Nonparametric Bayes estimation of gap-time distribution with recurrent event data

Nonparametric Bayes (NPB) estimation of the gap-time survivor function governing the time to occurrence of a recurrent event in the presence of censoring is considered. In our Bayesian approach, the gap-time distribution, denoted by F, has a Dirichlet process prior with parameter α. We derive NPB and nonparametric empirical Bayes (NPEB) estimators of the survivor function F?=1?F and construct point-wise credible intervals. The resulting Bayes estimator of F? extends that based on single-event right-censored data, and the PL-type estimator is a limiting case of this Bayes estimator. Through simulation studies, we demonstrate that the PL-type estimator has smaller biases but higher root-mean-squared errors (RMSEs) than those of the NPB and the NPEB estimators. Even in the case of a mis-specified prior measure parameter α, the NPB and the NPEB estimators have smaller RMSEs than the PL-type estimator, indicating robustness of the NPB and NPEB estimators. In addition, the NPB and NPEB estimators are smoother (in some sense) than the PL-type estimator.     

A smoothed Q‐learning algorithm for estimating optimal dynamic treatment regimes

In this paper, we propose a smoothed Q‐learning algorithm for estimating optimal dynamic treatment regimes. In contrast to the Q‐learning algorithm in which nonregular inference is involved, we show that, under assumptions adopted in this paper, the proposed smoothed Q‐learning estimator is asymptotically normally distributed even when the Q‐learning estimator is not and its asymptotic variance can be consistently estimated. As a result, inference based on the smoothed Q‐learning estimator is standard. We derive the optimal smoothing parameter and propose a data‐driven method for estimating it. The finite sample properties of the smoothed Q‐learning estimator are studied and compared with several existing estimators including the Q‐learning estimator via an extensive simulation study. We illustrate the new method by analyzing data from the Clinical Antipsychotic Trials of Intervention Effectiveness–Alzheimer's Disease (CATIE‐AD) study.     

Objective Bayesian Analysis of Skew‐t Distributions

Abstract. We study the Jeffreys prior and its properties for the shape parameter of univariate skew‐t distributions with linear and nonlinear Student's t skewing functions. In both cases, we show that the resulting priors for the shape parameter are symmetric around zero and proper. Moreover, we propose a Student's t approximation of the Jeffreys prior that makes an objective Bayesian analysis easy to perform. We carry out a Monte Carlo simulation study that demonstrates an overall better behaviour of the maximum a posteriori estimator compared with the maximum likelihood estimator. We also compare the frequentist coverage of the credible intervals based on the Jeffreys prior and its approximation and show that they are similar. We further discuss location‐scale models under scale mixtures of skew‐normal distributions and show some conditions for the existence of the posterior distribution and its moments. Finally, we present three numerical examples to illustrate the implications of our results on inference for skew‐t distributions.     

Nonparametric estimators for shift in quantal bioassay

Let F(x) and F(x+θ) be log dose-response curves for a standard preparation and a test preparation, respectively, in a parallel quantal bioassay designed to test the relative potency of a drug, toxicant, or some other substance, and suppose the form of F is unknown. Several estimators of the shift parameter θ or relative potency, are compared, including some generalized and trimmed Spearman-Kärber estimators and a non parametric maximum likelihood estimator. Both point and interval estimation are discussed. Some recommendations concerning the choices of estimators are offered.     

Bias-robust L-estimators of a scale parameter

We derive optimal bias-robust L-estimators of a scale parameter σ based on random samples from F(( ·?θ/σ), where θ and σ are unknown and F is an unknown member of a ε-contaminated neighborhood of a fixed symmetric error distribution F ₀. Within a very general class S of L-estimators which are Fisher-consistent at F, we solve for: (i) the estimator with minimax asymptotic bias over the ε-contamination neighborhood; and (ii) the estimator with minimum gross error sensitivity at F ₀ [the limiting case of (i) as ε → 0]. The solutions to problems (i) and (ii) are shown, using a generalized method of moment spaces, to be mixtures of at most two interquantile ranges. A graphical method is presented for finding the optimal bias-robust solutions, and examples are given.     

