Qiqing Yu

Consistency of the GMLE with Mixed Case Interval-censored Data

In this paper we consider an interval censorship model in which the endpoints of the censoring intervals are determined by a two stage experiment. In the first stage the value k of a random integer is selected; in the second stage the endpoints are determined by a case k interval censorship model. We prove the strong consistency in the L1(μ)‐topology of the non‐parametric maximum likelihood estimate of the underlying survival function for a measure μ which is derived from the distributions of the endpoints. This consistency result yields strong consistency for the topologies of weak convergence, pointwise convergence and uniform convergence under additional assumptions. These results improve and generalize existing ones in the literature.

Asymptotic properties of the GMLE with case 2 interval-censored data

In case 2 interval censoring the random survival time X of interest is not directly observable, but only known to have occurred before Y, between Y and Z, or after Z, where (Y, Z) is a pair of observable inspection times such that Y

On Consistency of the Self-Consistent Estimator of Survival Functions with Interval-Censored Data

The self-consistent estimator is commonly used for estimating a survival function with interval-censored data. Recent studies on interval censoring have focused on case 2 interval censoring, which does not involve exact observations, and double censoring, which involves only exact, right-censored or left-censored observations. In this paper, we consider an interval censoring scheme that involves exact, left-censored, right-censored and strictly interval-censored observations. Under this censoring scheme, we prove that the self-consistent estimator is strongly consistent under certain regularity conditions.

Asymptotic Properties of Self-Consistent Estimators with Mixed Interval-Censored Data

Mixed interval-censored (MIC) data consist of n intervals with endpoints Li and Ri, i = 1, ..., n. At least one of them is a singleton set and one is a finite non-singleton interval. The survival time Xi is only known to lie between Li and Ri, i = 1, 2, ..., n. Peto (1973, Applied Statistics, 22, 86–91) and Turnbull (1976, J. Roy. Statist. Soc. Ser. B, 38, 290–295) obtained, respectively, the generalized MLE (GMLE) and the self-consistent estimator (SCE) of the distribution function of X with MIC data. In this paper, we introduce a model for MIC data and establish strong consistency, asymptotic normality and asymptotic efficiency of the SCE and GMLE with MIC data under this model with mild conditions.

Estimation of a joint distribution function with multivariate interval-censored data when the nonparametric MLE is not unique

A nonparametric estimator of a joint distribution function F 0 of a d-dimensional random vector with interval-censored (IC) data is the generalized maximum likelihood estimator (GMLE), where d > 2. The GMLE of F 0 with univariate IC data is uniquely defined at each follow-up time. However, this is no longer true in general with multivariate IC data as demonstrated by a data set from an eye study. How to estimate the survival function and the covariance matrix of the estimator in such a case is a new practical issue in analyzing IC data. We propose a procedure in such a situation and apply it to the data set from the eye study. Our method always results in a GMLE with a nonsingular sample information matrix. We also give a theoretical justification for such a procedure. Extension of our procedure to Coxs regression model is also mentioned.

THE NPMLE of the joint distribution function with right-censored and masked competing risks data

Even though the right-censored competing risks data with masked failure cause have been studied for 30 years, the asymptotic properties of the nonparametric maximum-likelihood estimator (NPMLE) of the joint distribution function with such data have never been studied. We show that the solution to the NPMLE is not unique, and the NPMLE proposed in the current literature is inconsistent. Moreover, we construct a consistent NPMLE and establish its asymptotic normality. It is a non-trivial example in the survival analysis context that there exist an inconsistent NPMLE as well as another consistent NPMLE with the same data and under the same model. Our proofs do not need the symmetry assumption made by almost all researchers on such data. We present simulation results on the consistent NPMLE and apply the NPMLE to a data set in medical research.

Minimax invariant estimator of a continuous distribution function

Consider the problems of the continuous invariant estimation of a distribution function with a wide class of loss functions. It has been conjectured for long that the best invariant estimator is minimax for all sample sizes n≥1. This conjecture is proved in this short note.

Unbiasedness of the Theil–Sen estimator

We consider the simple linear regression model. The Theil–Sen estimator is a point estimator of the slope parameter in the model and has many nice properties, most of which are established by Sen. Thus, it is introduced in several classical textbooks on non-parametric statistics. Sen also gave a proof that the Theil–Sen estimator is unbiased under the assumption that the error distribution is continuous. The statement is incorrect. We construct several counterexamples. Furthermore, we show that the continuity assumption on the error distribution is not important to unbiasedness. In particular, if the sample size n = 2 or 3, then the Theil–Sen estimator is unbiased. Moreover, if either the error distribution or the covariates have certain symmetry, then the Theil–Sen estimator is also unbiased.

How to find all Buckley-James estimates instead of just one?

Consider the simple linear regression problem with right-censored data. The Buckley-James estimator (BJE) (1979) has been considered superior over the other available procedures. So far the existing algorithms fail to present all possible BJEs. In this note, we present a feasible non-iterative algorithm which can find all BJEs. We use the Stanford heart transplant data to show that if we do not find all BJEs (6 BJEs for the data), then the analysis may be misleading as is the case in Buckley and James (1979, p. 435) and in Miller (1981, p. 156). Extension to multiple linear regression is also discussed.

Estimation Under the Lehmann Regression Model with Interval-Censored Data

We consider the estimation problem under the Lehmann model with interval-censored data, but focus on the computational issues. There are two methods for computing the semi-parametric maximum likelihood estimator (SMLE) under the Lehmann model (or called Cox model): the Newton-Raphson (NR) method and the profile likelihood (PL) method. We show that they often do not get close to the SMLE. We propose several approach to overcome the computational difficulty and apply our method to a breast cancer research data set.