# Classical and Bayesian Inference for the Inverse Lomax Distribution Under Adaptive Progressive Type-II Censored Data with COVID-19

### Abstract

In this paper, we consider the classical and the Bayesian inferences for unknown parameters of inverse Lomax distribution and their corresponding survival characteristics under the adaptive progressive type-II censoring scheme. In the classical setup, first we obtain the maximum likelihood estimates for the unknown shape parameter of the distribution and its corresponding survival characteristics. Further, we consider symmetric and asymmetric loss functions for the estimation of shape parameter and its corresponding survival characteristics under the Bayesian paradigm. The performances of various derived estimators were recorded using Markov chain Monte Carlo simulation technique for different sample sizes. Finally, a COVID-19 mortality data set is provided to illustrate the computation of various estimators.

### Downloads

### References

Adegoke, T. M., Adegoke, G. K., Yahaya, A. M., Uthman, K. T., and Odigie, A. D. (2018). On Bayesian estimation of an exponential distribution. In Proceedings of 2nd International Conference, Professional Statisticians Society of Nigeria (pp. 328–332).

Ahsanullah, M. (1991). Record values of the Lomax distribution. Statistica Neerlandica, 45(1), 21–29.

Aitchison, J., Dunsmore, I.R., (1975). Statistical Prediction Analysis. Cambridge University Press, London.

Albalawi, O., Kabdwal, N. C., Azhad, Q. J., Hora, R., and Alsaedi, B. S. (2022). Estimation of the Generalized Logarithmic Transformation Exponential Distribution under Progressively Type-II Censored Data with Application to the COVID-19 Mortality Rates. Mathematics, 10(7), 1015.

Almetwally, E. M., Almongy, H. M., and ElSherpieny, E. A. (2019). Adaptive type-II progressive censoring schemes based on maximum product spacing with application of generalized Rayleigh distribution. Journal of Data Science, 17(4), 802–831.

Almongy, H. M., Almetwally, E. M., Aljohani, H. M., Alghamdi, A. S., and Hafez, E. H. (2021). A new extended rayleigh distribution with applications of COVID-19 data. Results in Physics, 23, 104012.

Balakrishnan, N., and Ahsanullah, M. (1994). Relations for single and product moments of record values from Lomax distribution. Sankhyā: The Indian Journal of Statistics, Series B, 140–146.

Balakrishnan, N., and Sandhu, R. A. (1995). A simple simulational algorithm for generating progressive Type-II censored samples. The American Statistician, 49(2), 229–230.

Basu, A. P., and Ebrahimi, N. (1991). Bayesian approach to life testing and reliability estimation using asymmetric loss function. Journal of statistical planning and inference, 29(1–2), 21–31.

Berger, J.O., (1980). Statistical Decision Theory Foundations, Concept, and Methods. Springer, New York.

Box GEP and Tiao GC (1973). Bayesian Inference in Statistical Analysis, Addison-Wesley, Mas-sachusetts.

Calabria, R., and Pulcini, G. (1996). Point estimation under asymmetric loss functions for left-truncated exponential samples. Communications in Statistics-Theory and Methods, 25(3), 585–600.

Chaudhary, A. K., and Kumar, V. (2020). A Bayesian Estimation and Predictionof Gompertz Extension Distribution Using the MCMC Method. Nepal Journal of Science and Technology, 19(1), 142–160.

Chen, M. H., and Shao, Q. M. (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8(1), 69–92.

Cohen, A. C. (1963). Progressively censored samples in life testing. Technometrics, 5(3), 327–339.

Edwards, W., Lindman, H., and Savage, L. J. (1963). Bayesian statistical inference for psychological research. Psychological review, 70(3), 193.

El-Din, M. M., Amein, M. M., El-Attar, H. E., and Hafez, E. H. (2017). Symmetric and Asymmetric Bayesian Estimation For Lindley Distribution Based on Progressive First Failure Censored Data. Math. Sci. Lett, 6(3), 255–260.

El-Din, M. M., Shafay, A. R., and Nagy, M. (2018). Statistical inference under adaptive progressive censoring scheme. Computational Statistics, 33(1), 31–74.

EL-Sagheer, R. M., Mahmoud, M. A., and Nagaty, H. (2019). Statistical inference for Weibull-exponential distribution using adaptive type-II progressive censoring. J Stat Appl Probab, 8(2), 1–13.

Ferguson, T.S., (1967). Mathematical Statistics: A Decision Theoretic Approach. Academic Press, New York.

Goyal, T., Rai, P. K., and Maury, S. K. (2019). Classical and Bayesian studies for a new lifetime model in presence of type-II censoring. Communications for Statistical Applications and Methods, 26(4), 385–410.

Hora, R., Kabdwal, N., Srivastava, P. (2021). Inference for the generalized inverse Lindley distribution under type-II censored data. International journal of statistics and applied mathematics, 6(5), 155–166. doi: 10.22271/maths.2021.v6.i5b.739.