Improved confidence estimators for Fieller's confidence sets

Fieller's confidence set C_F for ratios of location parameters, although of great importance in practice, is often cited as an example to criticize frequentist theory. The reason is that the set can consist of the whole parameter space and yet the confidence is γ = 1 – α in any case. In this paper, we study the problem of constructing data-dependent estimators better than γ₊, A reasonable estimator appears to be γ₊, which is one if C_F is the whole parameter space and γ otherwise. By using an estimated confidence approach and a squared-error loss, it is shown that γ₊ dominates γ. The risk improvement of γ₊ over γ can be sizable. Also, by numerically comparing γ₊ with a generalized Bayes estimator γ_L, which is shown to be admissible when one or two ratios are concerned, it is shown that γ₊ is nearly admissible. We also conclude that the common practice of reporting 1 – α only when C_F is not the whole parameter space is nearly admissible.     

Density Estimation by Total Variation Penalized Likelihood Driven by the Sparsity ℓ1 Information Criterion

Abstract. We propose a non‐linear density estimator, which is locally adaptive, like wavelet estimators, and positive everywhere, without a log‐ or root‐transform. This estimator is based on maximizing a non‐parametric log‐likelihood function regularized by a total variation penalty. The smoothness is driven by a single penalty parameter, and to avoid cross‐validation, we derive an information criterion based on the idea of universal penalty. The penalized log‐likelihood maximization is reformulated as an ?₁‐penalized strictly convex programme whose unique solution is the density estimate. A Newton‐type method cannot be applied to calculate the estimate because the ?₁‐penalty is non‐differentiable. Instead, we use a dual block coordinate relaxation method that exploits the problem structure. By comparing with kernel, spline and taut string estimators on a Monte Carlo simulation, and by investigating the sensitivity to ties on two real data sets, we observe that the new estimator achieves good L ₁ and L ₂ risk for densities with sharp features, and behaves well with ties.     

Testing the adequacy of a linear model VIA critical smoothing

An important problem for fitting local linear regression is the choice of the smoothing parameter. As the smoothing parameter becomes large, the estimator tends to a straight line, which is the least squares fit in the ordinary linear regression setting. This property may be used to assess the adequacy of a simple linear model. Motivated by Silverman's (1981) work in kernel density estimation, a suitable test statistic is the critical smoothing parameter where the estimate changes from nonlinear to linear, while linearity or non- linearity requires a more precise judgment. We define the critical smoothing parameter through the approximate F-tests by Hastie and Tibshirani (1990). To assess the significance, the “wild bootstrap” procedure is used to replicate the data and the proportion of bootstrap samples which give a nonlinear estimate when using the critical bandwidth is obtained as the p-value. Simulation results show that the critical smoothing test is useful in detecting a wide range of alternatives.     

Tests of symmetry based on transformed empirical processes

A probability distribution function F is said to be symmetric when 1 ‐ F(x) ‐ F(‐x) = 0 for all x∈ R. Given a sequence of alternatives contiguous to a certain symmetric F₀, the authors are concerned with testing for the null hypothesis of symmetry. The proposed tests are consistent against any nonsymmetric alternative, and their power with respect to the given sequence can easily be optimized. The tests are constructed by means of transformed empirical processes with an adequate selection of the underlying isometry, and the optimum power is obtained by suitably choosing the score functions. The test statistics are very easy to compute and their asymptotic distributions are simple.     

Estimation of non-occurrence probability in periodic model

Dimitrov and Khalil (1992) introduced a class of new probability distributions for modeling environmental evolution with periodic behavior. One of the key parameters in these distributions is α, the probability that the event being studied does not occur. In that article the authors derive an estimator for this parameter assuming a series of conditions. In this article it is shown that the estimator is valid under more general conditions, i.e. same of the assumptions are not necessary. It is shown that under the assumption that the elapsed time measured from the starting point of a period until the first occurrence time of the event given that the event occurred in this cycle is related to α, an approximate maximum likelihood estimator of a is proposed. The large sample properties of the estimator are discussed. Monte Carlo study is done for supporting the theoretical results.     

