Document Type : Research Article

Authors

• Ramin Kazemi
• Akram Kohansal

Department of Statistics, Imam Khomeini International University, Qazvin, Iran

Abstract

We consider the estimation of model parameters and prediction of unobserved records based on record statistics for the Basic Gompertz distribution (BGD) with parameter $\lambda$ using frequentist and Bayesian analysis. In frequentist analysis, we give the moment generating function of the $m$th record, the maximum likelihood estimation (MLE) of $\lambda$, the moment-based estimate (MBE)  of $\lambda$, the confidence interval for $\lambda$, and the prediction of future records.  Exactly, we show that $$\hat{\lambda}_{\text{MBE}}=\frac{m(m+1)}{2\sum_{i=1}^{m}(e^{Y_i}-1)}, \hat{\lambda}_{\text{MLE}}=\frac{m}{e^{y_m}-1},$$ where $m\geq 1 $ and $Y_m=\max(\min)\{ X_1,\ldots,X_m \}$. In Bayesian analysis, we obtain the Bayesian sample-based estimation and prediction. Exactly, we show that under the squared error loss (SEL) function, $$\hat{\lambda}_{BS}=\frac{m+a}{e^{y_m}+b-1}$$ and under the LINEX loss function, $$\hat{\lambda}_{BL}=-\frac{m+a}{c}\ln\bigg(  \frac{e^{y_m}+b-1}{e^{y_m}+(b+c)-1}\bigg).$$ Based on Monte Carlo simulations, the performances of the different methods of estimation and prediction are compared via MSEs and Biases. Finally, a real dataset has been analyzed for illustrative purposes.

Highlights

• Closed-form MLE and moment-based estimators for the Basic Gompertz distribution are derived from record statistics.
• Exact confidence and prediction intervals are constructed using a pivotal statistic and the highest conditional density method.
• Bayesian estimators under squared error and LINEX loss functions are obtained in closed form via a gamma conjugate prior.
• Monte Carlo simulations show Bayesian estimators outperform frequentist methods in MSE and bias.
• HPD credible and prediction intervals are computed and validated on a real dataset.

Keywords

• Basic Gompertz distribution
• Record statistics
• Bayesian estimation
• Maximum likelihood estimation (MLE)
• Moment-based estimation (MBE)
• prediction
• LINEX loss function

Main Subjects

• Mathematical Modeling