Variable Scale Kernel Density Estimation for Simple Linear Degradation Model


In this study, we proposed the variable scale kernel estimator for analyzing the degradation data. The properties of the proposed method are investigated and compared with the classical method such as; maximum likelihood and ordinary least square methods via simulation technique. The criteria bias and MSE are used for comparison. Simulation results showed that the performance of the variable scale kernel estimator is acceptable as a general estimator. It is nearly the best estimator when the assumption of the distribution is invalid. Application to real data set is also given.

DOI Code: 10.1285/i20705948v14n2p359

Keywords: Bandwidth selection; Classical kernel; Degradation; Failure time; Maximum likelihood; Ordinary least square; Variable scale kernel estimation.


Abramson, I. S. (1982). On bandwidth variation in kernel estimates-square root law. Ann. Statist., Vol 10,1217-1223.

Al-Haj Ebrahem M., Alodat M. and Arman A. (2009a). Estimating the Time-to-Failure Distribution of a Linear Degradation Model Using a Bayesian Approach. Applied Mathematical Sciences, Vol. 3, 2009, no. 1, 27 – 42.

Al-Haj Ebrahem M., Eidous, O. and Kmail, G. (2009b). Estimating Percentiles of Time-to-Failure Distribution Obtained from a Linear Degradation Model Using The Kernel Density Method. Communications in Statistics – Simulation and Computation.38 (9): 1811-1822

Alodat M. T and Al-Haj Ebrahem M. (2007). Ranked Set Sampling Technique to Estimate a Time-to-Failure Distribution of a Linear Degradation Model. Journal of Applied Statistical Science. 17(1): 143-149.

Ba Dakhen, L., Al-Haj Ebrahem, M and Eidous, O. (2017). Semi-parametric method to estimate time-to-failure distribution and its percentile for simple linear degradation model. Journal of Modern Applied Statistical methods. 16(2) :322-346

Eidous, o., Al-Haj Ebrahem M. and Ba Dakhen, L (2017). Estimating the time-to-failure distribution and its percentile for simple linear degradation model using Double kernel method. Journal of probability and Statistical science. 15(1):121-134

John W. Pratt, F. Y. Edgeworth and R. A. (1976). Fisher on the Efficiency of Maximum Likelihood Estimation. The Annals of Statistics, 4 (3): pp. 501–514

Lu, C.J. and Meeker, W. Q. (1993). Using Degradation Measure to Estimate a Time-Failure Distribution. Technometrics, 35, 161-174.

Meeker, W. Q. and Escobar, L. A. (1998). Statistical Method for Reliability Data. John Wiley and Sons, Inc. New York.

Meeker, W. Q., Escobar, L. A. and Lu, C. J. (1998). Accelerated Degradation Tests: Modeling and Analysis. Technometrics, 40, 89-99.

Robinson, M.E. and Crowder, M.J (2000). Bayesian methods for a growth-curve degradation model with repeated measures. Lifetime Data Analysis, 6, 357-374.

Shoukri, M. M., Mian, I. U. and Tracy, D. S. (1988). Sampling Properties of Estimators of the Log-Logistic Distribution with Application to Canadian Precipitation Data. The Canadian Journal of Statistics 16 (3): 223–236.

Silverman, B.W (1986). Density Estimation For Statistics and Data Analysis. Chapman and Hall, London.

Full Text: pdf

Creative Commons License
This work is licensed under a Creative Commons Attribuzione - Non commerciale - Non opere derivate 3.0 Italia License.