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Improved parameter estimation of the log-logistic distribution with applications

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Abstract

In this paper, we deal with parameter estimation of the log-logistic distribution. It is widely known that the maximum likelihood estimators (MLEs) are usually biased in the case of the finite sample size. This motivates a study of obtaining unbiased or nearly unbiased estimators for this distribution. Specifically, we consider a certain ‘corrective’ approach and Efron’s bootstrap resampling method, which both can reduce the biases of the MLEs to the second order of magnitude. As a comparison, the commonly used generalized moments method is also considered for estimating parameters. Monte Carlo simulation studies are conducted to compare the performances of the various estimators under consideration. Finally, two real-data examples are analyzed to illustrate the potential usefulness of the proposed estimators, especially when the sample size is small or moderate.

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References

  • Abbas K, Tang Y (2015) Objective Bayesian analysis for log-logistic distribution. Commun Stat Simul Comput. doi:10.1080/03610918.2014.925925

    MATH  Google Scholar 

  • Ashkar F, Mahdi S (2003) Comparison of two fitting methods for the log-logistic distribution. Water Resour Res 39:12–17

    Article  Google Scholar 

  • Ashkar F, Mahdi S (2006) Fitting the log-logistic distribution by generalized moments. J Hydrol 328:694–703

    Article  Google Scholar 

  • Balakrishnan N, Malik HJ (1987) Moments of order statistics from truncated log-logistic distribution. J Stat Plan Inference 17:251–267

    Article  MathSciNet  MATH  Google Scholar 

  • Chen Z (2006) Estimating the shape parameter of the log-logistic distribution. Int J Reliab Qual Saf Eng 13:257–266

    Article  Google Scholar 

  • Cordeiro GM, Klein R (1994) Bias correction in ARMA models. Stat Probab Lett 19:169–176

    Article  MathSciNet  MATH  Google Scholar 

  • Cox DR, Snell EJ (1968) A general definition of residuals. J R Stat Soc Series B Methodol 30:248–275

    MathSciNet  MATH  Google Scholar 

  • Efron B (1979) Bootstrap methods: another look at the Jackknife. Ann Stat 7:1–26

    Article  MathSciNet  MATH  Google Scholar 

  • Fisk PR (1961) The graduation of income distributions. Econometrica 29:171–185

    Article  MATH  Google Scholar 

  • Giles DE, Feng H, Godwin RT (2013) On the bias of the maximum likelihood estimator for the two-parameter Lomax distribution. Commun Stat Theory Methods 42:1934–1950

    Article  MathSciNet  MATH  Google Scholar 

  • Giles DE, Feng H, Godwin RT (2016) Bias-corrected maximum likelihood estimation of the parameters of the generalized Pareto distribution. Commun Stat Theory Methods 45:2465–2483

    Article  MathSciNet  MATH  Google Scholar 

  • Greenwood JA, Landwehr JM, Matalas NC, Wallis JR (1979) Probability weighted moments: definition and relation to paramters of several distributions expressible in inverse form. Water Resour Res 15:1049–1054

    Article  Google Scholar 

  • Gupta RC, Akman O, Lvin S (1999) A study of log-logistic model in survival analysis. Biometr J 41:431–443

    Article  MATH  Google Scholar 

  • Kantam RRL, Srinivasa GR (2002) Log-logistic distribution: modified maximum likelihood estimation. Gujarat Stat Rev 29:25–36

    Google Scholar 

  • Lemonte AJ (2011) Improved point estimation for the Kumaraswamy distribution. J Stat Comput Simul 81:41971–1982

    MathSciNet  MATH  Google Scholar 

  • MacKinnon J, Smith A (1998) Approximate bias correction in econometrics. J Econom 85:205–230

    Article  MathSciNet  MATH  Google Scholar 

  • Nelson NL (1982) Applied life data analysis. Wiley, New York

    Book  MATH  Google Scholar 

  • Pouzat C (2012) STAR: spike train analysis with R. R package version 0.3-7. http://CRAN.R-project.org/package=STAR

  • Schwartz J, Giles DE (2016) Bias-reduced maximum likelihood estimation of the zero-inflated Poisson distribution. Commun Stat Theory Methods 45:465–478

