A Note on Topp-Leone Odd Log-Logistic Inverse Exponential Distribution

Salman Abbas; Fakhar Mustafa; Syed Ali Taqi; Selen Cakmakyapan; Gamze Ozel

doi:10.2991/jsta.d.200827.002

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Volume 19, Issue 3, September 2020, Pages 397 - 407

A Note on Topp-Leone Odd Log-Logistic Inverse Exponential Distribution

Authors

Salman Abbas¹^{, *}, Fakhar Mustafa², Syed Ali Taqi¹, Selen Cakmakyapan³, Gamze Ozel⁴

¹Department of Statistics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan

²Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus, Sahiwal, Pakistan

³Department of Statistics, Istanbul Medeniyet University, İstanbul, Turkey

⁴Department of Statistics, Hacettepe University, Ankara, Turkey

^*Corresponding author. Email: salmanabbas@ciitwah.edu.pk

Corresponding Author

Salman Abbas

Received 10 March 2019, Accepted 16 January 2020, Available Online 7 September 2020.

DOI: 10.2991/jsta.d.200827.002 How to use a DOI?
Keywords: Inverse exponential distribution; Topp-Leone odd log-logistic family; Hazard function; Residual and reversed residual life functions; Unimodal
Abstract: The inverse exponential distribution is widely used in the field of reliability. In this article, we present a generalization of the inverse exponential distribution in formation of Topp-Leone odd log-logistic inverse exponential distribution. We provide a comprehensive account of some mathematical properties of the Topp-Leone odd log-logistic inverse exponential distribution. The possible shapes of the corresponding probability density function and hazard function are obtained and graphical demonstration are presented. The distribution is found to be unimodal. The results for moment, moment-generating function, and probability-generating function are computed. The residual and reversed residual functions are also obtained. The proposed method of maximum likelihood is used for the estimation of model parameters. The performance of the parameters is investigated through simulation. The usefulness of the proposed model is illustrated by means of a real data set.
Copyright: © 2020 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Keller and Kamath [1] introduced the inverse exponential (IEx) distribution to study the reliability of computer control (CNC) machine tools. Lin et al. [2] discussed the IEx distribution in term of different causes of failure for the machines. They obtained the maximum likelihood estimator and confidence limits for the parameter and the reliability function using complete samples. They also compared this model with the inverted Gaussian and log-normal distributions based on a maintenance data set. Sanku Dey [3] and Gyan Prakash [4] studied the IEx distribution according to Bayesian point of view. Abouammoh and Alshingiti [5] worked on the addition of an extra shape parameter to obtain the generalized IEx distribution. It is to be noted that this distribution originated from the exponentiated Frechet distribution studied by Nadarajah and Kotz [6].

Several interesting generalizations of the IEx distribution have been derived including transmuted-generalized IEx distribution by Elbatal [7], the Kumaraswamy IEx distribution by Oguntunde et al. [8], the transmuted IEx distribution by Oguntunde and Adejumo [9], the exponentiated-generalized IEx distribution by Oguntunde et al. [10], and the beta-generalized IEx distribution by Bakoban and Abu-Zinadah [11].

The most common exhibition of hazard rates appeared in practice is the bathtub shaped. A simple model form bathtub-shaped hazard rates is the Topp-Leone distribution introduced by Topp and Leone [12]. The quantile function of this model has closed-form and the model is also very flexible to generate the data.

We start by defining the class of probability distribution introduced by Alizadeh et al. [13] using the cumulative distribution function (cdf) of the odd log-logistic (OLL) distribution as a generator and named as the Topp-Leone odd log-logistic (TLOLL-G) family of distributions. The proposed family has two additional shape parameters. The distribution function (cdf) of the proposed class of distributions is given as follows:

F(x)=1−1−G(x;ξ)aG(x:ξ)a+Ḡ(xξ)a2b,(1.1)

and the corresponding density function (pdf) of the proposed family is given as

f(x)=2abg(x;ξ)G(x,ξ)a−1Ḡ(x;ξ)2a−1[G(x;ξ)a+Ḡ(x;ξ)a]31−1−G(x;ξ)aG(x:ξ)a+Ḡ(xξ)a2b−1,(1.2)

where a>0 and b>0 are the shape parameters, ξ is the vector parameter of the parent G(x), g(x) is the derivative of G(x), and G(x)¯=1−G(x).

