Isotonic Estimator Research Articles

Isotonic Estimation of Additive Covariate Effects Under Proportional Hazard Models

Isotonic Estimation of Additive Covariate Effects Under Proportional Hazard Models

Adaptive and efficient isotonic estimation in Wicksell's problem

We consider nonparametric estimation in Wicksell's problem, which has applications in astronomy for estimating the distribution of star positions in a galaxy and in material sciences for determining a material's 3D microstructure from 2D cross sections. We focus on the isotonised version of the plug-in estimator (IIE) for the cdf F of the spheres' squared radii. This estimator is fully automatic, requiring no tuning parameters, and we show it is adaptive to local smoothness properties of the distribution function F to be estimated. We also prove a local asymptotic minimax lower bound in this non-standard setting, with log ⁡ n / n -asymptotics and where the functional F to be estimated is not regular. Combined, our results prove that the isotonic estimator (IIE) is an adaptive, easy-to-compute, and efficient estimator.

Modified exponential ratio estimator for ranked set sampling in the presence of tie information and application on COVID-19 real data

In this article, we develop the isotonic ratio mean estimator proposed by Kocyigit and Kadilar (2020) within the case of tie information in the Ranked Set Sampling (RSS). In addition, we take into consideration the population size being large and small. We offer a generalized exponential ratio estimator using the modified isotonic estimator for this situation. Simulation results show that the proposed estimator is more efficient than the estimators in the literature. In the real data study, the 34 regions of China since China is the country where the pandemic started are taken as a small population, and the data of 212 countries on a world scale is taken as a large population, then the estimators are calculated. In the real data study, the proposed estimator gives better results than the existing estimators thus the results in the simulation study are similar to the real data study.

Meta.shrinkage: An R Package for Meta-Analyses for Simultaneously Estimating Individual Means

Meta-analysis is an indispensable tool for synthesizing statistical results obtained from individual studies. Recently, non-Bayesian estimators for individual means were proposed by applying three methods: the James–Stein (JS) shrinkage estimator, isotonic regression estimator, and pretest (PT) estimator. In order to make these methods available to users, we develop a new R package meta.shrinkage. Our package can compute seven estimators (named JS, JS+, RML, RJS, RJS+, PT, and GPT). We introduce this R package along with the usage of the R functions and the “average-min-max” steps for the pool-adjacent violators algorithm. We conduct Monte Carlo simulations to validate the proposed R package to ensure that the package can work properly in a variety of scenarios. We also analyze a data example to show the ability of the R package.

Ratio estimators for ranked set sampling in the presence of tie information

In this article, we study the situation of observations whose ranks cannot be determined for the auxiliary variable in the Ranked Set Sampling (RSS) method. Therefore, we examine the case of tie information for ratio estimators of the population mean. We propose a new exponential ratio estimator using the modified isotonic estimator for this situation. Simulation results show that the proposed estimator is more efficient than the other estimators in literature. In addition, when we examine the recent COVID-19 pandemic situation, we see that the data is suitable for this structure. We can also see from the real data that the proposed estimator gives better results.

Nonparametric inference for panel count data with competing risks

In survival and reliability studies, panel count data arise when we investigate a recurrent event process and each study subject is observed only at discrete time points. If recurrent events of several types are possible, we obtain panel count data with competing risks. Such data arise frequently from transversal studies on recurrent events in demography, epidemiology and reliability experiments where the individuals cannot be observed continuously. In the present paper, we propose an isotonic regression estimator for the cause specific mean function of the underlying recurrent event process of a competing risks panel count data. Further, a nonparametric test is proposed to compare the cause specific mean functions of the panel count competing risks data. Asymptotic properties of the proposed estimator and test statistic are studied. A simulation study is conducted to assess the finite sample behaviour of the proposed estimator and test statistic. Finally, the procedures developed are applied to a real data arising from skin cancer chemo prevention trial.

Nonparametric multi-samples test for simple stochastic ordering against unrestricted alternative

Statistical inferences under simple stochastic ordering for two continuous populations has a long and rich history. But further study for multiple populations is still expected. In this article, we consider samples test for simple stochastic ordering against unrestricted alternative. A test statistic is constructed with the empirical distribution functions and the isotonic regression estimates of distribution functions, and the null asymptotic distribution of the proposed test statistic is given. A bootstrap procedure is suggested for the critical values, and some simulation results are presented to illustrate the proposed test method.

Circumventing superefficiency: An effective strategy for distributed computing in non-standard problems

We propose a strategy for computing estimators in some non-standard M-estimation problems, where the data are distributed across different servers and the observations across servers, though independent, can come from heterogeneous sub-populations, thereby violating the identically distributed assumption. Our strategy fixes the super-efficiency phenomenon observed in prior work on distributed computing in (i) the isotonic regression framework, where averaging several isotonic estimates (each computed at a local server) on a central server produces super-efficient estimates that do not replicate the properties of the global isotonic estimator, i.e. the isotonic estimate that would be constructed by transferring all the data to a single server, and (ii) certain types of M-estimation problems involving optimization of discontinuous criterion functions where M-estimates converge at the cube-root rate. The new estimators proposed in this paper work by smoothing the data on each local server, communicating the smoothed summaries to the central server, and then solving a non-linear optimization problem at the central server. They are shown to replicate the asymptotic properties of the corresponding global estimators, and also overcome the super-efficiency phenomenon exhibited by existing estimators.

