Abstract
In this study, we will look at estimating the parameters of the Gompertz distribution. We know that the maximum likelihood technique is the most often used method in the literature for parameter estimation. However, it is well known that the maximum likelihood estimators (MLEs) are biased for small sample sizes. As a result, we are motivated to produce nearly unbiased estimators for the parameters of this distribution. To be more specific, we concentrate on two bias-correction strategies (analytical and bootstrap approaches) to minimize MLE biases to the second order of magnitude. Monte Carlo simulations are used to compare the performances of these estimators. Finally, two real-data examples are offered to demonstrate the utility of our proposed estimators in small sample sizes.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Gompertz [17] introduced the Gompertz distribution to fit mortality tables. This distribution is unimodal and positively skew, whereas the Gompertz hazard rate function increases monotonically. As a result, the Gompertz distribution is used to model phenomena that has an increasing failure rate. It is worth mentioning that it has some interesting relationships with well-known distributions like exponential, Weibull, Gumbel, generalized logistic and double exponential distributions (see, [39]). Garg et al. [13] studied the Maximum likelihood estimation of the parameters of the Gompertz survival function.
Ahuja and Nash [1] contains a survey and applications of the Gompertz distribution. Many researchers have contributed to the studies of characterization and statistical methodology of this distribution for analyzing a variety of real-world applications, including the analysis of medical, survival, behavioral, biological, environmental, and actuarial studies. For instance, [2,3,4, 8, 10, 12, 21, 22, 26, 27, 31,32,33, 35, 40, 42].
The maximum likelihood technique is well-known as the most popular estimating method. This is due to its appealing mathematical properties for large sample sizes, such as unbiasedness, consistency, efficiency and asymptotic normality. These properties, however, may not hold for small or even moderate sample sizes; see, for example, [14,15,16, 25, 34, 38] along with others.
In this study, we will look at two strategies. The first is a correction strategy known as the "analytical approach" which was presented by [7]. This method corrects the bias of MLEs to the second order of magnitude by subtracting it from the MLEs. Some researchers, including [20, 28, 36], developed software applications (even though they are limited) that allow users to compute the analytic Cox-Snell formula for bias corrections for various pre-specified distributions. Moreover, the second strategy is based on the bootstrap re-sampling procedure which may minimize bias to the second order, as described in the "bootstrap approach" developed by [9]. In both strategies, we shall refer to these corrected estimators as bias-corrected estimators. To demonstrate the performance of these estimators, Monte Carlo simulations and real-world applications are used.
If X follows the Gompertz distribution (denoted by Gomp(\(\alpha ,\beta \))), then the distribution function (cdf) and the probability density function (pdf) of X are given respectively by (see for example [5, 19, 23])
where \(\alpha >0\) is the shape parameter and \(\beta >0\) is the scale parameter as shown in Fig. 1. In addition, the role of the shape parameter \(\alpha \) can also be seen in the hazard function (h(x)) of this distribution
It is clear that as \(\alpha \rightarrow 0\), the Gompertz distribution approaches the exponential distribution.
The rest of the paper is organized as follows. The Maximum Likelihood Estimation(MLE) is introduced in Sect. 2. In Sect. 3, we give bias-corrected MLEs for both analytical and bootstrap approaches. In Sect. 4, we obtain simulation results to evaluate the performance of the maximum likelihood estimation method. Moreover, two examples based on real data are used to demonstrate the performance of these approaches in Sect. 5. lastly, some concluding remarks are given in Sect. 6.
2 Maximum Likelihood Estimation
Let \(X_1,\cdots ,X_n\) be a random sample of size n from Gomp(\(\alpha ,\beta \)). The log likelihood function (l) is
We maximize Eq. (2.1) with respect to \(\alpha \ and\ \beta \) in order to obtain the MLE’s (\({\widehat{\alpha }}\ and\ {\widehat{\beta }}\)) of \(\alpha \ and\ \beta \) respectively. Thus, we have the following equations:
The non-linear Eqs. (2.2) and (2.3) cannot be solved analytically. However, several programs such as R, Mathematica or Maple can be used to solve these equations numerically to estimate the MLEs denoted by \({{\widehat{\xi }}}_{MLE}\) where \(\xi =\left( \alpha ,\beta \right) ^{'}\). For small sample sizes, these MLEs will be biased as mentioned previously. Therefore, the bias may provide misleading results, influencing the interpretation of occurrences in real-world applications. As a result, this motivates us to examine roughly unbiased estimators to decrease the bias of these Gompertz distribution MLEs.