In this study, we introduce a generalization of the IEx distribution named as the Topp-Leone odd log-logistic inverse exponential (TLOLL-IEx) distribution. The motivation of this generalization of IEx distribution is to enhance its applicability and adaptability. The addition of two shape parameters makes it more compatible for modeling the real life scenarios.

The organization of this article is as follows: The proposed TLOLL-IEx distribution and its properties are given in Section 2. Estimation of the model parameters is computed in Section 3. The performance of the parameters is investigated through simulation in Section 4. The application of the proposed model is illustrated in Section 5. The discussion ends with some concluding remarks in Section 6.

2. TOLL-IEx DISTRIBUTION AND ITS PROPERTIES

In this section, we derive the density function of the TLOLL-IEx distribution. For this aim, we consider the cdf of IEx distribution as

G(x)=e−λx,x>0,λ>0,(2.1)

and the pdf is given by

g(x)=λx2e−λx,x>0,λ>0.(2.2)

Using (2.1) in (1.1), we have the cdf of TLOLL-IEx distribution as

F(x)=1−1−e−λxe−λx+1−e−λxa2b,x>0,a,b,λ>0.(2.3)

The corresponding density of the TLOLL-IEx distribution is obtained by differentiating Eq. (2.3) and given as

f(x)=2abλx2e−aλx1−e−λx2a−1e−aλx+1−e−λxa31−1−e−λxe−λx+1−e−λxa2b−1,x>0,a,b,λ>0.(2.4)

We study some useful and important properties of the new distribution. The distributional properties of the proposed distribution is obtained for real values of a and b.

2.1. Shape

The density function TLOLL-IEx distribution as form (2.4) is given as

f(x)=2abλx2e−aλx1−e−λx2a−1e−aλx+1−e−λxa31−1−e−λxe−λx+1−e−λxa2b−1.

For real b, we use following series representation given by Prudnikov et al. (1986) as

(1+x)α=∑j=0∞Γ(α+1)j!(α+1−j)cj,α>0.

Then, the pdf of TLOLL-IEx distribution is written as

f(x)=∑k−0∞dk+1γλx−2e−λxγ.(2.5)

Here, dk is given by

dk=∑i=0∞∑j=02i(−1)i+jbi2ijck,j.

We define ck,j as

ck,j=1b0,jak,j−1b0,j∑r=1kbk,jck−r,j,

where bk,j=hk(a,j) and ak,j=∑l=k∞(−1)l+kjallk. Hence, in (2.5) a and b are real numbers. The density of TLOLL-IEx distribution is a weighted sum of infinite IEx distribution or it can also be termed specifically as non-central IEx distribution. Figure 1 demonstrates some of the possible shapes of the density function for the selected values of a,b, and λ.

Quantile Function and Various Relevant Measures

The quantile function of TLOLL-IE distribution corresponding to (2.1) is given by

x=Q(u)=1λ1−1−u1b−121a1+1−1−u1b−121a,0<u<1.(1)

The above expression concede us to extract the following form of statistical measures for the proposed model i. The first quartile Q1, the second quartile Q2(median), and the third quartile Q3 of the TLOLL-IE distribution correspond to the values u = 0.25, 0.50, and 0.75, respectively. ii. The skewness and kurtosis can be calculated by using the following relations, respectively. Bowleys skewness is based on quartiles in Kenney and Keeping (1962) calculated by

sk3=Q3−2Q2+Q1Q3−Q1.(2)

Moors kurtosis (Moors, 1988) is based on octiles via the form

sk4=q78−q58−q38+q18q68−q28,(3)

where q(.) is the quantile function explain in (3.2).

Asymptotics

Corollary 3.1:

Let c=inf(x|G(x))>0. Then, the asymptotics of F(x),f(x) and h(x) when x→c are, respectively, given as follows:

F(x)∼2ae−λxbasx⟶c,f(x)∼b(2a)λλx2e−λxbasx⟶c,h(x)∼b2aλλx2(e−λx)basx⟶c.