Preliminary phi-divergence test estimators for linear restrictions in a logistic regression model

The problem of estimation of the parameters in a logistic regression model is considered under multicollinearity situation when it is suspected that the parameter of the logistic regression model may be restricted to a subspace. We study the properties of the preliminary test based on the minimum ϕ -divergence estimator as well as in the ϕ -divergence test statistic. The minimum ϕ -divergence estimator is a natural extension of the maximum likelihood estimator and the ϕ -divergence test statistics is a family of the test statistics for testing the hypothesis that the regression coefficients may be restricted to a subspace.     

One‐Way anova for Functional Data via Globalizing the Pointwise F‐test

In this paper, we propose and study a new global test, namely, GPF test, for the one‐way anova problem for functional data, obtained via globalizing the usual pointwise F‐test. The asymptotic random expressions of the test statistic are derived, and its asymptotic power is investigated. The GPF test is shown to be root‐n consistent. It is much less computationally intensive than a parametric bootstrap test proposed in the literature for the one‐way anova for functional data. Via some simulation studies, it is found that in terms of size‐controlling and power, the GPF test is comparable with two existing tests adopted for the one‐way anova problem for functional data. A real data example illustrates the GPF test.     

Simultaneous estimation of several CDF’s: homogeneity constraint

Let {x_ij(1 ? j ? n_i)|i = 1, 2, …, k} be k independent samples of size n_j from respective distributions of functions F_j(x)(1 ? j ? k). A classical statistical problem is to test whether these k samples came from a common distribution function, F(x) whose form may or may not be known. In this paper, we consider the complementary problem of estimating the distribution functions suspected to be homogeneous in order to improve the basic estimator known as “empirical distribution function” (edf), in an asymptotic setup. Accordingly, we consider four additional estimators, namely, the restricted estimator (RE), the preliminary test estimator (PTE), the shrinkage estimator (SE), and the positive rule shrinkage estimator (PRSE) and study their characteristic properties based on the mean squared error (MSE) and relative risk efficiency (RRE) with tables and graphs. We observed that for k ? 4, the positive rule SE performs uniformly better than both shrinkage and the unrestricted estimator, while PTEs works reasonably well for k < 4.     

On distribution function estimation with partially rank-ordered set samples: estimating mercury level in fish using length frequency data

We study the non-parametric estimation of a continuous distribution function F based on the partially rank-ordered set (PROS) sampling design. A PROS sampling design first selects a random sample from the underlying population and uses judgement ranking to rank them into partially ordered sets, without measuring the variable of interest. The final measurements are then obtained from one of the partially ordered sets. Considering an imperfect PROS sampling procedure, we first develop the empirical distribution function (EDF) estimator of F and study its theoretical properties. Then, we consider the problem of estimating F, where the underlying distribution is assumed to be symmetric. We also find a unique admissible estimator of F within the class of nondecreasing step functions with jumps at observed values and show the inadmissibility of the EDF. In addition, we introduce a smooth estimator of F and discuss its theoretical properties. Finally, we expand on various numerical illustrations of our results via several simulation studies and a real data application and show the advantages of PROS estimates over their counterparts under the simple random and ranked set sampling designs.     

Generating data from improper distributions: application to Cox proportional hazards models with cure

Cure rate models are survival models characterized by improper survivor distributions which occur when the cumulative distribution function, say F, of the survival times does not sum up to 1 (i.e. F(+∞)<1). The first objective of this paper is to provide a general approach to generate data from any improper distribution. An application to times to event data randomly drawn from improper distributions with proportional hazards is investigated using the semi-parametric proportional hazards model with cure obtained as a special case of the nonlinear transformation models in [Tsodikov, Semiparametric models: A generalized self-consistency approach, J. R. Stat. Soc. Ser. B 65 (2003), pp. 759–774]. The second objective of this paper is to show by simulations that the bias, the standard error and the mean square error of the maximum partial likelihood (PL) estimator of the hazard ratio as well as the statistical power based on the PL estimator strongly depend on the proportion of subjects in the whole population who will never experience the event of interest.     

