Further results involving Marshall–Olkin log-logistic distribution: reliability analysis, estimation of the parameter, and applications
- Arwa M. Alshangiti^{1}Email author,
- M. Kayid^{1} and
- B. Alarfaj^{1}
Received: 22 November 2015
Accepted: 15 March 2016
Published: 31 March 2016
Abstract
The purpose of this paper is to provide further study of the Marshall–Olkin log-logistic model that was first described by Gui (Appl Math Sci 7:3947–3961, 2013). This model is both useful and practical in areas such as reliability and life testing. Some statistical and reliability properties of this model are presented including moments, reversed hazard rate and mean residual life functions, among others. Maximum likelihood estimation of the parameters of the model is discussed. Finally, a real data set is analyzed and it is observed that the presented model provides a better fit than the log-logistic model.
Keywords
Background
Recently, Gui (2013) introduced and studied the M–O log logistic distribution, denoted by M–O log-logistic. The paper’s objectives are to investigate some statistical and reliability properties of M–O log-logistic distribution and to illustrate its applicability in different areas. The paper is organized into five sections. The density and the moment of the model are given in “Extended log-logistic distribution” section. In that section, we provide some new statistical and reliability functions (reversed hazard rate, mean residual life, mean inactivity time, etc.) and discuss their properties. Furthermore, maximum likelihood estimation problems are considered in “Maximum likelihood estimators” section. To indicate the adequacy of the model, some applications using a numerical example and an example with real data are discussed in “Fitting reliability data” section. Finally, in “Conclusion” section, we provide a brief conclusion and some remarks regarding the current and future research (Additional file 1).
Extended log-logistic distribution
Statistical and reliability properties
The next result provide the behavior of the RHR of the M–O log-logistic \((\alpha ,\beta ,\gamma )\) distribution, and can be verified using elementary calculus.
Lemma 1
Let \(X \sim\) M–O log-logistic \((\alpha ,\beta ,\gamma )\), then the reversed hazard rate is decreasing if \(\beta >-1\), independent of \(\alpha\) and \(\gamma\).
Mean residual life of M–O log-logistic
\(\alpha\) | \(\beta\) | \(\gamma\) | MRL at \(t=2\) |
---|---|---|---|
0.3 | 3 | 2 | 1.16808 |
0.7 | 3 | 2 | 1.36339 |
1.5 | 3 | 2 | 1.69029 |
2.5 | 3 | 2 | 2.02493 |
Mean inactivity time of M–O log-logistic
\(\alpha\) | \(\beta\) | \(\gamma\) | MIT at \(t=2\) |
---|---|---|---|
0.3 | 3 | 2 | 0.84578 |
0.7 | 3 | 2 | 0.703955 |
1.5 | 3 | 2 | 0.614498 |
2.5 | 3 | 2 | 0.574372 |
Strong mean inactivity time of M–O log-logistic
\(\alpha\) | \(\beta\) | \(\gamma\) | SMIT at \(t=2\) |
---|---|---|---|
0.3 | 3 | 2 | 2.48358 |
0.7 | 3 | 2 | 2.14164 |
1.5 | 3 | 2 | 1.911302 |
2.5 | 3 | 2 | 1.804316 |
Mean, variance
Median
Median of M–O log-logistic
\(\alpha\) | \(\beta\) | \(\gamma\) | Median |
---|---|---|---|
0.3 | 2 | 0.2 | 0.109545 |
0.7 | 2 | 0.2 | 0.167332 |
1.3 | 2 | 0.2 | 0.228035 |
2 | 2 | 0.2 | 0.282843 |
Renyi entropy
Renyi entropy of M–O log-logistic
\(\alpha\) | \(\beta\) | \(\gamma\) | Renyi entropy |
---|---|---|---|
0.3 | 1.5 | 1 | 0.12391 |
0.7 | 1.5 | 1 | 0.68877 |
1.5 | 1.5 | 1 | 1.19687 |
2 | 1.5 | 1 | 1.38866 |
Maximum likelihood estimators
In statistics, maximum-likelihood estimation (MLE) is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, MLE provides estimates for the model’s parameters. The method of maximum likelihood corresponds to many well-known estimation methods in statistics.