Corollary 3.2:

The asymptotics of F(x),f(x) and h(x) when x→∞ are, respectively, given by

1−F(x)∼b1−e−λx2a as x⟶0,f(x)∼2abλx2e−λx1−e−λx2a−1 as x⟶0,h(x)∼2aλx2e−λx1−e−λx as x⟶0.

2.2. Moment

The rth noncentral moment, μr′=E(Xr) of the TLOLL family is given as

μr′=E[xr]=∑k=0∞dk+1∫0∞xrπk+1(x),

where πk+1(x)=γg(x)G(x)γ−1. After some calculation, the rth moments of the TLLOL-IE distribution are derived as

μr′=∑k=0∞dk+1(γλ)rΓ(1−r).(2.6)

One can see, (2.6) is an infinite weighted sum of moments of the IEx distribution. The higher-order moments can be obtained by substituting r<0 in (2.6). The variance, skewness, and kurtosis measures can be calculated using the relations given by

Var(x)=E(x2)−[E(x)]2,

Skewness(x)=E(x3)−3E(x)E(x2)+2E3(x)Var32(x),

kurtosis(x)=E(x4)−4E(x)E(x3)+6E(x2)E2(x)−E3(x)Var2(x).

2.3. Moment-Generating Function and Probability-Generating Function

The moment-generating function (mgf) of the TLOLL-IEx distribution is obtained by

Mx(t)=∫−∞∞etxf(x),

where etx=∑h=0∞txxhh!. Using the pdf (2.4), the mgf of the TLOLL-IEx distribution is given as follows:

Mx(t)=∑k,h=0∞tmh!dk+1(γλ)1−hΓ(1−h),

Similarly, the probability-generating function (pgf) of the TLOLL-IEx distribution is defined as follows:

δx(t)=∫0∞tx∑k=0∞dk+1γλx2e−λxγdt,

where tm=∑m=0∞(lnt)mxmm!, by solving the above expression, the pgf of the TLOLL-IEx distribution is given as follows:

ρx(t)=∑k,m=0∞(lnt)mm!dk+1(γλ)1−mΓ(1−m).

Some reliability measure including hazard rate function also known as reliability function rf, residual life function, and reversed residual life function. The expression of survival function of the TLOLL-IEx distribution is given as follows:

R(x)=1−∑i,k=0∞∑j=02idke−λxγ,x>0,a,b,λ>0,(4)

2.4. Hazard Function

The hazard function for any probability distribution is given as

h(x)=f(x)1−f(x).

For TLLOL-IE distribution, the hazard function is obtained by

h(x)=∑k−0∞dk+1γλx−2e−λxe−λxγ−11−∑i,k=0∞∑j=02idke−λxγ,x>0,a,b,λ>0,

Figure 2 presents, the plots of the hazard function for TLOLL-IEx distribution with several values of parameters. As seen in Figure 2, the hazard function of the TLOLL-IEx distribution is very flexible. The reversed hazard function is obtained as follows:

r(x)=2abλx2e−aλx1−e−λx2a−1e−aλx+1−e−λxa31−1−e−λxe−λx+1−e−λxa2

and the cumulative hazard function is given by

H(x)=−log1−1−1−e−λxe−λx+1−e−λxa2b.

2.5. Moments for Residual and Reversed Residual Life

The s−th moment of the residual life, says ms(t)=E[(x−y)n|X>y|],n=1,2,…. is unusually obtained using cdf. The s−th moment of the residual life for X is given by

ms(t)=1R(y)∫y∞(x−y)sdF(x),

the final expression of the moment of the residual life is given by

ms(y)=1R(y)∑k,r=0∞(−y)s−ryksr∫y∞xrπk+1(x)=1R(y)∑k,r=0∞(−y)s−ryksrdk+1γ(1−r),γλy.