    Article  MathSciNet  MATH  Google Scholar 

  • Shoukri MM, Mian IUH, Tracy DS (1988) Sampling properties of estimators of the log-logistic distribution with application to Canadian precipitation data. Canad J Stat 16:223–236

  • Wang F (2000) A new model with bathtub-shaped failure rate using an additive Burr XII distribution. Reliab Eng Syst Saf 70:305–312

    Article  Google Scholar 

  • Wang M, Wang W (2017) Bias-corrected maximum likelihood estimation of the parameters of the weighted Lindley distribution. Commun Stat Simul Comput 46:530–545

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the editor and two reviewers for their constructive comments that led to an improvement of the paper. This work appeared in the first author’s Master report, which was supervised by the corresponding author.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA

    Joseph Reath, Jianping Dong & Min Wang

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  1. Joseph Reath
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  2. Jianping Dong
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  3. Min Wang
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Correspondence to Min Wang.

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Appendix

Appendix

To obtain the bias-corrected MLEs developed by Cox and Snell (1968), we need to calculate higher-order derivatives of the log-likelihood function of the log-logistic distribution. These derivatives, taken with respect to \(\alpha \) and \(\beta \), are given as follows

$$\begin{aligned} \dfrac{\partial ^2 \log (L)}{\partial \alpha ^2}&= \dfrac{n\beta }{\alpha ^2} - \dfrac{2\beta (\beta +1)}{\alpha ^2}\sum _{i=1}^n\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-1}\\&\quad +\dfrac{2\beta ^2}{\alpha ^2}\sum _{i=1}^n\left( \dfrac{X_{i}}{\alpha }\right) ^{2\beta }\left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-2},\\ \dfrac{\partial ^2 log L}{\partial \beta ^2}&= -\dfrac{n}{\beta ^2} - 2\sum _{i=1}^n\left[ \log \left( \dfrac{X_{i}}{\alpha }\right) \right] ^2\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-2},\\ \dfrac{\partial ^2 log (L)}{\partial \alpha \partial \beta }&= -\dfrac{n}{\alpha } + \dfrac{2}{\alpha }\sum _{i=1}^n\left[ \beta \left( \dfrac{X_{i}}{\alpha }\right) ^\beta \left[ \log \left( \dfrac{X_{i}}{\alpha }\right) \right] \left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-1}\right] \\&\quad +\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^\beta \left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-1}\right] \\&\quad -\sum _{i=1}^n\left[ \beta \left( \dfrac{X_{i}}{\alpha }\right) ^{2\beta }\left[ \log \left( \dfrac{X_{i}}{\alpha }\right) \right] \left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-2}\right] ,\\ \dfrac{\partial ^3 log (L)}{\partial \alpha ^3}&=-\dfrac{2n\beta }{\alpha ^3} + \dfrac{4\beta ^3}{\alpha ^3}\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^{3\beta }\left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-3}\right] \\&\quad -\dfrac{4\beta ^2}{\alpha ^3}\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^{2\beta }\left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-2}\right] \\&\quad - \dfrac{4\beta ^3}{\alpha ^3}\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^{2\beta }\left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-2}\right] \\&\quad - \dfrac{2\beta ^2}{\alpha ^3}\sum _{i=1}^n\left[ (1+\beta )\left( \dfrac{X_{i}}{\alpha }\right) ^{2\beta }\left[ 1+(\dfrac{X_{i}}{\alpha })^\beta \right] ^{-2}\right] \\&\quad +\dfrac{2\beta ^2}{\alpha ^3}(1+\beta )\sum _{i=1}^n\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-1}\\&\quad +\dfrac{4\beta \left( 1+\beta \right) }{\alpha ^3}\sum _{i=1}^n\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-1},\\ \dfrac{\partial ^3 log (L)}{\partial \alpha ^2 \partial \beta }&= \dfrac{n}{\alpha ^2} +\dfrac{4\beta }{\alpha ^2}\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^{2\beta }\left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-2}\right] \\ \end{aligned}$$
$$\begin{aligned}&\quad -\dfrac{2\beta }{\alpha ^2}\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^\beta \left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-1}\right] \\&\quad - \dfrac{2\left( 1+\beta \right) }{\alpha ^2}\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^\beta \left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-1}\right] \\&\quad - \dfrac{4\beta ^2}{\alpha ^2}\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^{3\beta }\left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-3}\log \left( \dfrac{X_{i}}{\alpha }\right) \right] \\&\quad + \dfrac{4\beta ^2}{\alpha ^2}\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^{2\beta }\left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-2}\log \left( \dfrac{X_{i}}{\alpha }\right) \right] \\&\quad + \dfrac{2\beta (1+\beta )}{\alpha ^2}\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^{2\beta }\left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-2}\log \left( \dfrac{X_{i}}{\alpha }\right) \right] \\&\quad - \dfrac{2\beta (1+\beta )}{\alpha ^2}\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^\beta \log \left( \dfrac{X_{i}}{\alpha }\right) \left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-1}\right] ,\\ \dfrac{\partial ^3 log (L)}{\partial \alpha \partial \beta ^2}&= -\dfrac{4}{\alpha }\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^{2\beta }\log \left( \dfrac{X_{i}}{\alpha }\right) \left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-2}\right] \\&\quad +\dfrac{4}{\alpha }\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^{\beta }\log \left( \dfrac{X_{i}}{\alpha }\right) \left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-1}\right] \\&\quad +\dfrac{4\beta }{\alpha }\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^{3\beta }\left[ \log \left( \dfrac{X_{i}}{\alpha }\right) \right] ^2\left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-3}\right] \\&\quad -\dfrac{6\beta }{\alpha }\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^{2\beta }\left[ \log \left( \dfrac{X_{i}}{\alpha }\right) \right] ^2\left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-2}\right] \\&\quad + \dfrac{2\beta }{\alpha }\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^{\beta }\left[ \log \left( \dfrac{X_{i}}{\alpha }\right) \right] ^2\left[ 1+\left( \dfrac{X_{i}}{\alpha }\right) ^\beta \right] ^{-1}\right] ,\\ \dfrac{\partial ^3 log (L)}{\partial \beta ^3}&= \dfrac{2n}{\beta ^3} + 4\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^{2\beta }\left[ \log \left( \dfrac{X_{i}}{\alpha }\right) \right] ^3\left[ 1+\dfrac{X_{i}}{\alpha }\right] ^{-3}\right] \\&\quad - 2\sum _{i=1}^n\left[ \left( \dfrac{X_{i}}{\alpha }\right) ^{\beta }\left[ \log \left( \dfrac{X_{i}}{\alpha }\right) \right] ^3\left[ 1+\dfrac{X_{i}}{\alpha }\right] ^{-2}\right] . \end{aligned}$$

