Evaluation and Comparison of Estimators in the Gompertz Distribution

Abstract

This article addresses the different methods of estimation of the probability density function and the cumulative distribution function for the Gompertz distribution. Following estimation methods are considered: maximum likelihood estimators, uniformly minimum variance unbiased estimators, least squares estimators, weighted least square estimators, percentile estimators, maximum product of spacings estimators, Cramér–von-Mises estimators, Anderson–Darling estimators. Monte Carlo simulations are performed to compare the behavior of the proposed methods of estimation for different sample sizes. Finally, one real data set and one simulated data set are analyzed for illustrative purposes.

Appendix

Suppose that $X_1, X_2, \ldots , X_n$ denotes a random sample of size n drawn from the Gompertz distribution as defined in (1.1). Then we estimate unknown parameters of the $G(c, \theta )$ distribution using following estimation methods.

1.1 Maximum Likelihood Estimation

The log-likelihood function of $(c, \theta )$ is given by

$$\begin{aligned} l(c , \theta )=n\ln \theta +c\sum _{i=1}^n x_{(i:n)}-\frac{\theta }{c} \sum _{i=1}^n(e^{cx_{(i:n)}}-1). \end{aligned}$$

The corresponding MLEs $(\hat{c},\hat{\theta })$ of $(c , \theta )$ is obtained by simultaneously solving the following non-linear equations

$$\begin{aligned} \frac{\partial l}{\partial c}= & {} \sum _{i=1}^n x_{(i:n)} +\frac{\theta }{c^2} \sum _{i=1}^n (e^{cx_{(i:n)}}-1)-\frac{\theta }{c}\sum _{i=1}^n x_{i:n}e^{cx_{(i:n)}}=0\\ \frac{\partial l}{\partial \theta }= & {} \frac{n}{\theta }-\frac{1}{c}(e^{cx_{(i:n)}}-1)=0. \end{aligned}$$

1.2 Least Squares Estimation

The LSEs $(\tilde{c}_{ls},\tilde{\theta }_{ls})$ of $(c , \theta )$ can be obtained by minimizing

$$\begin{aligned} \sum _{j=1}^n \left[ 1-\exp \left\{ -\frac{\theta }{c}(e^{cx_{(j:n)}}-1)\right\} -\frac{j}{n+1}\right] ^2 \end{aligned}$$

with respect to c and $\theta$.

1.3 Weighted Least Square Estimation

The WLSEs $(\tilde{c}_{wls},\tilde{\theta }_{wls})$ of $(c , \theta )$ can be obtained by minimizing

$$\begin{aligned} \sum _{j=1}^n \frac{(n+1)^2(n+2)}{j(n-j+1)} \left[ 1-\exp \left\{ -\frac{\theta }{c}(e^{cx_{(j:n)}}-1)\right\} -\frac{j}{n+1}\right] ^2 \end{aligned}$$

with respect to c and $\theta$.

1.4 Percentile Estimation

Let $(\tilde{\alpha },\tilde{\beta })$ are the PCEs of $(c , \theta )$, then $(\tilde{c},\tilde{\theta })$ can be obtained by minimizing

$$\begin{aligned} \sum _{i=1}^n [x_{(i:n)} - c^{-1} \ln {(1-c \theta ^{-1} \ln {(1-p_i)})}]^2 \end{aligned}$$

with respect to c and $\theta$.

1.5 Method of Maximum Product Spacing

The MPS estimates $(\tilde{c},\tilde{\theta })$ of $(c, \theta )$ can be obtained by solving

$$\begin{aligned} \frac{\partial H(c ,\theta )}{\partial c}= & {} \frac{1}{n+1}\sum _{i=1}^{n+1}\frac{1}{D _i(c ,\theta )}\left[ \Delta _1(x_{(i:n)}\mid c ,\theta )- \Delta _1(x_{(i-1:n)}\mid c ,\theta )\right] =0\\ \frac{\partial H(c ,\theta )}{\partial \theta }= & {} \frac{1}{n+1}\sum _{i=1}^{n+1}\frac{1}{D _i(c ,\theta )}\left[ \Delta (x_{(i:n)}\mid c ,\theta )- \Delta (x_{(i-1:n)}\mid c ,\theta )\right] =0 \end{aligned}$$

where $H(c , \theta )\!=\! \frac{1}{n+1}\sum _{i=1}^{n+1} \log D_i(c, \theta )$ and $D_i(c, \theta )\!=\!F(x_{(i:n)}\mid c, \theta )-F(x_{(i-1:n)}\mid c, \theta ).$ Also $\Delta _1 \left( x_{(i:n)}\mid \theta , c \right) =\frac{\theta }{c}e^{-\theta /c (e^{cx_{(i:n)}}-1)}\left[ x_ie^{cx_{(i:n)}}-\frac{1}{c}(e^{cx_{(i:n)}}-1)\right]$ and $\Delta \left( x_{(i:n)}\mid \theta , c \right) =- \frac{1}{c}(e^{c x_{i}}-1)e^{-\frac{\theta }{c}(e^{c x_{(i:n)}}-1)}.$

1.6 Method of Cramér–von-Mises

The CVM estimates of c and $\theta$, say $\tilde{c}_{cvm}$ and $\tilde{\theta }_{cvm}$, can be obtained by solving

$$\begin{aligned} \sum _{i=1}^n \left( F(x_{(i:n)}\mid c, \theta )-\frac{2i-1}{2n}\right) \Delta _1(x_{(i:n)}\mid c, \theta )= & {} 0\\ \sum _{i=1}^n \left( F(x_{(i:n)}\mid c, \theta )-\frac{2i-1}{2n}\right) \Delta (x_{(i:n)}\mid c, \theta )= & {} 0, \end{aligned}$$

where $\Delta _1$ and $\Delta$ are specified previously.

1.7 Method of Anderson–Darling and Modified Anderson–Darling

The AD estimates of c and $\theta$, say $\tilde{c}_{AD}$ and $\tilde{\theta }_{AD}$, can be obtained by solving

$$\begin{aligned} \sum _{i=1}^n(2i-1)\left[ \frac{\Delta _1(x_{(i:n)}\mid c, \theta )}{F(x_{i}\mid c, \theta )}-\frac{\Delta _1(x_{(n+1-i:n)}\mid c, \theta )}{\bar{F}(x_{(n+1-i:n)}\mid c, \theta )}\right]= & {} 0\\ \sum _{i=1}^n(2i-1)\left[ \frac{\Delta (x_{(i:n)}\mid c, \theta )}{F(x_{(i:n)}\mid c, \theta )}-\frac{\Delta (x_{(n+1-i:n)}\mid c, \theta )}{\bar{F}(x_{(n+1-i:n)}\mid c, \theta )}\right]= & {} 0. \end{aligned}$$