The mean residual life of X shows the expectation of additional life length for a unit which survive at age y. It can be easily computed by setting the value s=1. The p−th moment of the reversed residual life says Mp(t)=E[(y−1)n|X≤y|], for y>0 and p=1,2,…. is uniquely computed using F(y) and given by

Mp(y)=1F(y)∫0y(y−x)ndF(x).(2.7)

The p−th moment of the reversed residual life is determine as

Mp(Y)=1F(y)∑k,r=0∞(−1)yknrtp−r∫y∞xrπk+1(x),=1F(y)∑k.r=0∞(−1)ykpryp−rdk+1γ(1−r),γλy.(2.8)

3. INFERENCE

In this section, the method of maximum likelihood is considered for the estimation of parameters of TLOLL-IEx distribution. Let, x1,….,xn be the random sample from the TLOLL-IEx distribution with shape parameters a, b, and p∗1 is baseline vector parameter λ. The log-likelihood function for Θ=(a,b,λT)T, say L=L(Θ), is obtained by

L=nlog(2)+nlog(a)+nlog(b)+∑i=0nλloge−λxx2+(a−1)∑i=0nloge−λx+(2a−1)∑i=0nlog1−e−λx−3∑i=0nloge−λxa+1−e−λxa+(b−1)∑i=1nlog1−1−e−λxae−λxa+1−e−λxa2.(3.1)

We can maximize the above equation using R (optim function), SAS (PROCNLMIXED), and Ox program (MaxBFGS sub-routine) or by solving the nonlinear likelihood equations by differentiation. The score vector components, say I(Θ)=(∂L∂a,∂L∂b,∂L∂λT)T=(Ia,Ib,IλT)T, are given as follows:

Ia=na+∑i=0nlogw+2∑i=0nlog1−w−3∑i=0nwalogw+1−walog1−wwa+1−wa+(b−1)×∑i=1n−21−wawa+1−wawawalogw+1−walog1−wwa+1−wa2−walogwwa+1−wa1−1−wawa+1−wa2

Ib=nb+∑i=1nlog1−1−wawa+1−wa2

Iλ=−3∑i=0naw1−wa−1x−awaxwa+1−wa+(2a−1)∑i=0nwx1−w−(a−1)∑i=0n1x−λ∑i=0n1xx2+∑i=0nlogwx2+(b−1)×∑i=1n−21−wawa+1−waaw1−wa−1x−awaxwawa+1−wa2+awaxwa+1−wa1−1−wawa+1−wa2.

Here, w=e−λx. We can find the maximum likelihood estimations (MLEs) by setting the above equations equal to 0 and solving them iteratively. The Fisher information matrix for the parameters of the TLOLL-IEx distribution is obtained by

âb̂λ̂∼Nabλ,ĴaaĴabĴaλĴbbĴbλĴλλ

1J=−EJaaJabJaλJbbJbλJλλ.

By determining the inverse dispersion matrix, the asymptotic variances and covariances of the MLEs for a,b, and λ can be obtained. Using above equations, approximate 100(1−λ)% confidence intervals for a,b, and λ are, respectively, obtained as

â±Zγ2Ĵaa,b^±Zγ2Ĵbb,λ^±Zγ2Ĵλλ,

where Zγ is the upper 100γth quantile of the standard normal distribution.

4. SIMULATION STUDY

In this section, we conduct a simulation study. We generate 10,000 samples of size, n = 50, 100, 250, 500 and n = 1000 of the TLOLL-IEx model. The evaluation of estimates is based on the mean, the mean squared error (MSE), and We use R software for computation. The results in Table 1 show that the estimates are closer to the true values of the parameters from all sample size which clearly indicates that estimate are quite suitable. As the value of n increases, we can see that the estimates tend to move toward their true values which justifies the fact of asymptotic normality.

	b=1		Mean			MSE			AL
λ	a	n	λ	b	a	λ	b	a	λ	b	a
		50	0.661	1.321	0.783	0.312	0.215	1.315	0.831	0.737	0.749
		100	0.601	1.135	0.632	0.234	0.177	1.103	0.557	0.811	0.861
0.5	0.5	250	0.549	1.075	0.554	0.133	0.121	0.311	0.313	0.853	0.882
		500	0.531	1.041	0.516	0.031	0.037	0.108	0.104	0.873	0.931
		1000	0.503	1.009	0.505	0.011	0.015	0.036	0.023	0.941	0.953