Jan, U., and Ahmad, S. P. (2017). Bayesian analysis of inverse Lomax distribution using approximation techniques. Bayesian Analysis, 7(7), 1–12.

Kleiber, C. (2004). Lorenz ordering of order statistics from log-logistic and related distributions. Journal of Statistical Planning and Inference, 120(1–2), 13–19.

Kleiber, C., and Kotz, S. (2003). Statistical size distributions in economics and actuarial sciences (Vol. 470). John Wiley & Sons.

Kumar, R., Srivastava, A. K., and Kumar, V. (2012). Analysis of Gumbel model for software survival using Bayesian paradigm. International Journal of Advanced Research in Artificial Intelligence, 1(9), 39–45.

Lee, M. Y., and Lim, E. H. (2009). Characterizations of the Lomax, exponential and Pareto distributions by conditional expectations of record values. Journal of the Chungcheong Mathematical Society, 22(2), 149–153.

Lomax, K. S. (1954). Business failures: Another example of the analysis of failure data. Journal of the American Statistical Association, 49(268), 847–852.

Mahmoud, M. A., Soliman, A. A., Abd Ellah, A. H., and El-Sagheer, R. M. (2013). Estimation of generalized Pareto under an adaptive type-II progressive censoring.

Mohan, R., and Chacko, M. (2021). Estimation of parameters of Kumaraswamy-exponential distribution based on adaptive type-II progressive censored schemes. Journal of Statistical Computation and Simulation, 91(1), 81–107.

Mubarak, A.E.S.A.E.G., Almetwally E.M. (2021). A new extension exponential distribution with applications of COVID-19 data. J Fin Business Res. 22:444–60. doi: 10.21608/jsst.2021.51484.1178.

Ng, H. K. T., Kundu, D., and Chan, P. S. (2009). Statistical analysis of exponential lifetimes under an adaptive Type-II progressive censoring scheme. Naval Research Logistics (NRL), 56(8), 687–698.

Rahman, J., Aslam, M., and Ali, S. (2013). Estimation and prediction of inverse Lomax model via Bayesian approach. Caspian Journal of Applied Sciences Research, 2(3), 43–56.

Riad, F. H., and Hafez, E. H. (2020). Point and Interval Estimation for Frechet Distribution Based on Progressive First Failure Censored Data. J. Stat. Appl. Pro, 9, 181–191.

Sewailem, M. F., and Baklizi, A. (2019). Inference for the log-logistic distribution based on an adaptive progressive type-II censoring scheme. Cogent Mathematics & Statistics, 6(1), 1684228.

Shrama, A., and Kumar, P. (2020). Estimation of Parameters of Inverse Lomax Distribution under Type-II Censoring Scheme. Journal of Statistics Applications & Probability, 10(1), 85–102.

Singh, S. K., Singh, U., and Kumar, D. (2013). Bayes estimators of the survival function and parameter of inverted exponential distribution using informative and non-informative priors. Journal of Statistical computation and simulation, 83(12), 2258–2269.

Singh, S. K., Singh, U., and Yadav, A. S. (2016). Survival estimation for inverse Lomax distribution under type Π censored data using Markov chain Monte Carlo method. International Journal of Mathematics and Statistics, 17(1), 128–146.

Sinha, S. K. (1987). Bayesian estimation of the parameters and survival function of a mixture of Weibull life distributions. Journal of statistical planning and inference, 16, 377–387.

Sobhi, M. M. A., and Soliman, A. A. (2016). Estimation for the exponentiated Weibull model with adaptive Type-II progressive censored schemes. Applied Mathematical Modelling, 40(2), 1180–1192.

Soliman, A. A., Abd Ellah, A. H., Abou-Elheggag, N. A., and Modhesh, A. A. (2013). Estimation from Burr type XII distribution using progressive first-failure censored data. Journal of Statistical Computation and Simulation, 83(12), 2270–2290.

Srivastava, A. K. (2020). Estimation of Parameters and Survival Function of Log Gompertz Model: Bayesian Approach under Gamma Prior. International Journal of Advanced Research in Science, Communication and Technology, 10(2), 183–195.

Varian, H.R., (1975). A Bayesian approach to real estate assessment. In: Stephen, E.F., Zellner,A. (Eds.), Studies in Bayesian Econometrics and Statistics in Honor of Leonard J. Savage. North-Holland, Amsterdam, pp. 195–208.

Yadav, A. S., Singh, S. K., and Singh, U. (2016). On hybrid censored inverse Lomax distribution: application to the survival data. Statistica, 76(2), 185–203.

Ye, Z. S., Chan, P. S., Xie, M., and Ng, H. K. T. (2014). Statistical inference for the extreme value distribution under adaptive Type-II progressive censoring schemes. Journal of Statistical Computation and Simulation, 84(5), 1099–1114.

Zellner, A., and Geisel, M. S. (1968). Sensitivity of control to uncertainty and form of the criterion function. In: Donald, G.W. (Ed.), The Future of Statistics. Academic Press, New York, 269–289.