Distribution-free properties of isotonic regression

It is well known that the isotonic least squares estimator is characterized as the derivative of the greatest convex minorant of a random walk. Provided the walk has exchangeable increments, we prove that the slopes of the greatest convex minorant are distributed as order statistics of the running averages. This result implies an exact non-asymptotic formula for the squared error risk of least squares in homoscedastic isotonic regression when the true sequence is constant that holds for every exchangeable error distribution.

Central limit theorems for the $L_{p}$-error of smooth isotonic estimators

We investigate the asymptotic behavior of the $L_{p}$-distance between a monotone function on a compact interval and a smooth estimator of this function. Our main result is a central limit theorem for the $L_{p}$-error of smooth isotonic estimators obtained by smoothing a Grenander-type estimator or isotonizing the ordinary kernel estimator. As a preliminary result we establish a similar result for ordinary kernel estimators. Our results are obtained in a general setting, which includes estimation of a monotone density, regression function and hazard rate. We also perform a simulation study for testing monotonicity on the basis of the $L_{2}$-distance between the kernel estimator and the smoothed Grenander-type estimator.

Limit Theory in Monotone Function Estimation

We give an overview of the different concepts and methods that are commonly used when studying the asymptotic properties of isotonic estimators. After introducing the inverse process, we illustrate its use in establishing weak convergence of the estimators at a fixed point and also weak convergence of global distances, such as the $\mathbb{L}_{p}$-distance and supremum distance. Furthermore, we discuss the developments on smooth isotonic estimation.

New types of shrinkage estimators of Poisson means under the normalized squared error loss

ABSTRACTIn estimating p( ⩾ 2) independent Poisson means, Clevenson and Zidek (1975) have proposed a class of estimators that shrink the unbiased estimator to the origin and dominate the unbiased one under the normalized squared error loss. This class of estimators was subsequently enlarged in several directions. This article deals with the problem and proposes new classes of dominating estimators using prior information pertinently. Dominance is shown by partitioning the sample space into disjoint subsets and averaging the loss difference over each subset. Estimation of several Poisson mean vectors is also discussed. Further, simultaneous estimation of Poisson means under order restriction is treated and estimators which dominate the isotonic regression estimator are proposed for some types of order restrictions.

The asymptotic distribution of the isotonic regression estimator over a general countable pre-ordered set

We study the isotonic regression estimator over a general countable pre-ordered set. We obtain the limiting distribution of the estimator and study its properties. It is proved that, under some general assumptions, the limiting distribution of the isotonized estimator is given by the concatenation of the separate isotonic regressions of the certain subvectors of an unrestrecred estimator’s asymptotic distribution. Also, we show that the isotonization preserves the rate of convergence of the underlying estimator. We apply these results to the problems of estimation of a bimonotone regression function and estimation of a bimonotone probability mass function.

Isotonized smooth estimators of a monotone baseline hazard in the Cox model

We consider two isotonic smooth estimators for a monotone baseline hazard in the Cox model, a maximum smooth likelihood estimator and a Grenander-type estimator based on the smoothed Breslow estimator for the cumulative baseline hazard. We show that they are both asymptotically normal at rate nm∕(2m+1), where m≥2 denotes the level of smoothness considered, and we relate their limit behavior to kernel smoothed isotonic estimators studied in Lopuhaä and Musta (2016). It turns out that the Grenander-type estimator is asymptotically equivalent to the kernel smoothed isotonic estimators, while the maximum smoothed likelihood estimator exhibits the same asymptotic variance but a different bias. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behavior of the two methods.

Successive comparisons between ordered normal means using isotonic regression estimators

ABSTRACTIn this study we discuss multiple comparison procedures for checking differences among a sequence of normal means with ordered restriction. Lee and Spurrier (1995) proposed a multiple comparison procedure which tests the difference between two adjacent means using the difference of sample means. In this study we propose a multiple comparison procedure modifying Lee and Spurrier's (1995) procedure using isotonic regression estimators instead of sample means. We determine the critical value for pairwise comparisons for a specified significance level. Furthermore, we formulate the power of the test. Finally, we give some numerical examples regarding critical values and power of the test intended to compare our procedure with Lee and Spurrier's (1995) procedure.

Carbetocin at Cesarean delivery for labour arrest: a sequential allocation trial to determine the effective dose.