		50	0.537	1.319	2.417	0.578	0.315	2.112	1.731	0.821	0.886
		100	0.522	1.217	2.313	0.441	0.257	1.011	1.335	0.851	0.922
0.5	2	250	0.519	1.033	2.139	0.361	0.113	0.713	0.716	0.885	0.936
		500	0.509	1.015	2.101	0.131	0.057	0.317	0.318	0.911	0.948
		1000	0.501	1.003	2.011	0.053	0.013	0.103	0.107	0.942	0.953

		50	0.631	1.353	1.619	0.485	0.553	0.972	2.538	0.831	0.841
		100	0.539	1.172	1.761	0.349	0.331	0.609	1.549	0.866	0.883
3	2.5	250	0.526	1.105	1.837	0.111	0.145	0.363	0.649	0.913	0.914
		500	0.513	1.051	1.935	0.018	0.051	0.131	0.341	0.937	0.938
		1000	0.501	1.011	1.992	0.001	0.012	0.015	0.131	0.944	0.948

Table 1

Estimated biases, mean squared errors (MSEs), average lenghts(ALS) for the several parameter values.

5. APPLICATION

In this part, we discuss the applicability of TLOLL-IEx model in real-life phenomena. We consider the data of the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20 mm given by Chwastiak etal. [14 ]. The data is given in Table 2. Then, fitted distributions and their abbreviations are presented in Table 3.

1.312, 1.314, 1.479, 1.552, 1.700, 1.803, 1.861, 1.865, 1.944, 1.958, 1.966, 1.997, 2.006,

2.021, 2.027, 2.055, 2.063, 2.098, 2.140, 2.179, 2.224, 2.240, 2.253, 2.270, 2.272, 2.274,

2.301, 2.301, 2.359, 2.382, 2.382, 2.426, 2.434, 2.435, 2.478, 2.490, 2.511, 2.514, 2.535,

2.554, 2.566, 2.570, 2.586, 2.629, 2.633, 2.642, 2.648, 2.684, 2.697, 2.726, 2.770, 2.773,

2.800, 2.809, 2.818, 2.821, 2.848, 2.880, 2.954, 3.012, 3.067, 3.084, 3.090, 3.096, 3.128,

3.233, 3.433, 3.585, 3.585

Table 2

Carbon fiber data.

Distribution	Abbreviation	References
Topp-Leone OLL IEx	TLOLL-IEx	Proposed
IEx	IE	[7]
Exponential	E	[15]
Lindley	L	[9]
Generalized Lindley	GL	[9]
Generalized Gamma	GG	[9]
Exponential Power	EP	[15]
Power Quasi Lindley	PQL	[16]
Weibull	W	[17]
Sujatha	S	[10]
Generalized Sujatha	GS	[10]

Table 3

Fitted distributions and their abbreviations.

The unknown parameters of each distribution are estimated using the maximum likelihood method. AIC (Akaike Information Criterion), CAIC (Consistent Akaike Information Criterion), and BIC (Bayesian Information Criterion) are used as goodness of fit measure. The estimated parameters of the fitted models, AIC, CAIC, and BIC are presented in Table 4. The values in this table clearly justify that the TLOLL-IEx distribution provides better fits than other models.

Distribution	Estimated Parameters			AIC	CAIC	BIC
TLOLL(a,b,λ)	6.4855	0.5668	2.0404	103.8239	104.1931	110.5262
IE(λ)	2.343			264.0296	264.0893	266.2637
E(β)	2.4513			263.7352	263.7949	265.9693
L(θ)	0.6545			240.3805	240.4402	242.6146
GL(θ,α,β)*	9.3907	22.7198	4.771	106.0848	106.454	112.7871
GG(θ,α,β)*	3.5861	2.6483	0.3044	103.9877	104.3569	110.69
EP(α,β)*	2.992	3.7061		111.204	111.3858	115.6722
W(α,λ)*	3.843	0.088		275.8682	276.05	280.3364
S(θ)*	0.0956			243.5	243.63	244.93
GS(θ,α)*	0.0972	14.473		244.54	244.68	243.97

MLE, maximum likelihood estimation; AIC, Akaike information criterion; CAIC, consistent Akaike information criterion; BIC, Bayesian information criterion.