Note that if X has the log-logistic distribution with parameters \(\alpha \) and \(\beta \), then \(Y = \left( {X}/{\alpha }\right) ^\beta \sim f(y) = {1}/{(1+y)^2}, ~y > 0.\) To find the joint cumulants of the log-logistic distribution, we follow the results of Abbas and Tang (2015) and obtain the following expectations

$$\begin{aligned}&{\mathbb {E}}\left\{ \left( \dfrac{X}{\alpha }\right) ^\beta \left[ 1+\left( \dfrac{X}{\alpha }\right) ^\beta \right] ^{-1}\right\} = \int _{0}^{\infty }y[1+y]^{-3}dy = \dfrac{1}{2}, \\&{\mathbb {E}}\left\{ \left( \dfrac{X}{\alpha }\right) ^{2\beta } \left[ 1+\left( \dfrac{X}{\alpha }\right) ^\beta \right] ^{-2}\right\} = \int _{0}^{\infty }y^2[1+y]^{-4}dy = \dfrac{1}{3},\\&\beta {\mathbb {E}}\left\{ \left( \dfrac{X}{\alpha }\right) ^\beta \left[ \log \left( \dfrac{X}{\alpha }\right) \right] \left[ 1+\left( \dfrac{X}{\alpha }\right) ^\beta \right] ^{-1}\right\} = \int _{0}^{\infty }y[\log (y)][1+y]^{-3}dy = \dfrac{1}{2},\\&\beta ^2 {\mathbb {E}}\left\{ \left( \dfrac{X}{\alpha }\right) ^\beta \left[ \log \left( \dfrac{X}{\alpha }\right) \right] ^2\left[ 1+\left( \dfrac{X}{\alpha }\right) ^\beta \right] ^{-2}\right\} \\&\quad = \int _{0}^{\infty }y[\log (y)]^2[1+y]^{-4}dy = \dfrac{1}{18}(-6+\pi ^2),\\&\beta {\mathbb {E}}\left\{ \left( \dfrac{X}{\alpha }\right) ^{2\beta } \left[ \log \left( \dfrac{X}{\alpha }\right) \right] \left[ 1+\left( \dfrac{X}{\alpha }\right) ^\beta \right] ^{-2}\right\} = \int _{0}^{\infty }y^2[\log (y)][1+y]^{-4}dy = \dfrac{1}{2}. \end{aligned}$$

