival function) of the Marshall and Olkin (MO) family is defined by
(4)
using (2) into (4), we obtain
by the Marshall and Olkin right truncated Fréchet-inverted Weibull distribution (MORTFIWD) as
(5)
The pdf corresponding to (5) is given by
(6)
and its hazard function reduces to
(7)
The cumulative distribution function
of the Marshall and Olkin of the right truncated Fréchet-inverted Weibull distribution (MORTFIWD) is denoted by
(8)
by substituting (5) in (8), we find
(9)
Figure 1 and Figure 2 outlined the manner of the density function and interprets the susctability and elasticity of the pattern graphically. The pdf plot shows that for some different values
. Figure 3 and Figure 4 outlined the manner of
Figure 1. The density function for different values of the parameters.
Figure 2. The density function for different values of the parameters.
Figure 3. The hazard rate function for different values of the parameters.
Figure 4. The hazard rate function for different values of the parameters.
hazard rate function. Figure 5 and Figure 6 outlined the manner of reserved hazard rate function which is monocular at different parameter combinations.
2. Moments
The rth moments
of MORTFIWD
(10)
Figure 5. The reserved hazard rate function for different values of the parameters.
Figure 6. The reserved hazard rate function for different values of the parameters.
since
, for
we have
(11)
by substituting (11) in (10), we find
(12)
where
(13)
(14)
(15)
Setting
we find (14) becomes as follows
where
by using the definition of the incomplete gamma function
So we find the rth moments
of MORTFIWD
(16)
3. Moment Generting Function
We need to compute the moment generting function of the Marshall and Olkin of the right truncated Fréchet-inverted Weibull distribution (MORTFIWD). If X a random variable has the distribution MORTFIWD where
and
is a positive integer, then the moment generting function of MORTFIWD is denoted by
(17)
From (11), we find
(18)
where
(19)
(20)
4. Inverse Moments
In this section, inverse moments
(21)
substituting by (11) in (10), we have
(22)
(23)
(24)
where
(25)
(26)
setting
hence (14) becomes as follows
(27)
where
by using the definition of the incomplete gamma function
so the inverse rth moments
about MORTFIWD
(28)
(29)
(30)
5. The Mean Time to Failure
We need to compute the mean time to failure of the Marshall and Olkin of the right truncated Fréchet-inverted Weibull distribution (MORTFIWD). If X a random variable has the distribution MORTFIWD where
and
is a positive integer, then the mean time to failure of MORTFIWD is given by
putting
in Equation (16), we find
If X a random variable has the distribution MORTFIWD. We find the mean time to failure of the Marshall and Olkin of the right truncated Fréchet-inverted Weibull distribution (MORTFIWD)
(31)
6. Maximum Likehood Method
The maximum likelihood assessment method is essentially used and extends the maximum acquaintance about the properties of the assessment parameters. Furthermore, the natural approximation of the estimators may truthfully and mathematically for huge sample theory. Accordingly, the Maximum likehood assessment has depended to guess the unknown parameters (
and b) of the MORTFIWD distribution. Let random variable X from the observed distribution and have the parameters (
and b)T with size n. The sample likelihood function is
The log-likehood function is
(32)
(33)
(34)
7. Application
The potentiality and flexibility of the new distribution is introduced in this section. The new model has contrasted with some other existing life-time models such as:
(i) Hinkley [10] introduced the first real-life data. Data compounds on 30 notices of the March precipitation in Minneapolis/St Paul. The values are
0.77 1.74 0.81 1.20 1.95 1.20 0.47 1.43 3.37 2.20 2.81 1.87
3.00 3.09 1.51 2.10 0.52 1.62 1.31 0.32 0.59 0.81 1.18 1.35
4.75 2.48 0.96 1.89 0.90 2.05
(ii) The second data also compounds on 30 values for the failure time which be repairable objects used by Murthy et al. [11]. The values are
1.43 0.11 0.71 0.77 2.63 1.49 3.46 2.46 0.59 1.97
0.74 1.23 0.94 4.36 0.40 1.74 4.73 2.23 0.45 1.86
0.70 1.06 1.46 0.30 1.82 2.37 0.63 1.23 1.24 1.17
(iii) Bhaumik [12] introduced the third real data life The values are
5.1 1.2 1.3 0.6 0.5 2.4 0.5 1.1 8.0 0.8 0.4 0.6
0.9 0.4 2.0 0.5 5.3 3.2 2.7 2.9 2.5 2.3 1.0 0.2
0.10 0.1 1.8 0.9 2.0 4.0 6.8 1.2 0.4 0.2
The new generated Marshall and Olkin of the right truncated Fréchet-inverted Weibull distribution (MORTFIWD) is compared with the right truncated Fréchet-inverted Weibull distribution, Marshall-Olkin extended inverted Kumaraswamy (MOEIK) distribution with the Generalized Inverted Kumaraswamy (GIK), Transmuted Exponentiated Inverse Rayleigh (TEIR), Logistic Weibull (LW), Transmuted Power Lindley (TPL), Marshall Olkin Frechet (MOFr) and Inverted Kumaraswamy (IK) distribution for these data sets.