3 Bias-Corrected MLEs
This section examines two bias-correction techniques. The first is [7]’s "analytical approach", which is described in Sect. 3.1, and the second is [9]’s "bootstrap approach", which is presented in Sect. 3.2.
3.1 Analytical Approach
Assume \(l(\xi )\) is the log likelihood function based on n observations with a p-dimensional parameter vector represented as \(\xi ={\left( {\xi }_1,\dots ,{\xi }_p\right) }^{'}\) and \(l(\xi )\) is regular with respect to all derivatives up to the third order.
The joint cumulants of \(l=l(\xi )\) derivatives are therefore defined as
These joint cumulants’ derivatives are denoted by
In addition, the expressions in Eqs. (3.1) through (3.4) are assumed to be of order O(n).
Assume \(\varPhi =\left[ -{\eta }_{ij}\right] \) denotes the Fisher information matrix of \(\xi \), with \(i,j=1,\cdots p\). For non-identical independent samples, [7] demonstrated that the bias of the \(s^{th}\) element of the MLE of \(\xi \) is
where \({\eta }^{ij}\) denotes the \({\left( i,j\right) }^{th}\) member of the information matrix’s inverse. Thereafter, [6] demonstrated that Eq. (3.5) remains valid even if the sample data are not identical and non-independent observations, given that all expressions in Eq. (3.1) through Eq. (3.4) are of order O(n). Equation (3.5) was expressed as
Because the term of the form in Eq. (3.3) was eliminated from Eq. (3.6), this bias expression in Eq. (3.6) is often easier to compute than that of Eq. (3.5).
Let \({\omega }^{(k)}_{ij}={\eta }^{(k)}_{ij}-\frac{1}{2}{\eta }_{ijk}\,\ i,j,k=1,\dots ,p\). As a result, we have these two matrices (\(\varPsi \) and \({\varOmega }^{(k)}\)) as
Hence, \({\widehat{\xi }}\)’s bias expression may be written in matrix form as
where vec is an operation that stacks a matrix’s column vectors one on top of the other. Therefore, the \(\xi \) bias-corrected MLE, represented as \({{\widehat{\xi }}}_{BCMLE}\), is given by
where \({\widehat{\xi }}\) is the MLE of \(\xi \), \({\widehat{\varPhi }}=\varPhi |_{\xi ={\widehat{\xi }}}\) and \({\widehat{\varPsi }}=\varPsi |_{\xi ={\widehat{\xi }}}\). It should be noted that the bias of \({{\widehat{\xi }}}_{BCMLE}\) is \(O\left( n^{-2}\right) \).
We have \(\xi ={\left( \alpha , \beta \right) }^{'}\)and \(p=2\) because we are studying the Gompertz distribution. To get the bias-corrected MLEs, we must first compute the higher-order derivatives of the log-likelihood function for the Gompertz distribution with respect to \(\alpha \) and \(\beta \) as shown below
Notice that if X follows Gomp(\(\alpha ,\beta \)) then \(Y=\frac{\alpha }{\beta }\left( e^{\beta X}-1\right) \ \sim \ f\left( y\right) =e^{-Y}\ \ for\ y>0\). Therefore, Y follows exponential distribution. The following formulas will be needed:
The upper incomplete gamma function is defined as
These derivatives of the upper incomplete gamma function are straightforward to be obtained as
The Eqs. (3.17) and (3.22) can be implemented easily in any mathematical or statistical programs. Moreover,
see equation (4.358.1) in [18]. Hence, using the preceding Eqs. (3.17)–(3.23), we obtain
Similarly,
As a result,
Similarly,
Refer to A for the joint cumulants of the derivatives of the log-likelihood function. As previously stated, any computer program may be used to compute the bias-corrected MLE of \(\xi \) (\({{\widehat{\xi }}}_{BCMLE}\) ) provided by
where \(\varPhi \) is the Fisher information matrix of \(\xi \) and \(\varPsi =\left[ {\varOmega }^{\left( 1\right) }|\ {\varOmega }^{\left( 2\right) }\right] \ such\ that\ \ {\varOmega }^{(k)}=\left[ {\omega }^{(k)}_{ij}\right] \,\ i,j,k=1,2\).
