Electronic Journal of Statistics

Parametric conditional variance estimation in location-scale models with censored data

Abstract

Suppose the random vector $(X,Y)$ satisfies the regression model $Y=m(X)+\sigma (X)\varepsilon$, where $m(\cdot)=E(Y|\cdot),$ $\sigma^{2}(\cdot)=\mbox{Var}(Y|\cdot)$ belongs to some parametric class $\{\sigma _{\theta}(\cdot):\theta \in \Theta\}$ and $\varepsilon$ is independent of $X$. The response $Y$ is subject to random right censoring and the covariate $X$ is completely observed. A new estimation procedure is proposed for $\sigma _{\theta}(\cdot)$ when $m(\cdot)$ is unknown. It is based on nonlinear least squares estimation extended to conditional variance in the censored case. The consistency and asymptotic normality of the proposed estimator are established. The estimator is studied via simulations and an important application is devoted to fatigue life data analysis.

Article information

Source
Electron. J. Statist., Volume 11, Number 1 (2017), 148-176.

Dates
First available in Project Euclid: 1 February 2017

https://projecteuclid.org/euclid.ejs/1485939611

Digital Object Identifier
doi:10.1214/16-EJS1139

Mathematical Reviews number (MathSciNet)
MR3604021

Zentralblatt MATH identifier
1356.62172

Subjects
Primary: 62N01: Censored data models 62N02: Estimation

Citation

Heuchenne, Cédric; Laurent, Géraldine. Parametric conditional variance estimation in location-scale models with censored data. Electron. J. Statist. 11 (2017), no. 1, 148--176. doi:10.1214/16-EJS1139. https://projecteuclid.org/euclid.ejs/1485939611

References

• [1] Beran, R. (1981). Nonparametric regression with randomly censored survival data. Technical Report, Univ. California, Berkeley.
• [2] Dai, J. and Sperlich, S. (2010). Simple and effective boundary correction for kernel densities and regression with an application to the world income and Engel curve estimation., Comput. Statist. Data Anal., 54, 2487–2497.
• [3] Dette, H. and Heuchenne, C. (2012). Scale checks in censored regression., Scand. J. Statist., 39, 323–339.
• [4] Efron, B. (1981). Censored data and the bootstrap., J. Amer. Statist. Assoc., 76, 312–319.
• [5] Heuchenne, C. and Van Keilegom, I. (2007). Polynomial regression with censored data based on preliminary nonparametric estimation., Ann. Instit. Statist. Math., 59, 273–298.
• [6] Heuchenne, C. and Van Keilegom, I. (2008). Nonlinear regression with censored data., Technometrics, 49, 34–44.
• [7] Heuchenne, C. and Van Keilegom, I. (2010). Estimation in nonparametric location-scale regression models with censored data., Ann. Inst. Statist. Math., 62, 439–463.
• [8] Jennrich, R.I. (1969). Asymptotic properties of nonlinear least squares estimators., Ann. Math. Statist., 40, 633–643.
• [9] Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete observations., J. Amer. Statist. Assoc., 53, 457–481.
• [10] Levenberg, K. (1944). A method for the solution of certain problems in least squares., Quart. Appl. Math., 2, 164–168.
• [11] Li, G. and Datta, S. (2001). A bootstrap approach to non-parametric regression for right censored data., Ann. Inst. Statist. Math., 53, 708–729.
• [12] Marquardt, D. (1963). An algorithm for least-squares estimation of nonlinear parameters., SIAM J. Appl. Math., 11, 431–441.
• [13] Nelson, W. (1984). Fitting of fatigue curves with nonconstant standard deviation to data with runouts., J. Testing Eval., 12,
• [14] Pascual, F.G. and Meeker, W.Q. (1997). Analysis of fatigue data with runouts based on a model with nonconstant standard deviation and a fatigue limit parameter., J. Testing Eval., 25, 292–301.
• [15] Pascual, F.G. and Meeker, W.Q. (1999). Estimating fatigue curves with the random fatigue-limit model., Technometrics, 4, 277–290.
• [16] Shen, C.L. (1994)., Statistical analysis of fatigue data. Unpublished Ph.D. dissertation, University of Arizona, Department of Aerospace and Mechanical Engineering.
• [17] Van der Vaart, A.W. (1998)., Asymptotic statistics. Cambridge University Press, Cambridge.
• [18] Van Keilegom, I. and Akritas, M. G. (1999). Transfer of tail information in censored regression models., Ann. Statist., 27, 1745–1784.
• [19] Van Keilegom, I. and Veraverbeke, N. (1997). Estimation and bootstrap with censored data in fixed design nonparametric regression., Ann. Inst. Statist. Math., 49, 467–491.
• [20] Wei, S. X. (2002). A censored-GARCH model of asset returns with price limits., Journal of Empirical Finance, 9, 197–223.