A PROFILE LIKELIHOOD THEORY FOR THE CORRELATED GAMMA-FRAILTY MODEL WITH CURRENT STATUS FAMILY DATA

A profile likelihood inference is made for the regression coefficient and frailty parameters in the correlated gamma-frailty model for current status family data. With the introduction of an identifiability assumption, the identifiability of the parameters and the existence of the nonparametric maximum likelihood estimate (NPMLE) are established, the consistency and convergence rate of the NPMLE are obtained, the invertibility of the efficient Fisherinformation matrix is proved, and a quadratic expansion of the profile likelihood is established. From these, we show that the NPMLE of the parameters of interest is asymptotically normal and efficient, its covariance matrix can be estimated consistently by means of the profile likelihood, and the likelihood ratio test is asymptotically chi-squared. A simulation study is carried out to illustrate the numerical performance of the likelihood ratio test.

[1]  Susan A. Murphy,et al.  Observed information in semi-parametric models , 1999 .

[2]  S A Murphy,et al.  Observed Information in Semiparametric Models , 2022 .

[3]  Ya-Xiang Yuan,et al.  A Nonlinear Conjugate Gradient Method with a Strong Global Convergence Property , 1999, SIAM J. Optim..

[4]  A. V. D. Vaart.  Asymptotic Statistics: Delta Method , 1998 .

[5]  Jian Huang,et al.  Efficient Estimation for the Cox Model with Interval Censoring Efficient Estimation for the Cox Model with Interval Censoring , 1994 .

[6]  On Profile Likelihood: Comment , 2000 .

[7]  A. W. van der Vaart,et al.  On Profile Likelihood , 2000 .

[8]  Jim Hefferon,et al.  Linear Algebra , 2012 .

[9]  P. Gänssler.  Weak Convergence and Empirical Processes - A. W. van der Vaart; J. A. Wellner. , 1997 .

[10]  A. Yashin,et al.  Correlated individual frailty: an advantageous approach to survival analysis of bivariate data. , 1995, Mathematical population studies.

[11]  Jian Huang,et al.  Asymptotic normality of the NPMLE of linear functionals for interval censored data, case 1 , 1995 .

[12]  Jon A. Wellner,et al.  A Hybrid Algorithm for Computation of the Nonparametric Maximum Likelihood Estimator from Censored Data , 1997 .

[13]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[14]  K. Do,et al.  Efficient and Adaptive Estimation for Semiparametric Models. , 1994 .

[15]  Susan A. Murphy,et al.  Semiparametric likelihood ratio inference , 1997 .

[16]  A. W. van der Vaart.  onbekend [Review of: Efficient and adaptive estimation for semiparametric models.] , 2000 .

[17]  Erik T. Parner,et al.  Asymptotic theory for the correlated gamma-frailty model , 1998 .

[18]  C. Hsiung,et al.  An asymptotic theory for the nonparametric maximum likelihood estimator in the Cox gene model , 2005 .

[19]  W. Greub.  Linear Algebra , 1981 .

[20]  W. Rudin.  Principles of mathematical analysis , 1964 .

[21]  A Tsodikov,et al.  Semiparametric models: a generalized self‐consistency approach , 2003, Journal of the Royal Statistical Society. Series B, Statistical methodology.