3.2 Bootstrap Approach
To create pseudo-samples from the original sample, [9] invented the bootstrap resampling technique. We deduct the estimated bias from these samples from the original MLEs in the following manner to produce the bias-corrected MLEs:
A random sample of size n from F, the distribution function (cdf), is represented by the formula x \(={\left( x_1,\cdots ,x_n\right) }^{'}\). Let \(\nu =t\left( F\right) \) be a function of F, and let \({\widehat{\nu }}\) be the estimator of \(\nu \). By creating observations using replacement, we resample the original sample, x, into pseudo-samples of size n, x \(^{*}\) \(={\left( x^*_1,\dots ,x^*_n\right) }^{'}\). From these pseudo-samples, indicated by \({{\widehat{\nu }}}^*=g({{{\textbf{x}}}}^{{\mathbf *}})\), we get the bootstrap replicates of \({\widehat{\nu }}\). The cdf of \({\widehat{\nu }}\), \(F_{{\widehat{\nu }}}\) may be estimated using the empirical cdf (ecdf) of \({{\widehat{\nu }}}^*\). This following equation may be used to estimate the bootstrap bias of the estimator \({\widehat{\nu }}=g({{\textbf{x}}})\) as
Since it is a consistent estimator, we may substitute F with \(F_{{\widehat{\nu }}}\) in Equation (3.32) to get the estimated bootstrap bias as
If there are N bootstrap estimates, for example, \(\left( {{\widehat{\nu }}}^{*\left( 1\right) },\dots ,{{\widehat{\nu }}}^{*\left( N\right) }\right) \), and N is sufficiently large then the estimator \({{{\mathbb {E}}}}_{F_{{\widehat{\nu }}}}\left[ {{\widehat{\nu }}}^*\right] \) in Equation (3.33) may be roughly expressed as
Consequently, the estimated bootstrap bias is
Therefore, using Efron’s bootstrap resampling approach, the bias-corrected estimator (\({\nu }^B\)) equals
As a result, we in this instance let \({{\widehat{\xi }}}_{BCBOOT}={\nu }^B\).
4 A Simulation Study
In this part, using the cdf and pdf supplied in Eqs. (1.1) and (1.2), respectively, we run Monte Carlo simulations to evaluate the efficacy of the various Gompertz distribution estimators that have been taken into consideration. We select random samples of size n = 5, 10, 20, 35, 50, 75, 100, 200 with \(\alpha \) = 0.1, 0.5, 1, 1.5, 3 and \(\beta \) = 0.1, 0.5, 1, 1.5, 3. For each combination of (n, \(\alpha \), \(\beta \)), we simulated using \(M=5000\) Monte Carlo replications and \(B=1000\) bootstrap replications.
We compute the average bias and RMSEs for an estimator \({\hat{\xi }}\) of the parameter \(\xi =(\alpha ,\beta )'\), which are indicated by Bias(\({\widehat{\xi }}\))=\(\frac{1}{M}\sum ^M_{i=1}{\left( {{\widehat{\xi }}}_i-\xi \right) }\) and RMSE(\({\widehat{\xi }}\))=\(\sqrt{\frac{1}{M}\sum ^M_{i=1}{{\left( {{\widehat{\xi }}}_i-\xi \right) }^2}}\), respectively.
Figures 2 and 3 show the average biases and RMSEs of the estimations of \(\alpha \) and \(\beta \) across sample sizes. Some inferences that can be drawn are the ones below.
-
1.
