• Bulletin of the Korean Mathematical Society
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• v.48 no.3
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• pp.523-537
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• 2011

A receiver operating characteristic (ROC) curve plots the true positive rate of a classier against its false positive rate, both of which are accuracy measures of the classier. The ROC curve has several interesting geometrical properties, including concavity which is a necessary condition for a classier to be optimal. In this paper, we study the nonparametric maximum likelihood estimator (NPMLE) of a concave ROC curve and its modification to reduce bias. We characterize the NPMLE as a solution to a geometric programming, a special type of a mathematical optimization problem. We find that the NPMLE is close to the convex hull of the empirical ROC curve and, thus, has smaller variance but positive bias at a given false positive rate. To reduce the bias, we propose a modification of the NPMLE which minimizes the $L_1$ distance from the empirical ROC curve. We numerically compare the finite sample performance of three estimators, the empirical ROC curve, the NMPLE, and the modified NPMLE. Finally, we apply the estimators to estimating the optimal ROC curve of the variance-threshold classier to segment a low depth of field image and to finding a diagnostic tool with multiple tests for detection of hemophilia A carrier.

• The Korean Journal of Applied Statistics
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• v.6 no.2
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• pp.341-353
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• 1993

As an estimator of the conditional probability of discovering a new species at the next observation after a sample of certain size is taken, the one proposed by Good(1953) has been most widely used. Recently, Clayton and Frees(1987) showed via simulation that their nonparametric maximum likelihood estimator(NPMLE) has smaller MSE than Good's estimator when the population is relatively nonuniform. Lee(1989) proved that their conjecture is asymptotically true for truncated geometric population distributions. One shortcoming of the NPMLE, however, is that it has a considerable amount of negative bias. In this study we proposed a bias-corrected version of the NPMLE for virtually all realistic population distributions. We also showed that it has a smaller asymptotic MSE than Good's extimator except when the population is very uniform. A Monte Carlo simulation was performed for small sample sizes, and the result supports the asymptotic results.

• 한국데이터정보과학회:학술대회논문집
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• 2002.06a
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• pp.51-73
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• 2002

Empirical likelihood ratio method is a new technique in nonparametric inference developed by A. Owen (1988, 2001). Sometimes empirical likelihood has difficulties to define itself. As such a case in point, we discuss the way to define a modified empirical likelihood for the location of symmetry using well-known points of symmetry as a side conditions. The side condition of symmetry is defined through a finite subset of the infinite set of constraints. The modified empirical likelihood under symmetry studied in this paper is to construct a constrained parameter space $\theta+$ of distributions imposing known symmetry as side information. We show that the usual asymptotic theory (Wilks theorem) still hold for the empirical likelihood ratio on the constrained parameter space and the asymptotic distribution of the empirical NPMLE of difference of two symmetric points is obtained.

• The Korean Journal of Applied Statistics
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• v.25 no.6
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• pp.947-958
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• 2012

When modeling event times in biomedical studies, the outcome might be incompletely observed. In this paper, we assume that the outcome is recorded as current status failure time data. Despite well-developed literature the routine practical use of many current status data modeling methods remains infrequent due to the lack of specialized statistical software, the difficulty to assess model goodness-of-fit, as well as the possible loss of information caused by covariate grouping or discretization. We propose a model based on pseudo-observations that is convenient to implement and that allows for flexibility in the choice of the outcome. Parameter estimates are obtained based on generalized estimating equations. Examples from studies in bile duct hyperplasia and breast cancer in conjunction with simulated data illustrate the practical advantages of this model.

• Communications for Statistical Applications and Methods
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• v.25 no.6
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• pp.633-645
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• 2018

Gaussian error distributions are a common choice in traditional regression models for the maximum likelihood (ML) method. However, this distributional assumption is often suspicious especially when the error distribution is skewed or has heavy tails. In both cases, the ML method under normality could break down or lose efficiency. In this paper, we consider the log-concave and Gaussian scale mixture distributions for error distributions. For the log-concave errors, we propose to use a smoothed maximum likelihood estimator for stable and faster computation. Based on this, we perform comparative simulation studies to see the performance of coefficient estimates under normal, Gaussian scale mixture, and log-concave errors. In addition, we also consider real data analysis using Stack loss plant data and Korean labor and income panel data.

• Korea journal of population studies
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• v.26 no.1
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• pp.31-57
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• 2003

This study is intended to examine (l)whether the value of son-for example, old age security and succession of family lineage- causing son preference in the traditional society can be explained at the individual level, (2)whether women without son in the son preference country continue her childbearing until having at least one son or give up the desire of having a son at a certain level. To accomplish these purposes, the 1974 Korean National Fertility Survey data are analyzed by the quadratic hazard models controlling unobserved heterogeneity. Unlike ordinary regression model, even omitted variables that affect hazard rates and are uncorrelated with the included independent variables can distort the parameter estimates in the hazard model. Therefore the nonparametric maximum likelihood estimator(NPMLE) of a mixing distribution developed by Heckman and Singer is used to control unobserved heterogeneity. Based on the statistical result in this study, the value of son causing son preference is determined at the societal level, not at the individual level. And Korean women without a son did not continue endlessly childbearing during child bearing ages until having a son. In general, they gave up the desire having a son when she had born six daughters continuously. Thus, 30-40 years ago, the number of daughters that women without a son giving up the desire of son was six, which is about the level of total fertility rate during 1960s. In these days, we can often see many women who have only two or three daughters and do not any son. This means that the level of giving up the desire of son, which is one factor representing the strength of son preference, becomes lower. If the strength of son preference did not become much weaker, then the fertility rates in Korea could not reach the below replacement level.