ABSTRACT. The likelihood ratio statistic for testing pointwise hypotheses about the survival time distribution in the current status model can be inverted to yield confidence intervals (CIs). One advantage of this procedure is that CIs can be formed without estimating the unknown parameters that figure in the asymptotic distribution of the maximum likelihood estimator (MLE) of the distribution function. We discuss the likelihood ratio-based CIs for the distribution function and the quantile function and compare these intervals to several different intervals based on the MLE. The quantiles of the limiting distribution of the MLE are estimated using various methods including parametric fitting, kernel smoothing and subsampling techniques. Comparisons are carried out both for simulated data and on a data set involving time to immunization against rubella. The comparisons indicate that the likelihood ratio-based intervals are preferable from several perspectives. Key words: asymptotic distribution, bootstrap, confidence interval, current status data, kernel smoothing, quantile estimation, rubella data, subsampling

The behavior of maximum likelihood estimates (MLE’s) and the likelihood ratio statistic in a family of problems involving pointwise nonparametric estimation of a monotone function is studied. This class of problems differs radically from the usual parametric or semiparametric situations in that the MLE of the monotone function at a point converges to the truth at rate n 1/3 (slower than the usual √ n rate) with a non-Gaussian limit distribution. A framework for likelihood based estimation of monotone functions is developed and limit theorems describing the behavior of the MLE’s and the likelihood ratio statistic are established. In particular, the likelihood ratio statistic is found to be asymptotically pivotal with a limit distribution that is no longer χ 2 but can be explicitly characterized in terms of a functional of Brownian motion. 1

Limit distributions for the greatest convex minorant and its derivative are considered for a general class of stochastic processes including partial sum processes and empirical processes, for independent, weakly dependent and long range dependent data. The results are applied to isotonic regression, isotonic regression after kernel smoothing, estimation of convex regression functions, and estimation of monotone and convex density functions. Various pointwise limit distributions are obtained, and the rate of convergence depends on the self similarity properties and on the rate of convergence of the processes considered. 1. Introduction. Let {xn}n≥1

We consider estimation in a particular semiparametric regression model for the mean of a counting process under the assumption of "panel count" data. The basic model assumption is that the conditional mean function of the counting process is of the form E{N(t)|Z} = exp (θ' Z)Λ(t) where Z is a vector of covariates and Λ is the baseline mean function. The "panel count" observation scheme involves observation of the counting process N for an individual at a random number K of random time points; both the number and the locations of these time points may dier across individuals. We study maximum pseudo-likelihood and maximum likelihood estimators ... and ... of the regression parameter θ. The pseudo-likelihood estimators are fairly easy to compute, while the full maximum likelihood estimators pose more challenges from the computational perspective. We derive expressions for the asymptotic variances of both estimators under the proportional mean model. Our primary aim is to understand when the pseudo-likelihood estimators have very low efficiency relative to the full maximum likelihood estimators. The upshot is that the pseudo-likelihood estimators can have arbitrarily small efficiency relative to the full maximum likelihood estimators when the distribution of K, the number of observation time points per individual, is very heavy-tailed.

∗ corresponding author In this paper we introduce three natural “score statistics ” for testing the hypothesis that F(t0) takes on a fixed value in the context of nonparametric inference with current status data. These three new test statistics have natural intepretations in terms of certain (weighted) L2 distances, and are also connected to natural “one-sided ” scores. We compare these new test statistics with an analogue of the classical Wald statistic and the likelihood ratio statistic introduced in Banerjee and Wellner (2001) for the same testing problem. Under classical “regular ” statistical problems the likelihood ratio, score, and Wald statistics all have the same chi-squared limiting distribution under the null hypothesis. In sharp contrast, in this non-regular problem all three statistics have different limiting distributions under the null hypothesis. Thus we begin by establishing the limit distribution theory of the statistics under the null hypothesis, and discuss calculation of the relevant critical points for the test statistics. Once the null distribution theory is known, the immediate question becomes that of power. We establish the limiting behavior of the three types of statistics under local alternatives. We have also compared the rejection probabilities of these five different statistics via a limited Monte-Carlo study. Our conclusions are: (a) the Wald statistic is less powerful than the likelihood ratio and score statistics; and (b) one of the score statistics may have more power than the likelihood ratio statistic for some alternatives.

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Abstract. In this paper we study the nonparametric MLE and LSE of a convex hazard function. Our estimators are shown to be consistent and to converge at rate n 2/5. Moreover we establish the pointwise asymptotic distribution theory of both estimators under the assumption that the true hazard function is positive with positive second derivative at the fixed point. The same problems for a convex hazard function under right censoring and for the Poisson process with a convex rate are also considered briefly. 1.

We address the problem of pointwise estimation of a regression function under certain shape constraints, using a number of different statistics that can be viewed as measures of discrepancy from a postulated null hypothesis. Pointwise confidence sets are obtained via the usual inversion technique that exploits the duality between construction of confidence sets for a parameter of interest and testing pointwise hypotheses about that parameter. Monotonicity, unimodality and U–shapes are considered. A major advantage of these proposed methods lies in the fact that the statistics of interest are approximately pivotal for large sample sizes and therefore enable inference to be carried out without the need to estimate difficult nuisance parameters. Multivariate generalizations are briefly discussed. 1.1. Introduction and Background Function estimation is a ubiquitous and consequently well–studied problem in nonparametric statistics. In several scientific problems, qualitative background knowledge about the function is available, in which case it is sensible to incorporate such information in the statistical analysis. Shape–restrictions are typical examples of such qualitative knowledge and appear in a large body of applications. In particular, monotonicity is a

We investigate the problem of finding confidence sets for split points in decision trees (CART). Our main results establish the asymptotic distribution of the least squares estimators and some associated residual sum of squares statistics in a binary decision tree approximation to a smooth regression curve. Cube-root asymptotics with nonnormal limit distributions are involved. We study various confidence sets for the split point, one calibrated using the subsampling bootstrap, and others calibrated using plug-in estimates of some nuisance parameters. The performance of the confidence sets is assessed in a simulation study. A motivation for developing such confidence sets comes from the problem of phosphorus pollution in the Everglades. Ecologists have suggested that split points provide a phosphorus threshold at which biological imbalance occurs, and the lower endpoint of the confidence set may be interpreted as a level that is protective of the ecosystem. This is illustrated using data from a Duke University Wetlands Center phosphorus dosing study in the Everglades.

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