The MLE estimators of \(\alpha \) appear to have a positive bias for each simulation being taken into account. This demonstrates how, generally speaking, they overestimate the value of the parameter \(\alpha \), especially when the sample size is small. Additionally, the MLE estimators commonly show a negative bias, i.e., they consistently underestimate the true value of the parameter \(\beta \) for different sample sizes, when the real value of the parameter \(\beta \) is equal to or greater than one. On the other hand, the MLE estimators frequently seem to have a positive bias when the true value of the parameter \(\beta \) is less than one. This means that for various sample sizes, they on average underestimate the true value of the parameter \(\beta \).
-
2.
In most simulations for various sample sizes, the MLE estimators underperformed the BCMLEs of \(\alpha \) and \(\beta \) in terms of bias and RMSE, i.e., the BCBOOTs of \(\alpha \) and \(\beta \) beat the MLE estimators. Therefore, the BCMLEs would be ideal or better options for estimating \(\alpha \) and \(\beta \) if bias is a concern.
-
3.
The biases and RMSEs of all examined estimators will decrease as sample size n increases, as expected. This is mostly because, according to statistical theory, most estimators function more effectively as sample size n increases. For the bias-corrected estimators for small sample sizes, the reductions in bias and RMSE are quite considerable, as shown above. For example, where n = 10, \(\alpha \) = 0.1, and \(\beta \) =1, we have Bias(\({\hat{\alpha }}\) MLE)=0.3229, Bias(\({\hat{\alpha }}\) BCMLE)=0.0676, Bias(\({\hat{\alpha }}\) BCBOOT)= 0.0576, Bias(\({\hat{\beta }}\) MLE) = \(-\)0.0435, Bias(\({\hat{\beta }}\) BCMLE) = 0.0066, Bias(\({\hat{\beta }}\) BCBOOT) = \(-\)0.0122, RMSE(\({\hat{\alpha }}\) MLE) = 0.8725, RMSE(\({\hat{\alpha }}\) BCMLE) = 0.4399, RMSE(\({\hat{\alpha }}\) BCBOOT)= 0.5036, RMSE(\({\hat{\beta }}\) MLE) = 0.4422, RMSE(\({\hat{\beta }}\) BCMLE) = 0.3101, RMSE(\({\hat{\beta }}\) BCBOOT) = 0.3437.
5 Illustrative Applications
The maximum likelihood estimator, denoted by \({{\widehat{\xi }}}_{\textrm{MLE}}\), the bias-corrected maximum likelihood estimator using an analytical approach, denoted by \({{\widehat{\xi }}}_{\textrm{BCMLE}}\), and the bias-corrected estimator using a bootstrap approach, denoted by \({{\widehat{\xi }}}_{\textrm{BCBOOT}}\), are taken into consideration. In order to compare the performance of these estimators, we take into account two real data sets.
The data set represents the lifetimes of 20 electronic components, see [30], page 100. Also, it was studied by [37]. It is provided, for convenience, as follows: 0.03, 0.12, 0.22, 0.35, 0.73, 0.79, 1.25, 1.41, 1.52, 1.79, 1.8, 1.94, 2.38, 2.4, 2.87, 2.99, 3.14, 3.17, 4.72 and 5.09.
Table 1 contains the estimated values for the parameters of the Gompertz distribution. Table 1 demonstrates that the bias-corrected MLE and bootstrap estimates of \(\alpha \) are less than the MLE estimate, indicating that the MLE approach overestimates this parameter. The Gompertz distribution’s pdf and cdf are shown in Fig. 4 after being assessed using the \(\alpha \) and \(\beta \) estimations in Table 1. We suggest using bias-corrected MLEs for this data set since the density shape based on the MLE approach may be deceptive, as illustrated in this Figure.
The second data set represents the failure times (in minutes) for a sample of 15 electronic components in an accelerated life test. This data was taken from [24], page 204. Additionally, researchers like [29, 11], and [41] analyzed this data as well. We provide it here for convenience: 1.4, 5.1, 6.3, 10.8, 12.1, 18.5, 19.7, 22.2, 23, 30.6, 37.3, 46.3, 53.9, 59.8 and 66.2.
