Nonparametric Maximum Likelihood Estimation Based on Ranked Set Samples

Consider independent observations $$(X_i,R_i)$$ with random or fixed ranks $$R_i$$, while conditional on $$R_i$$, the random variable $$X_i$$ has the same distribution as the $$R_i$$-th order statistic within a random sample of size k from an unknown distribution function F. Such observation schemes are well known from ranked set sampling and judgment post-stratification. Within a general, not necessarily balanced setting we derive and compare the asymptotic distributions of three different estimators of the distribution function F: a stratified estimator, a nonparametric maximum-likelihood estimator and a moment-based estimator. Our functional central limit theorems generalize and refine previous asymptotic analyses. In addition, we discuss briefly pointwise and simultaneous confidence intervals for the distribution function with guaranteed coverage probability for finite sample sizes. The methods are illustrated with a real data example, and the potential impact of imperfect rankings is investigated in a small simulation experiment.

The goal of the present work is to estimate the nonlinear correlation between two random variables when the sample is drawn from a Farlie-Gumbel-Morgenstern (FGM)-type bivariate gamma distribution.In the context of estimating the dependence parameter, a maximum likelihood (ML) methodology is used.Thus, the present work offers ML estimators based on simple random sampling (SRS) and ranked set sampling (RSS).Additionally, we consider generalized modified RSS (GMRSS), which only requires a single rank to obtain a sample.Using GMRSS, we aim to observe the effect of the rth order statistic and its concomitant on the ML estimator.According to the Monte Carlo simulation, it is clearly seen that RSS provides an ML estimator as efficient as the ML estimator based on SRS.On the other hand, it appears that the ML estimator based on GMRSS (with minimum or maximum ranked pairs) is the best option among the studied ML estimators.Moreover, these findings are made even more meaningful by the fact that GMRSS is easier to obtain than SRS and RSS.

We develop the non-parametric maximum likelihood estimator (MLE) of the full Mbh capture-recapture model which utilizes both initial capture and recapture data and permits both heterogeneity (h) between animals and behavioural (b) response to capture. Our MLE procedure utilizes non-parametric maximum likelihood estimation of mixture distributions (Lindsay, 1983; Lindsay and Roeder, 1992) and the EM algorithm (Dempsteret al., 1977). Our MLE estimate provides the first non-parametric estimate of the bivariate capture-recapture distribution. Since non-parametric maximum likelihood estimation exists for submodels Mh (allowing heterogeneity only), Mb (allowing behavioural response only) and M0 (allowing no changes), we develop maximum likelihood-based model selection, specifically the Akaike information criterion (AIC) (Akaike, 1973). The AIC procedure does well in detecting behavioural response but has difficulty in detecting heterogeneity.

Maximum likelihood estimation of dependence parameter using ranked set sampling

The ranked set sampling (RSS) technique has been shown to be superior to classical simple random sampling (SRS) in the sense that it always provides a more precise estimator of the population mean. However, it is quite often that some measurements are below the limit of detection and hence become censored. In such situations, the superiority of RSS over SRS may no longer be held. In this article we consider the problem of estimating the population mean and standard deviation based on a ranked set sample with some data being censored. Maximum likelihood estimators are proposed when the data are assumed to follow a lognormal distribution. In the case where the distribution is unknown, a variant of the Kaplan–Meier estimator is proposed in the estimation of the population mean. A simulation study is conducted to compare the performance of the proposed RSS estimators with the corresponding SRS estimators. The impact of imperfect judgment ranking is also discussed. The proposed methods are applied to a real data set on mercury concentration in swordfish. Copyright © 2002 John Wiley & Sons, Ltd.

In statistical parameter estimation problems, how well the parameters are estimated largely depends on the sampling design used. In this article, a modified best linear unbiased estimator of the shape parameter β from log-logistic distribution is considered when scale parameter α is known and when α is unknown under simple random sampling (SRS) and ranked set sampling (RSS). In addition, a modified BLUE of β, when α is known using an RSS version based on the order statistic that maximizes the Fisher information for a fixed set size, will be considered. Theoretical properties of the suggested estimators are compared with its counterpart estimators under SRS. It is found that these estimators under RSS can be real competitors against those under SRS.