*

Estimated parameter values are obtained from related references.

Table 4

MLEs and the values of AIC, CAIC, and BIC statistics.

We also obtain a visual comment with histogram given in Figure 3 for the best model.

6. CONCLUSION

In this paper, the discussion has been carried out through the generalization of the IEx distribution. We motivate from the TLOLL family of distributions and named the proposed model as TLOLL-IEx distribution. Several structural characteristics of the proposed distribution are derived and discussed. The maximum likelihood method is employed for estimating the model parameters. For effectiveness of the derived model, we consider a real-life data set and the results are compared with some well-known existing distributions. The TLOLL-IEx distribution is found unimodal. We hope that this generalization will applicable in the fields of reliability and lifetime analysis.

CONFLICTS OF INTEREST

The authors declare no potential conflict of interests.

AUTHORS' CONTRIBUTIONS

All authors contributes equally.

ACKNOWLEDGMENTS

We are very thankful to all referees for their precious time for the evaluation of the article. We also appreciate the correspondence of the editor.

REFERENCES

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3.S. Dey, Data Sci. J., Vol. 6, 2007, pp. 107-113.

4.G. Prakash, J. Modern Appl. Stat. Methods, Vol. 11, 2012, pp. 16.

5.A. Abouammoh and A.M. Alshingiti, J. Stat. Comput. Simulation, Vol. 79, 2009, pp. 1301-1315.

6.S. Nadarajah and S. Kotz, Interstat Electron. J., Vol. 14, 2003, pp. 01-07.

7.I. Elbatal, Econ. Quality Control, Vol. 28, 2013, pp. 125-133.

8.P. Oguntunde, O. Babatunde, and A. Ogunmola, Int. J. Stat. Appl., Vol. 4, 2014, pp. 113-116.

9.P. Oguntunde and A. Adejumo, J. Adv. Stat. Probability, Vol. 3, 2015, pp. 1-7.

10.P. Oguntunde, A. Adejumo, and O. Balogun, Appl. Math., Vol. 4, 2014, pp. 47-55.

11.R. Bakoban and H.H. Abu-Zinadah, REVSTAT Stat. J., Vol. 15, 2017, pp. 65-88.

12.C.W. Topp and F.C. Leone, J. Am. Stat. Assoc., Vol. 50, 1955, pp. 209-219.

13.E. Brito, G. Cordeiro, H. Yousof, M. Alizadeh, and G. Silva, J. Stat. Comput. Simulation, Vol. 87, 2017, pp. 3040-3058.

14.S. Chwastiak, J.B. Barr, and R. Didchenko, Carbon, Vol. 17, 1979, pp. 49-53.

15.N. Akdam, I. Kinaci, and B. Saracoglu, Hacettepe J. Stat., Vol. 46, 2017, pp. 239-253.

16.R.P. Covert and G.C. Philip, Vol. AIIE Trans. 5, 1973, pp. 323-326.

17.S. Mondal and D. Kundu, Sankhya Ser. B, Vol. 81, 2019, pp. 1-25.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 19 - 3
Pages: 397 - 407
Publication Date: 2020/09/07
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.200827.002 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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ris enw bib

TY  - JOUR
AU  - Salman Abbas
AU  - Fakhar Mustafa
AU  - Syed Ali Taqi
AU  - Selen Cakmakyapan
AU  - Gamze Ozel
PY  - 2020
DA  - 2020/09/07
TI  - A Note on Topp-Leone Odd Log-Logistic Inverse Exponential Distribution
JO  - Journal of Statistical Theory and Applications
SP  - 397
EP  - 407
VL  - 19
IS  - 3
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.200827.002
DO  - 10.2991/jsta.d.200827.002
ID  - Abbas2020
ER  -

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