Consistent maximum likelihood estimation of a unimodal density using shape restrictions

It is well known that the unimodal maximum likelihood estimator of a density is consistent everywhere but at the mode. The authors review various ways to solve this problem and propose a new estimator that is concave over an interval containing the mode; this interval may be chosen by the user or through an algorithm. The authors show how to implement their solution and compare it to other approaches through simulations. They show that the new estimator is consistent everywhere and determine its rate of convergence in the Hellinger metric.     

Pitman closeness,monotonicity and consistency of best linear unbiased and invariant estimators for exponential distribution under Type II censoring

Comparisons of best linear unbiased estimators with some other prominent estimators have been carried out over the last 50 years since the ground breaking work of Lloyd [E.H. Lloyd, Least squares estimation of location and scale parameters using order statistics, Biometrika 39 (1952), pp. 88–95]. These comparisons have been made under many different criteria across different parametric families of distributions. A noteworthy one is by Nagaraja [H.N. Nagaraja, Comparison of estimators and predictors from two-parameter exponential distribution, Sankhyā Ser. B 48 (1986), pp. 10–18], who made a comparison of best linear unbiased (BLUE) and best linear invariant (BLIE) estimators in the case of exponential distribution. In this paper, continuing along the same lines by assuming a Type II right censored sample from a scaled-exponential distribution, we first compare BLUE and BLIE of the exponential mean parameter in terms of Pitman closeness (nearness) criterion. We show that the BLUE is always Pitman closer than the BLIE. Next, we introduce the notions of Pitman monotonicity and Pitman consistency, and then establish that both BLUE and BLIE possess these two properties.     

An F-type test for detecting departure from monotonicity in a functional linear model

When studying associations between a functional covariate and scalar response using a functional linear model (FLM), scientific knowledge may indicate possible monotonicity of the unknown parameter curve. In this context, we propose an F-type test of monotonicity, based on a full versus reduced nested model structure, where the reduced model with monotonically constrained parameter curve is nested within an unconstrained FLM. For estimation under the unconstrained FLM, we consider two approaches: penalised least-squares and linear mixed model effects estimation. We use a smooth then monotonise approach to estimate the reduced model, within the null space of monotone parameter curves. A bootstrap procedure is used to simulate the null distribution of the test statistic. We present a simulation study of the power of the proposed test, and illustrate the test using data from a head and neck cancer study.     

Progressive censoring with fixed censoring times

In this paper, we revisit the progressive Type-I censoring scheme as it has originally been introduced by Cohen [Progressively censored samples in life testing. Technometrics. 1963;5(3):327–339]. In fact, original progressive Type-I censoring proceeds as progressive Type-II censoring but with fixed censoring times instead of failure time based censoring times. Apparently, a time truncation has been added to this censoring scheme by interpreting the final censoring time as a termination time. Therefore, not much work has been done on Cohens's original progressive censoring scheme with fixed censoring times. Thus, we discuss distributional results for this scheme and establish exact distributional results in likelihood inference for exponentially distributed lifetimes. In particular, we obtain the exact distribution of the maximum likelihood estimator (MLE). Further, the stochastic monotonicity of the MLE is verified in order to construct exact confidence intervals for both the scale parameter and the reliability.     

Estimation and Inference Procedures for Semiparametric Distribution Models with Varying Linear‐Index

More flexible semiparametric linear‐index regression models are proposed to describe the conditional distribution. Such a model formulation captures varying effects of covariates over the support of a response variable in distribution, offers an alternative perspective on dimension reduction and covers a lot of widely used parametric and semiparameteric regression models. A feasible pseudo likelihood approach, accompanied with a simple and easily implemented algorithm, is further developed for the mixed case with both varying and invariant coefficients. By justifying some theoretical properties on Banach spaces, the uniform consistency and asymptotic Gaussian process of the proposed estimator are also established in this article. In addition, under the monotonicity of distribution in linear‐index, we develop an alternative approach based on maximizing a varying accuracy measure. By virtue of the asymptotic recursion relation for the estimators, some of the achievements in this direction include showing the convergence of the iterative computation procedure and establishing the large sample properties of the resulting estimator. It is noticeable that our theoretical framework is very helpful in constructing confidence bands for the parameters of interest and tests for the hypotheses of various qualitative structures in distribution. Generally, the developed estimation and inference procedures perform quite satisfactorily in the conducted simulations and are demonstrated to be useful in reanalysing data from the Boston house price study and the World Values Survey.     

On perfect simulation for some mixtures of distributions

This paper studies the implementation of the coupling from the past (CFTP) method of Propp and Wilson (1996) in the set-up of two and three component mixtures with known components and unknown weights. We show that monotonicity structures can be exhibited in both cases, but that CFTP can still be costly for three component mixtures. We conclude with a simulation experiment exhibiting an almost perfect sampling scheme where we only consider a subset of the exhaustive set of starting values.     

Oligarchy for Social Choice Correspondences and Strategy-Proofness

We study the existence of a group of individuals which has some decisive power for social choice correspondences that satisfy a monotonicity property which we call modified monotonicity. And we examine the relation between modified monotonicity and strategy-proofness of social choice correspondences according to the definition by Duggan and Schwartz (2000). We will show mainly the following two results. (1) Modified monotonicity implies the existence of an oligarchy. An oligarchy is a group of individuals such that it has some decisive power (semi-decisiveness), and at least one of the most preferred alternatives of every its member is always chosen by any social choice correspondence. (2) Strategy-proofness of social choice correspondences is equivalent to modified monotonicity.     

Impartial Nominations for a Prize

A group of peers must choose one of them to receive a prize; everyone cares only about winning, not about who gets the prize if someone else. An award rule is impartial if one's message never influences whether or not one wins the prize. We explore the consequences of impartiality when each agent nominates a single (other) agent for the prize. On the positive side, we construct impartial nomination rules where both the influence of individual messages and the requirements to win the prize are not very different across agents. Partition the agents in two or more districts, each of size at least 3, and call an agent a local winner if he is nominated by a majority of members of his own district; the rule selects a local winner with the largest support from nonlocal winners, or a fixed default agent in case there is no local winner. On the negative side, impartiality implies that ballots cannot be processed anonymously as in plurality voting. Moreover, we cannot simultaneously guarantee that the winner always gets at least one nomination, and that an agent nominated by everyone else always wins.     

On exact confidence intervals in a competing risks model with generalized hybrid type-I censored exponential data

In a recent paper by Mao, Shi and Sun that appeared in Journal of Statistical Computation and Simulation, the authors discuss, among other approaches, the construction of exact confidence intervals for the underlying parameters by ‘pivoting the cumulative distribution functions’ of the corresponding maximum likelihood estimators (MLEs). The authors assume that this method is applicable without providing the appropriate justification. In this short note the two requirements for the applicability of this method are discussed, namely, the stochastic monotonicity of the MLEs and the existence of solutions to the equations defining the exact confidence interval's endpoints.     

Estimation of partially linear single-index additive hazards model with current status data

In this paper, we introduce a partially linear single-index additive hazards model with current status data. Both the unknown link function of the single-index term and the cumulative baseline hazard function are approximated by B-splines under a monotonicity constraint on the latter. The sieve method is applied to estimate the nonparametric and parametric components simultaneously. We show that, when the nonparametric link function is an exact B-spline, the resultant estimator of regression parameter vector is asymptotically normal and achieves the semiparametric information bound and the rate of convergence of the estimator for the cumulative baseline hazard function is optimal. Simulation studies are presented to examine the finite sample performance of the proposed estimation method. For illustration, we apply the method to a clinical dataset with current status outcome.