Full likelihood inference in the Cox model with LTRC data when covariates are discrete

For the regression parameter β in the Cox model, there have been several estimates based on different types of approximated likelihood. For right-censored data, Ren and Zhou [Full likelihood inferences in the Cox model: an empirical approach. Ann Inst Statist Math. 2011;63:1005–1018] derive the full likelihood function for (β, F₀), where F₀ is the baseline distribution function in the Cox model. In this article, we extend their results to left-truncated and right-censored data with discrete covariates. Using the empirical likelihood parameterization, we obtain the full-profile likelihood function for β when covariates are discrete. Simulation results indicate that the maximum likelihood estimator outperforms Cox's partial likelihood estimator in finite samples.     

A Family of Goodness‐of‐Fit Tests for Copulas Based on Characteristic Functions

A general class of rank statistics based on the characteristic function is introduced for testing goodness‐of‐fit hypotheses about the copula of a continuous random vector. These statistics are defined as L ₂ weighted functional distances between a nonparametric estimator and a semi‐parametric estimator of the characteristic function associated with a copula. It is shown that these statistics behave asymptotically as degenerate V ‐statistics of order four and that the limit distributions have representations in terms of weighted sums of independent chi‐square variables. The consistency of the tests against general alternatives is established and an asymptotically valid parametric bootstrap is suggested for the computation of the critical values of the tests. The behaviour of the new tests in small and moderate sample sizes is investigated with the help of simulations and compared with a competing test based on the empirical copula. Finally, the methodology is illustrated on a five‐dimensional data set.     

The General Expressions for the Moments of the Stochastic Shrinkage Parameters of the Liu Type Estimator

ABSTRACT

One of the problems with the Liu estimator is the appropriate value for the unknown biasing parameter d. In this article we consider the optimum value for d and give upper bound for the expected value of the estimator of this biasing parameter. We also derive the general expressions for the moments of the stochastic shrinkage parameters of the Liu estimator and the generalized Liu estimator. Numerical calculations are carried out to illustrate the behavior of the mean and variance of the biasing parameter. Also, a numerical example is given to illustrate the effect of the biasing parameter d, on the mean square error of the Liu estimator.     

Combining poisson means

The problem of estimating the Poisson mean is considered based on the two samples in the presence of uncertain prior information (not in the form of distribution) that two independent random samples taken from two possibly identical Poisson populations. The parameter of interest is λ₁ from population I. Three estimators, i.e. the unrestricted estimator, restricted estimator and preliminary test estimator are proposed. Their asymptotic mean squared errors are derived and compared; parameter regions have been found for which restricted and preliminary test estimators are always asymptotically more efficient than the classical estimator. The relative dominance picture of the estimators is presented. Maximum and minimum asymptotic efficiencies of the estimators relative to the classical estimator are tabulated. A max-min rule for the size of the preliminary test is also discussed. A Monte Carlo study is presented to compare the performance of the estimator with that of Kale and Bancroft (1967).     

Robust M-estimators of scale: Minimax bias versus maximal variance

In the location-scale estimation problem, we study robustness properties of M-estimators of the scale parameter under unknown ?-contamination of a fixed symmetric unimodal error distribution F₀. Within a general class of M-estimators, the estimator with minimax asymptotic bias is shown to lie within the subclass of α-interquantile ranges of the empirical distribution symmetrized about the sample median. Our main result is that as ? → 0, the limiting minimax asymptotic bias estimator is sometimes (e.g., when F_o is Cauchy), but not always, the median absolute deviation about the median. It is also shown that contamination in the neighbourhood of a discontinuity of the influence function of a minimax bias estimator can sometimes inflate the asymptotic variance beyond that achieved by placing all the ?-contamination at infinity. This effect is quantified by a new notion of asymptotic efficiency that takes into account the effect of infinitesimal contamination of the parametric model for the error distribution.