The events of interest in any survival analysis study are regularly subject to censoring. There are various censoring schemes, including right or left censoring, and interval censoring. The most frequent censoring scheme is the right censoring, where subjects might drop out of the study or simply because not all events of interest occur before the end of the study. Moreover, for each subject, additional information referred to as covariates is registered at the beginning or throughout the study, such as age, sex, undergoing treatment, etc. The classical model to study the distribution of the events of interest, while accounting for additional information, is the Cox model. The Cox model expresses the hazard function of a subject given a set of covariates in terms of a baseline hazard, for which all covariates are zero, and an exponential function of the covariates and corresponding regression parameters. The baseline hazard can be left completely unspecified while estimating the regression parameters. Nonetheless, in practice, there are numerous studies in which the baseline hazard appears to be monotone. Time to death or to the onset of a disease are observed to have a nondecreasing baseline hazard, while the survival or recovery time after a successful medical treatment usually exhibit a nonincreasing baseline hazard. The aim of this thesis is to study the behavior of nonparametric baseline hazard and baseline density estimators in the Cox model under monotonicity constraints. The event times are assumed to be right censored and the censoring mechanism is assumed to be independent of the event of interest and non-informative. The covariates are assumed to be time-independent, usually recorded at the beginning of the study. In addition to point estimates, interval estimates of a monotone baseline hazard will be provided, based on a likelihood ratio method, along with testing at a fixed point. Furthermore, kernel smoothed estimates of a monotone baseline hazard will be defined and their behavior will be investigated. In Chapter 2, we propose several nonparametric monotone estimators of a baseline hazard or a baseline density within the Cox model. We derive the nonparametric maximum likelihood estimator of a nondecreasing baseline hazard and we consider a Grenander-type estimator, defined as the left-hand slope of the greatest convex minorant of the Breslow estimator. The two estimators are then shown to be strongly consistent and asymptotically equivalent. Moreover, we derive their common limit distribution at a fixed point. The two equivalent estimators of a nonincreasing baseline hazard and their asymptotic properties are acquired similarly. Furthermore, we introduce a Grenander-type estimator of a nonincreasing baseline density, defined as the left-hand slope of the least concave majorant of an estimator of the baseline cumulative distribution function derived from the Breslow estimator. This estimator is proven to be strongly consistent and its asymptotic distribution at a fixed point is derived. Chapter 3 provides an asymptotic linear representation of the Breslow estimator of the baseline cumulative hazard function in the Cox model. This representation can be used to derive the asymptotic distribution of the Grenander type estimator of a monotone baseline hazard estimator. The representation consists of an average of independent random variables and a term involving the difference between the maximum partial likelihood estimator and the underlying regression parameter. The order of the remainder term is arbitrarily close to n^-1. Chapter 4 focuses on interval estimation and on testing whether a monotone baseline hazard function in the Cox model has a particular value at a fixed point, via a likelihood ratio method. Nonparametric maximum likelihood estimators under the null hypothesis are defined for both nondecreasing and nonincreasing baseline hazard functions. These characterizations, along with those of the monotone nonparametric maximum likelihood estimators provide the asymptotic distribution of the likelihood ratio test. This asymptotic distribution enables, via inversion, the construction of pointwise confidence intervals. This method of constructing confidence intervals avoids the issue of estimating the nuisance parameters, as in the case of confidence intervals based on the asymptotic distribution of the estimators. Simulations indicate that the two methods yield confidence intervals with comparable coverage probabilities. Nonetheless, the confidence intervals based on the likelihood ratio are smaller, on average. Finally, in chapter 5 we consider smooth baseline hazard estimators. The estimators are obtained by kernel smoothing the maximum likelihood and Grenander-type estimators of a monotone baseline hazard function. Three different estimators are proposed for a nondecreasing baseline hazard, which are provided by the interchange of the smoothing and isotonization step. With this respect, we define a smoothed maximum likelihood estimator (SMLE), as well as a smoothed Grenander type (SG) estimator and a Grenander type smoothed (GS) estimator. All estimators are shown to be strongly pointwise or uniformly consistent.