Similarly, Table 2 lists the estimated values of the parameters of the Gompertz distribution. Table 2 demonstrates that the bias-corrected MLE and bootstrap estimates of \(\alpha \) are less than the MLE estimate, indicating that the MLE approach overestimates this parameter. The Gompertz distribution’s pdf and cdf are shown in Fig. 5 after being assessed using the \(\alpha \) and \(\beta \) estimations in Table 2. Given that the density shape based on the MLE technique may be deceiving, as shown in this Figure, we advise utilizing bias-corrected MLEs for this data set.
6 Concluding Remarks
Based on the "analytical technique" created by [7], we were able to obtain the second-order bias-corrected MLEs of the Gompertz distribution. In addition to having straightforward formulas, bias-corrected MLEs simultaneously reduce the bias and root mean square errors (RMSEs) of the parameters of the Gompertz distribution. We also assessed the resampling technique known as the "bootstrap approach" described by [9] for parameters estimation. Bias-corrected MLEs should be advised for use in practical applications, particularly when the sample size is small or moderate, according to the numerical findings of both simulation studies and real-data applications.
Data Availability
All data sets analyzed during this study are included in this article.
Abbreviations
- MLE:
-
Maximum Likelihood Estimation
- GOMP:
-
Gompertz distribution
- CDF:
-
Cumulative distribution function
- PDF:
-
Probability density function
- HF:
-
Hazard function
- LL:
-
Log likelihood function
- ECDF:
-
Empirical cumulative distribution function
- BCMLE:
-
Bias-corrected MLE
- BCBOOT:
-
Bias-corrected bootstrap
- RMSE:
-
Root mean square error
References
Ahuja, J.C., Nash, S.W.: The generalized gompertz-verhulst family of distributions. Sankhyā: The Indian Journal of Statistics, Series A (1961-2002) 29(2), 141–156 (1967). http://www.jstor.org/stable/25049460
Al-Shomrani, A.A.: An improvement in maximum likelihood estimation of the burr xii distribution parameters. AIMS Math 7(9), 17444–17460 (2022). https://doi.org/10.3934/math.2022961
Alhassan, A.: Gompertz ampadu class of distributions: properties and applications. J Probab Stat 2022(330), 1104 (2022). https://doi.org/10.1155/2022/1104330
Asadi, M., Di Crescenzo, A., Sajadi, F.A., Spina, S.: A generalized gompertz growth model with applications and related birth–death processes. Ricerche Mat. (2020). https://doi.org/10.1007/s11587-020-00548-y
Balakrishnan, N., Cramer, E.: The Art of Progressive Censoring: Applications to Reliability and Quality, 1st ed. edn. Birkhäuser New York, NY (2014). https://doi.org/10.1007/978-0-8176-4807-7
Cordeiro, G.M., Klein, R.: Bias correction in arma models. Stat. Probab. Lett. 19(3), 169–176 (1994). https://doi.org/10.1016/0167-7152(94)90100-7
Cox, D.R., Snell, E.J.: A general definition of residuals. J. R. Stat. Soc.: Ser. B (Methodol.) 30(2), 248–265 (1968). https://doi.org/10.1111/j.2517-6161.1968.tb00724.x
Dey, S., Kayal, T., Tripathi, Y.M.: Evaluation and comparison of estimators in the gompertz distribution. Ann. Data Sci. 5(2), 235–258 (2018). https://doi.org/10.1007/s40745-017-0126-z
Efron, B.: Bootstrap methods: another look at the jackknife. Ann. Stat. 7(1), 1–26 (1979). (http://www.jstor.org/stable/2958830)
El-Gohary, A., Alshamrani, A., Al-Otaibi, A.N.: The generalized gompertz distribution. Appl. Math. Model. 37(1), 13–24 (2013). https://doi.org/10.1016/j.apm.2011.05.017. www.sciencedirect.com/science/article/pii/S0307904X11003118
Eliwa, M.S., Alshammari, F.S., Abualnaja, K.M., El-Morshedy, M.: A flexible extension to an extreme distribution. Symmetry 13(5) (2021). https://doi.org/10.3390/sym13050745