In addition, we have computed the following expectations

$$\begin{aligned}&{\mathbb {E}}\left\{ \left( \dfrac{X}{\alpha }\right) ^{3\beta }\left[ 1+\left( \dfrac{X}{\alpha }\right) ^\beta \right] ^{-3}\right\} = \int _{0}^{\infty }y^3[1+y]^{-5}dy = \dfrac{1}{4},\\&\beta {\mathbb {E}}\left\{ \left( \dfrac{X}{\alpha }\right) ^{3\beta }\left[ \log \left( \dfrac{X}{\alpha }\right) \right] \left[ 1+\left( \dfrac{X}{\alpha }\right) ^\beta \right] ^{-3}\right\} = \int _{0}^{\infty }y^3[\log (y)][1+y]^{-5}dy = \dfrac{11}{24},\\&\beta ^2 {\mathbb {E}}\left\{ \left( \dfrac{X}{\alpha }\right) ^{3\beta }\left[ \log \left( \dfrac{X}{\alpha }\right) \right] ^2\left[ 1+\left( \dfrac{X}{\alpha }\right) ^\beta \right] ^{-3}\right\} \\&\quad = \int _{0}^{\infty }y^3[\log (y)]^2[1+y]^{-5}dy = \dfrac{1}{12}(6+\pi ^2),\\&\beta ^2 {\mathbb {E}}\left\{ \left( \dfrac{X}{\alpha }\right) ^{2\beta }\left[ \log \left( \dfrac{X}{\alpha }\right) \right] ^2\left[ 1+\left( \dfrac{X}{\alpha }\right) ^\beta \right] ^{-2}\right\} \\&\quad = \int _{0}^{\infty }y^2[\log (y)]^2[1+y]^{-4}dy = \dfrac{1}{9}(3+\pi ^2),\\&\beta ^2 {\mathbb {E}}\left\{ \left( \dfrac{X}{\alpha }\right) ^{\beta }\left[ \log \left( \dfrac{X}{\alpha }\right) \right] ^2\left[ 1+\left( \dfrac{X}{\alpha }\right) ^\beta \right] ^{-1}\right\} \\&\quad = \int _{0}^{\infty }y[\log (y)]^2[1+y]^{-3}dy = \dfrac{\pi ^2}{6},\\&\beta ^3 {\mathbb {E}}\left\{ \left( \dfrac{X}{\alpha }\right) ^{2\beta }\left[ \log \left( \dfrac{X}{\alpha }\right) \right] ^3\left[ 1+\left( \dfrac{X}{\alpha }\right) ^\beta \right] ^{-3}\right\} \\&\quad = \int _{0}^{\infty }y^2[\log (y)]^3[1+y]^{-5}dy = \dfrac{1}{24}(-6 + \pi ^2), \end{aligned}$$

and

$$\begin{aligned} \beta ^3 {\mathbb {E}}\left\{ \left( \dfrac{X}{\alpha }\right) ^{\beta }\left[ \log \left( \dfrac{X}{\alpha }\right) \right] ^3\left[ 1+\left( \dfrac{X}{\alpha }\right) ^\beta \right] ^{-2}\right\} = \int _{0}^{\infty }y[\log (y)]^3[1+y]^{-4}dy&= 0.&\end{aligned}$$

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Reath, J., Dong, J. & Wang, M. Improved parameter estimation of the log-logistic distribution with applications. Comput Stat 33, 339–356 (2018). https://doi.org/10.1007/s00180-017-0738-y

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