There are many situations where it is costly to measure the variable under the study, but this variable can be easily ranked. Ranked set sampling (RSS) utilizes the information of the ranks in addition to the measurements. It was proved that RSS is more efficient than simple random sampling (SRS) when estimating the population arithmetic mean for the same sample size. Because of its efficiency, RSS was employed to a wide range of applications and becoming an interesting method for many researchers. It was studied parametrically and non-parametrically and several estimators based on different statistical distributions were investigated. This research adopts Rician distribution and several measurements from this distribution were investigated based on RSS. The estimations include the arithmetic, geometric, harmonic, quadratic means as well as the median, variance, mean deviation, coefficient of variation, coefficient of skewness and kurtosis. Simulation study shows that the estimators based on RSS are more efficient than those based on SRS. However, the gain in the efficiency with variance and coefficient of kurtosis is not too much and even worse for small sample size.

Ranked Set Sampling and Ranked Set Sampling concomitant are more efficient than Simple Random Sampling. This can be determined by calculating the Relative Precision which is a ratio value from the variance of the mean from each sampling technique. From the research of Ranked Set Sampling, obtained , and so Ranked Set Sampling is more efficient than Simple Random Sampling. For the research of Ranked Set Sampling concomitant, obtained , and so Ranked Set Sampling concomitant is more efficient than Simple Random Sampling, and for simple linear regression analysis obtained , , , so simple linear regression model of Ranked Set Sampling is more efficient than simple linear regression model of Simple Random Sampling

This article studies the properties of the maximum likelihood estimator of the population proportion in ranked set sampling with extreme ranks. The maximum likelihood estimator is described and its asymptotic distribution is derived. Finite sample size properties of the estimator are investigated using simulation studies. It turns out that the proposed estimator is substantially more efficient than its simple random sampling and ranked set sampling analogs, as the true population proportion tends to zero/unity. The method is illustrated using data from the National Health and Nutrition Examination Survey.

The maximum livelihood estimator (MLE) using a ranked set sample (RSS) usually has no closed expression because the maximum likelihood equation involves both hazard and inverse hazard functions, and may no longer be efficient when the judgment ranking is imperfect. In this paper, we consider a modified MLE (MMLE) using RSS for general parameters, which has the same expression as the MLE using a simple random sample (SRS), except that the SRS in the MLE is replaced by the RSS. The results show that, for the location parameter, the MMLE is always more efficient than the MLE using SRS, and for the scale parameter, the MMLE is at least as efficient as the MLE using SRS, when the same sample size is used. Under the perfect judgment ranking, numerical examples also show that the MMLE has good efficiency relative to the MLE based on RSS. When the judgment error is present, we conduct simulations to show that the MMLE is more robust than the MLE using RSS.

ABSTRACTRecently, a hybrid ranked set sampling (HRSS) scheme has been proposed in the literature. The HRSS scheme encompasses several existing ranked set sampling (RSS) schemes, and it is a cost-effective alternative to the classical RSS and double RSS schemes. In this paper, we propose an improved estimator for estimating the cumulative distribution function (CDF) using HRSS. It is shown, both theoretically and numerically, that the CDF estimator under HRSS scheme is unbiased and its variance is always less than the variance of the CDF estimator with simple random sampling (SRS). An unbiased estimator of the variance of CDF estimator using HRSS is also derived. Using Monte Carlo simulations, we also study the performances of the proposed and existing CDF estimators under both perfect and imperfect rankings. It turns out that the proposed CDF estimator is by far a superior alternative to the existing CDF estimators with SRS, RSS and L-RSS schemes. For a practical application, a real data set is considered on the bilirubin level of babies in neonatal intensive care.

In this paper, a new modification of ranked set sampling (RSS) is suggested, namely; unified ranked set sampling (URSS) for estimating the population mean and variance. The performance of the empirical mean and variance estimators based on URSS are compared with their counterparts in ranked set sampling and simple random sampling (SRS) via Monte Carlo simulation. Simulation results indicate that the URSS estimators perform better than their counterparts using RSS and SRS designs when the ranking is perfect. When the ranking is imperfect, the URSS estimators still are superior than their counterparts in ranked set sampling and simple random sampling methods. Finally, an illustrative example is provided to show the efficiency of the new method in practice.

This study is focused on ranked set sampling theory which is constructive and cost efficient. Estimation for population mean for different sample sizes based on new ranked set sampling (NRSS) scheme is explored. A simulation study has been conducted and a comparison has been made with simple random sampling (SRS), ranked set sampling (RSS), extreme ranked set sampling (ERSS) and quartile ranked set sampling (QRSS) for normal and non – normal distributions.

Parametric estimation for the scale parameter for scale distributions using moving extremes ranked set sampling