Franses, P.H.: Fitting a gompertz curve. J. Oper. Res. Soc. 45(1), 109–113 (1994). https://doi.org/10.2307/2583955
Garg, M.L., Rao, B.R., Redmond, C.K.: Maximum-likelihood estimation of the parameters of the gompertz survival function. J. R. Stat. Soc. Ser. C (Appl. Stat.) 19(2), 152–159 (1970). https://doi.org/10.2307/2346545
Giles, D.E.: Bias reduction for the maximum likelihood estimators of the parameters in the half-logistic distribution. Commun. Stat. Theory Methods 41(2), 212–222 (2012). https://doi.org/10.1080/03610926.2010.521278
Giles, D.E., Feng, H., Godwin, R.T.: On the bias of the maximum likelihood estimator for the two-parameter lomax distribution. Commun. Stat. Theory Methods 42(11), 1934–1950 (2013). https://doi.org/10.1080/03610926.2011.600506
Giles, D.E., Feng, H., Godwin, R.T.: Bias-corrected maximum likelihood estimation of the parameters of the generalized pareto distribution. Communications in Statistics - Theory and Methods 45(8), 2465–2483 (2016). https://doi.org/10.1080/03610926.2014.887104
Gompertz, B.: On the nature of the function expressive of the law of human mortality and on a new mode of determining the value of life contingencies. Philos. Trans. R. Soc. Lond. 115, 513–583 (1825). https://doi.org/10.1098/rstl.1825.0026
Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. Academic Press (2007)
Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous univariate distributions, 2nd edition, vol. 2. John wiley & sons (1995)
Johnson, P.H., Qi, Y., Chueh, Y.C.: Csck mle bias calculation (2012)
Jong-Wuu Wu, W.L.H., Tsai, C.H.: Estimation of parameters of the gompertz distribution using the least squares method. Appl. Math. Comput. 158(1), 133–147 (2004). https://doi.org/10.1016/j.amc.2003.08.086
Khan, M.S., King, R., Hudson, I.L.: Transmuted gompertz distribution: properties and estimation. Pak. J. Stat. 32(3), 161–182 (2016). (https://www.researchgate.net/publication/306223810_Transmuted_gompertz_distribution_Properties_and_estimation)
Klein, J.P., Moeschberger, M.L.: Survival Analysis Techniques for Censored and Truncated Data, 2nd ed. edn. Springer New York, NY (2014). https://doi.org/10.1007/b97377
Lawless, J.F.: Statistical models and methods for lifetime data, 2 edn. John Wiley & Sons, Inc. (2003)
Ling, X., Giles, D.E.: Bias reduction for the maximum likelihood estimator of the parameters of the generalized rayleigh family of distributions. Communications in Statistics - Theory and Methods 43(8), 1778–1792 (2014). https://doi.org/10.1080/03610926.2012.675114
Lv, Q., Hua, R., Gui, W.: Statistical inference of gompertz distribution under general progressive type ii censored competing risks sample. Communications in Statistics - Simulation and Computation , 1–20 (2022). https://doi.org/10.1080/03610918.2022.2028834
Makany, R.: A theoretical basis for gompertz’s curve. Biom. J. 33(1), 121–128 (1991). https://doi.org/10.1002/bimj.4710330115
Mazucheli, J., Menezes, A.F.B., Nadarajah, S.: mle.tools: An R Package for Maximum Likelihood Bias Correction. The R Journal 9(2), 268–290 (2017). https://doi.org/10.32614/RJ-2017-055
MirMostafaee, S., Qomi, M.N., Fernández, A.J.: Tolerance limits for minimal repair times of a series system with rayleigh distributed component lifetimes. Applied Mathematical Modelling 40(4), 3153–3163 (2016) https://doi.org/10.1016/j.apm.2015.09.104. www.sciencedirect.com/science/article/pii/S0307904X15006472
Murthy, D.P., Xie, M., Jiang, R.: Weibull models, vol. 505. John Wiley & Sons (2004)
Pelinovsky, E., Kokoulina, M., Epifanova, A., Kurkin, A., Kurkina, O., Tang, M., Macau, E., Kirillin, M.: Gompertz model in covid-19 spreading simulation. Chaos, Solitons & Fractals 154, 111,699 (2022).https://doi.org/10.1016/j.chaos.2021.111699. https://www.sciencedirect.com/science/article/pii/S0960077921010535
