© Huda H. Qubbaj, Husam A. Bayoud, Hisham M. Hilow. Article available under the CC BY-SA 4.0 licence

This paper proposes nonparametric estimates for the two information measures extropy and entropy when a progressively Type-I interval censored data is available. Different nonparametric approaches are used for deriving the estimates, including: moments of the empirical cumulative distribution function and linear regression. The performance of the proposed estimates is studied under various censoring schemes via simulation studies. Furthermore, different real data sets are analyzed for illustrative purposes.The estimates based on linear approximation J^2 and H^2 outperform the other estimate in the majority of studied cases.

entropy; extropy; mean square error; nonparametric statistics; Monte Carlo simulation; Type-I interval censoring

Aggarwala, R., (2001). Progressive interval censoring: some mathematical results with applications to inference. Commun Stat: Theory Methods, 30, pp. 1921–1935.

Alotaibi, R., Rezk, H., Dey, S., and Okasha, H., (2021). Bayesian estimation for Dagum distribution based on progressive type I interval censoring. PLOS ONE, 16(6), DOI: 10.1371/journal.pone.0252556

Awad, A. M. and Alawneh, A., (1987). Application of entropy of a life time model. IMA J. Math. Control Inf., 4, pp. 143–147.

Balakrishnan, N., Cramer, E., (2014). The Art of Progressive Censoring: Applications to Reliability and Quality. Boston.

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

Correa, J. C., (1995). A new estimator of entropy. Communications in Statistics Theory and Methods, 24, pp. 2439–2449.

Du Y., Guo Y. and Gui W., (2018). Statistical Inference for the Information Entropy of the Log-Logistic Distribution under Progressive Type-I Interval Censoring Schemes. Symmetry, 10, p. 445.

Hazeb, R., Raqab, M. and Bayoud, H., (2021a). Non-parametric estimation of the extropy and the entropy measures based on progressive type-II censored data with testing uniformity. Journal of Statistical Computation and Simulation, 91 (11), pp. 2178–2210.

Hazeb, R., Bayoud, H. and Raqab, M., (2021b). Kernel and CDF-Based Estimation of Extropy and Entropy from Progressively Type-II Censoring with Application for Goodness of Fit Problems. Stochastic and Quality Control, 36 (1), pp. 73–83.

Kittaneh, O. A., Khan, M. A., Akbar, M., and Bayoud, H. A. (2016). Average entropy: a new uncertainty measure with application to image segmentation. The American Statistician, 70 (1), pp. 18–24.

Lad, F., Sanfilippo, G. and Agro, G., (2015). Extropy: Complementary Dual of Entropy. Statistical Science, 30, pp. 40–58.

Lio, YL., Chen, D. G. and Tasi, T. R., (2011). Parameter estimations for generalized exponential distribution under progressive type-I interval censoring.Comput. Stat. Data Anal., 54, pp. 1581–1591.

Nelson, W., (1982). Applied Life Data Analysis, Wiley, New York.

Ng, H. K. T., Wang, Z., (2009). Statistical estimation for the parameters of Weibulldistribution based on progressively Type-I interval censored sample.J.Stat.Comput.Simulat, 79(2), pp. 145–159.

Noughabi, H. A., Jarrahiferiz, J., (2019). On the estimation of extropy. Journal of Nonparametric Statistics, 31(1), pp. 88–99, DOI: 10.1080/10485252.2018.1533133.

Qiu, G., (2017). The Extropy of Order Statistics and Record values. Statistics and Probability Letters, 120, pp. 52–60.

Qiu, G. and Jia, K., (2018a). The Residual Extropy of Order statistics. Statistics and Probability Letters, 133, pp. 15–22.

Qiu, G. and Jia, K., (2018b). Extropy Estimators with Applications in Testing Uniformity. Journal of Nonparametric Statistics, 30(1), pp. 182–196, DOI: 10.1080/1048252.2017. 1404063.

Rao, M., Chen, Y., Vemuri, B. C., and Wang, F., (2004). Cumulative residual entropy: a new measure of information. IEEE transactions on Information Theory, 50 (6), pp. 1220–1228.

Raqab, M. Z. and Qiu, G., (2019). On extropy properties of ranked set sampling. Statistics, 53(1), pp. 210–226, DOI: 10.1080/02331888.2018.1533963.

Renyi, A., (1961). On measures of entropy and information. Stat. and Prob., 1, pp. 547– 561.

Shannon, C. E., (1948). A mathematical theory of communications. Bell System Tech. J., 27(3), pp. 379–423.

Singh, S, Tripathi, Y. M., (2016). Estimating the parameters of an inverse Weibull distribution under progressive type-I interval censoring. Stat. Pap., 59, pp. 21–56.

Tsallis, C., (1988). Possible generalization of Boltzmann Gibbs statistics. J. Stat. Phys., 52, pp. 470–487.

Vasicek, O., (1976). A test for normality based on sample entropy. Journal of the Royal Statistical Society B, 38, pp. 54–59.