The aim of this study was to estimate the effective dose 90% (ED90) of carbetocin to provide adequate uterine tone at Cesarean delivery (CD) for labour arrest. We conducted a double-blind dose-finding study of carbetocin using a biased-coin up-and-down design in women undergoing CD for labour arrest under epidural anesthesia. Forty healthy term pregnant women who had received at least three hours of oxytocin infusion during labour were recruited for the study. Carbetocin was administered intravenously upon delivery of the anterior shoulder of the fetus. The first patient received 20µg, and the dose for the subsequent patient was determined according to the response of the previous patient as per the biased-coin allocation scheme using increments or decrements of 20µg (maximum 140µg). Uterine tone was assessed by the obstetrician and rated as satisfactory or unsatisfactory throughout the intraoperative period. The primary outcome was satisfactory uterine tone with no need for additional uterotonic drugs intraoperatively. Secondary outcomes included use of additional uterotonic drugs postoperatively in the first 24hr, estimated blood loss, and adverse effects. The ED90 of carbetocin to produce adequate uterine tone was estimated at 121µg (95% confidence interval [CI]: 111 to 130; 99% CI: 108 to 133) using the truncated Dixon and Mood (DM) method. The isotonic estimator of ED90 was 140µg; however, the observed response rate across all doses was<90%. Also, the 95% CI of the DM estimator is likely to have lower than expected coverage, while the 99% CI may have about 90% coverage. Therefore, these results should be interpreted with caution. The overall median (range) estimated blood loss was 1,014 (104-2,436)mL. The overall incidence of hypotension and tachycardia were 45% and 57.5%, respectively. At a dose of 140µg, the incidence of tachycardia and intraoperative arrhythmias was 76% and 14%, respectively. The ED90 of carbetocin at CD for labour arrest, as determined in our study, should be interpreted with caution since it may be underestimated. This dose is higher than the currently recommended dose of 100µg at elective CD and should not be used routinely given the uncertainty regarding its efficacy and the high incidence of arrhythmias at higher doses. This trial was registered at ClinicalTrials.gov, number: NCT01725243.

On the Properties of Estimates of Monotonic Mean Vectors for Multivariate Normal Distributions

Problems concerning estimation of parameters and determination the statistic, when it is known a priori that some of these parameters are subject to certain order restrictions, are of considerable interest. In the present paper, we consider the estimators of the monotonic mean vectors for two dimensional normal distributions and compare those with the unrestricted maximum likelihood estimators under two different cases. One case is that covariance matrices are known, the other one is that covariance matrices are completely unknown and unequal. We show that when the covariance matrices are known, under the squared error loss function which is similar to the mahalanobis distance, the obtained multivariate isotonic regression estimators, motivated by estimators given in Robertson et al. (1988), which are the estimators given by Sasabuchi et al. (1983) and Sasabuchi et al. (1992), have the smaller risk than the unrestricted maximum likelihood estimators uniformly, but when the covariance matrices are unknown and unequal, the estimators have the smaller risk than the unrestricted maximum likelihood estimators only over some special sets which are defined on the covariance matrices. To illustrate the results two numerical examples are presented.

Testing monotonicity of a hazard: Asymptotic distribution theory

Two test statistics are introduced to test the null hypotheses that the sampling distribution has an increasing hazard rate on a specified interval $[0,a]$. These statistics are empirical $L_{1}$-type distances between the isotonic estimates, which use the monotonicity constraint, and either the empirical distribution function or the empirical cumulative hazard. They measure the excursions of the empirical estimates with respect to the isotonic estimates, owing to local non-monotonicity. Asymptotic normality of the test statistics, if the hazard is strictly increasing on $[0,a]$, is established under mild conditions. This is done by first approximating the global empirical distance by a distance with respect to the underlying distribution function. The resulting integral is treated as sum of increasingly many local integrals to which a central limit theorem can be applied. The behavior of the local integrals is determined by a canonical process, the difference between the stochastic process $x\mapsto W(x)+x^{2}$, where $W$ is standard two-sided Brownian motion, and its greatest convex minorant.

A Semiparametric Sequential Algorithm for Estimation of Dose–Response Curve

Sequential designs for binary regression models are used for efficient estimation of quantiles of interest of the dose–response curve given by a link function. In many applications the link function is assumed to be known up to a location and a scale. We perform an empirical study to illustrate how the quantile estimates may be affected when the link function is misspecified. We propose nonparametric estimation of the link function via an isotonic smoothing spline estimator and incorporate the estimator into the sequential allocation scheme. This makes the procedure more robust against link function misspecification and at the same time maintains the objective of efficient estimation of the parameters.

Semiparametric analysis of isotonic errors-in-variables regression models with randomly right censored response

This paper considers the estimation of a semiparametric isotonic regression model when the covariates are measured with additive errors and the response is randomly right censored by a censoring time. The authors show that the proposed estimator of the regression parameter is rootn consistent and asymptotically normal. The authors also show that the isotonic estimator of the functional component, at a fixed point, is cubic root-n consistent and converges in distribution to the slope at zero of the greatest convex minorant of the sum of a two-sided standard Brownian motion and the square of the time parameter. A simulation study is carried out to investigate the performance of the estimators proposed in this article.