Pollard, J.H., Valkovics, E.J.: The gompertz distribution and its applications. Genus 48(3/4), 15–28 (1992)
Read, C.: Gompertz distribution. Encyclopedia of statistical sciences. Wiley, New York (1983). https://doi.org/10.1002/9781118445112.stat00995
Reath, J., Dong, J., Wang, M.: Improved parameter estimation of the log-logistic distribution with applications. Comput. Statistics 33(1), 339–356 (2018). https://doi.org/10.1007/s00180-017-0738-y
Roozegar, R., Tahmasebi, S., Jafari, A.A.: The McDonald Gompertz distribution: Properties and applications. Communications in Statistics - Simulation and Computation 46(5), 3341–3355 (2017). https://doi.org/10.1080/03610918.2015.1088024
Stoši’c, B.D., Cordeiro, G.M.: Using maple and mathematica to derive bias corrections for two parameter distributions. J. Stat. Comput. Simul. 79(6), 751–767 (2009). https://doi.org/10.1080/00949650801911047
Teimouri, M., Gupta, A.K.: On the three-parameter Weibull distribution shape parameter estimation. Journal of Data Science 11(3), 403–414 (2021). https://doi.org/10.6339/JDS.2013.11(3).1110
Wang, M., Wang, W.: Bias-corrected maximum likelihood estimation of the parameters of the weighted lindley distribution. Communications in Statistics - Simulation and Computation 46(1), 530–545 (2017). https://doi.org/10.1080/03610918.2014.970696
Willekens, F.: Gompertz in Context: The Gompertz and Related Distributions. In: Tabeau, E., van den Berg Jeths, A., Heathcote, C. (eds) Forecasting Mortality in Developed Countries. European Studies of Population, vol. 9. Springer, Dordrecht (2001). https://doi.org/10.1007/0-306-47562-6_5
Wu, J.W., Lee, W.C.: Characterization of the mixtures of gompertz distributions by conditional expectation of order statistics. Biometrical Journal: Journal of Mathematical Methods in Biosciences 41(3), 371–381 (1999). https://doi.org/10.1002/(SICI)1521-4036(199906)41:3<371::AID-BIMJ371>3.0.CO;2-M
Yadav, A.S., Saha, M., Shukla, S., Tripathi, H., Dey, R.: Reliability test plan based on logistic-exponential distribution and its application. Journal of Reliability and Statistical Studies 14, 695–724 (2021). https://doi.org/10.13052/10.13052/jrss0974-8024.14215
Younis, F., Aslam, M., Bhatti, M.I.: Preference of prior for two-component mixture of lomax distribution. Journal of Statistical Theory and Applications 20, 407–424 (2021). https://doi.org/10.2991/jsta.d.210616.002
Acknowledgements
The author would like to thank the editor and referees for their insightful comments that helped to enhance the paper.
Funding
The author received no funding for this research.
Author information
Authors and Affiliations
Contributions
This paper was written by a single author who also created the figures and tables and reviewed and approved it.
Corresponding author
Ethics declarations
Conflict of interest
The author declares that he has no conflict of interest.
Ethical Approval and Consent to participate
Not applicable.
Consent for publication
The Author hereby consents to publication of his work upon its acceptance.
Human and Animal Ethics
Not applicable.
A The Joint Cumulants of the Log-Likelihood Function Derivatives
A The Joint Cumulants of the Log-Likelihood Function Derivatives
Thus, \({\omega }^{(k)}_{ij}={\eta }^{(k)}_{ij}-\frac{1}{2}{\eta }_{ijk}\),
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Al-Shomrani, A.A. An Improvement in Maximum Likelihood Estimation of the Gompertz Distribution Parameters. J Stat Theory Appl 22, 98–115 (2023). https://doi.org/10.1007/s44199-023-00057-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s44199-023-00057-5