The new distribution is compared with other distributions. By using some numerous fineness of fit measures such as the log likelihood function (2l), Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Consistent Akaike Information Criterion(CAIC), Hanna-Quinn Information Criterion (HQIC). The assessment parameters and fineness of fit measures of the pervious real life data sets. The fineness of fit measures such as AIC, BIC, CAIC, HQIC [13].
The guess of maximum likehood method for parameters with the fineness of fit measures (March precipitation).
Models Estimates −2l AIC BIC CAIC HQIC
MORTFIWD (α, β, b) 13.320 1.185136 4.75 75.0588 81.0589 81.982 85.2625 82.3036
RTFIWD (α, β) 1.38587 1.14316 - 75.7754 81.7754 82.6985 85.975 83.1202
MOEIK (α, β, λ) 4.3228 6.5798 6.9226 76.69 82.69 86.89 84.79 84.03
GIK (α, β, γ) 1.9552 3.9501 1.4202 78.63 84.63 88.84 86.77 85.98
TEIR (α, θ, λ) 6.5630 0.0958 −0.6700 84.20 90.20 94.41 92.30 91.55
LW (α, β, λ) 2.7709 0.3635 1.0061 77.86 83.86 88.06 85.96 85.21
TPL (α, θ, λ) 1.5965 0.4801 0.5812 77.30 83.31 89.05 86.18 86.19
MOFr (α, β, σ) 42.598 2.6975 0.3548 77.60 83.60 87.80 85.70 84.94
IK (α, β) 2.9872 8.5899 - 78.85 83.85 87.65 85.25 84.74
The guess of maximum likehood method for parameters with the fineness of fit measures (failure time data).
Models Estimates −2l AIC BIC CAIC HQIC
MORTFIWD (α, β, b) 30.3832 0.0614346 4.73 76.9488 82.9488 87.5279 83.7488 84.5104
RTFIWD (α, β) 1.39561 0.680631 − 79.1164 85.1164 89.32 86.0395 86.4611
MOEIK (α, β, λ) 3.8031 2.1861 9.6185 79.60 85.60 89.80 87.70 86.94
GIK (α, β, γ) 1.1623 1.4462 1.8481 81.64 87.64 91.84 89.74 88.98
TEIR (α, θ, λ) 7.2940 0.0254 −0.8803 117.0 123.0 127.2 125.1 124.3
LW (α, β, λ) 2.8927 0.5531 0.07775 80.83 86.83 91.03 88.93 88.17
TPL (α, θ, λ) 1.3380 0.6301 0.5356 80.73 86.73 90.93 88.83 88.07
MOFr (α, β, σ) 63.165 2.1070 0.1669 81.53 87.53 91.73 89.63 88.87
IK (α, β) 2.4609 4.1716 − 82.48 86.48 90.28 88.88 87.37
The guess of maximum likehood method for parameters with the fineness of fit measures (chloride data).
Models Estimates −2l AIC BIC CAIC HQIC
MORTFIWD (α, β, b) 2.27479 0.401845 8 106.265 112.265 113.065 116.844 113.826
RTFIWD (α, β) 1.09489 0.602397 − 106.279 112.28 113.08 116.859 113.841
MOEIK (α, β, λ) 2.1193 1.7026 2.2471 110.9 116.9 121.5 119.2 118.5
GIK (α, β, γ) 2.0236 2.6369 0.8858 111.5 117.5 122.1 119.8 119.1
TEIR (α, θ, λ) 6.9784 0.0138 −0.7798 170.8 176.8 181.4 179.1 178.3
LW (α, β, λ) 2.7709 0.7056 0.5526 111.9 117.9 122.5 120.2 119.5
TPL (α, θ, λ) 0.9265 0.7453 0.4094 111.2 117.2 122.5 120.5 119.7
MOFr (α, β, σ) 29.053 1.4730 0.1124 111.4 117.4 121.9 119.7 119.3
IK (α, β) 1.7409 2.1058 − 112.5 117.5 122.6 120.1 119.6
8. Conclusion
A new probability distribution is introduced by using Marshall and Olkin transformation. Some of its properties such as moments, moment generating function, order statistics and reliability functions are derived. The method of maximum likelihood is used to estimate the model parameters.