MLE of the parameter \({\small \alpha }\)
\({\alpha }\) | n | Estimate | Bias | MSE |
---|---|---|---|---|
0.3 | 20 | 0.296275 | −0.00372454 | 0.0237631 |
50 | 0.231197 | −0.0688027 | 0.0182828 | |
70 | 0.252118 | −0.047882 | 0.0128916 | |
150 | 0.285643 | −0.0143573 | 0.00492491 | |
1.2 | 20 | 1.02572 | −0.174276 | 0.291608 |
50 | 1.20359 | 0.00358953 | 0.103632 | |
70 | 1.12847 | −0.0715284 | 0.119474 | |
150 | 1.19823 | −0.00177408 | 0.0339848 | |
2.5 | 20 | 2.34865 | −0.151351 | 1.2994 |
50 | 2.59336 | 0.0933644 | 0.517638 | |
70 | 2.57025 | 0.0702524 | 0.392246 | |
150 | 2.53202 | 0.0320212 | 0.151326 |
MLE of the parameter \({\beta }\)
\(\beta\) | n | Estimate | Bias | MSE |
---|---|---|---|---|
\({\alpha =0.3}\) | 20 | 1.91064 | −0.0893629 | 0.394056 |
50 | 2.02933 | 0.0293271 | 0.0744106 | |
70 | 1.9975 | −0.00249889 | 0.0850081 | |
150 | 2.01348 | 0.0134826 | 0.0190133 | |
\({\alpha =1.2}\) | 20 | 1.96364 | −0.036358 | 0.341544 |
50 | 2.03782 | 0.0378182 | 0.0577503 | |
70 | 2.02835 | 0.0283514 | 0.0409679 | |
150 | 2.01348 | 0.0134826 | 0.0190133 | |
\({\alpha =2.5}\) | 20 | 1.94389 | −0.0561107 | 0.372105 |
50 | 1.9975 | −0.00249889 | 0.0850081 | |
70 | 2.02563 | 0.0283514 | 0.0409679 | |
150 | 2.01095 | 0.0109501 | 0.0227298 |
Hence the variance covariance matrix would be \(I^{-1}( \theta )\). The approximate \((1-\delta )100\,\%\) confidence intervals (CIs) for the parameters \(\alpha\), \(\beta\) and \(\gamma\) are \(\hat{\alpha }\pm Z_{\frac{ \delta }{2}}V(\hat{\alpha })\), \(\hat{\beta }\pm Z_{\frac{\delta }{ 2}}V(\hat{\beta })\) and \(\hat{\gamma }\pm Z_{\frac{\delta }{2} }V( \hat{\gamma })\) respectively, where \(V(\hat{\alpha } )\), \(V(\hat{\beta })\) and \(V( \hat{\gamma })\) are the variances of \(\hat{\alpha }\), \(\hat{\beta }\) and \(\hat{\gamma }\), which are given by the diagonal elements of \(I^{-1}(\theta )\), and \(Z_{\frac{\delta }{2}}\) is the upper \((\delta {/}2)\) percentile of standard normal distribution.
Fitting reliability data
In this section, we provide two data sets analysis to show how the model works in practice.
First data set
Data set
389 | 18 | 22 | 10 | 112 | 63 | 100 | 13 | 151 | 467 |
162 | 117 | 122 | 33 | 42 | 99 | 283 | 80 | 314 | 112 |
Some properties of data set
E(X) | Var(X) | Kurtosis | Skewness |
---|---|---|---|
135.45 | 16,735.1 | 0.526142 | 1.26934 |
MLE for data set
Parameter | MLE |
---|---|
\({\alpha }\) | 10.833 |
\({\beta }\) | 1.58621 |
\({\gamma }\) | 19.7451 |
The K–S and p value of data set
K–S | p value |
---|---|
0.103124 | 0.131 |
The result of likelihood ratio test
Log-likelihood | \(\Lambda\) | p value |
---|---|---|
\(-118.851\) | 32.7814 | 1.03127 × 10^{−8} |
We note that the calculated LRT statistic is greater than the critical point for this test, which is 6.635, and also that the p value is very small. According to the LRT, we conclude that this data fits the M–O log-logistic much better than the log-logistic distribution.
Second data set
Data set
10 | 13 | 13 | 14 | 14 | 15 | 15 | 16 | 25 | 26 | 26 | 27 | 38 | 53 |
17 | 17 | 17 | 17 | 18 | 18 | 18 | 19 | 27 | 27 | 28 | 28 | 42 | |
21 | 21 | 21 | 22 | 22 | 23 | 24 | 25 | 30 | 34 | 35 | 35 | 42 |
Some properties of data set
E(X) | Var(X) | Kurtosis | Skewness |
---|---|---|---|
23.825 | 86.1481 | 0.934659 | 1.05974 |
MLE for data set
Parameter | MLE |
---|---|
\({\alpha }\) | 0.457038 |
\({\beta }\) | 4.70536 |
\({\gamma }\) | 26.0741 |
The K–S and p value of data set
K–S | p value |
---|---|
0.0982563 | 0.212 |
The result of likelihood ratio test
Log-likelihood | \(\Lambda\) | p value |
---|---|---|
−141.405 | 7.84139 | 0.00510632 |
We note that the calculated LRT statistic is greater than the critical point for this test, which is 6.635, and also that the p value is very small. According to the LRT, we conclude this data fits the M–O log-logistic much better than the log-logistic distribution.
Conclusion
- 1.
Discuss the Bayesian analysis of the model.
- 2.
Introduce and study a new class of weighted M–O bivariate log-logistic distribution.
Declarations
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Acknowledgements
This research project was supported by a grant from the “Research Center of the Female Scientific and Medical Colleges” , Deanship of Scientific Research, King Saud University.
Competing interests
All authors declare that there is no competing interests regarding the publication of this paper.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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