Estimating the Mean and Variance of Normal Populations from Singly Truncated and Doubly Truncated Samples

Cohen, Arthur

doi:10.1214/aoms/1177729751

Cited by 216 publications

(114 citation statements)

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“…Normal Case Gupta (1952) and Cohen (1950) independently examined the situation of a normally sampled population censored above some value. From this censored sample, Gupta developed a likelihood function and subsequently, MLE's for the mean, μ, and standard deviation, σ. Utilizing this method for a biomarker censored below a fixed d, the log likelihood function for the normally distributed non-diseased population is found to be (Gupta, 1952):…”

Section: Appendixmentioning

confidence: 99%

Youden Index and Optimal Cut‐Point Estimated from Observations Affected by a Lower Limit of Detection

et al. 2008

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SummaryThe receiver operating characteristic (ROC) curve is used to evaluate a biomarker's ability for classifying disease status. The Youden Index (J), the maximum potential effectiveness of a biomarker, is a common summary measure of the ROC curve. In biomarker development, levels may be unquantifiable below a limit of detection (LOD) and missing from the overall dataset. Disregarding these observations may negatively bias the ROC curve and thus J. Several correction methods have been suggested for mean estimation and testing; however, little has been written about the ROC curve or its summary measures. We adapt non-parametric (empirical) and semi-parametric (ROC-GLM [generalized linear model]) methods and propose parametric methods (maximum likelihood (ML)) to estimate J and the optimal cut-point (c*) for a biomarker affected by a LOD. We develop unbiased estimators of J and c* via ML for normally and gamma distributed biomarkers. Alpha level confidence intervals are proposed using delta and bootstrap methods for the ML, semi-parametric, and non-parametric approaches respectively. Simulation studies are conducted over a range of distributional scenarios and sample sizes evaluating estimators' bias, root-mean square error, and coverage probability; the average bias was less than one percent for ML and GLM methods across scenarios and decreases with increased sample size. An example using polychlorinated biphenyl levels to classify women with and without endometriosis illustrates the potential benefits of these methods. We address the limitations and usefulness of each method in order to give researchers guidance in constructing appropriate estimates of biomarkers' true discriminating capabilities.

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Section: Appendixmentioning

confidence: 99%

Youden Index and Optimal Cut‐Point Estimated from Observations Affected by a Lower Limit of Detection

et al. 2008

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“…The maximum likelihood estimation for singly truncated and doubly truncated normal distribution was considered by Cohen (1950Cohen ( , 1991. Numerical solutions to the estimators of the mean and variance for singly truncated samples were computed with an auxiliar function which is tabulated in Cohen (1961).…”

Section: Introductionmentioning

confidence: 99%

The singly truncated normal distribution: A non-steep exponential family

Castillo

1994

Ann Inst Stat Math

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Abstract. This paper is concerned with the maximum likelihood estimation problem for the singly truncated normal family of distributions. Necessary and suficient conditions, in terms of the coefficient of variation, are provided in order to obtain a solution to the likelihood equations. Furthermore, the maximum likelihood estimator is obtained as a limit case when the likelihood equation has no solution.

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“…The sub-families defined by  = 0,  > 0 and  < 0 correspond, respectively, to the Gumbel, Frechet, and Weibull families [45]. Maximum Likelihood Estimates (MLE) [46,47] is applied to estimate GEV parameters for a given sample. For visual comparison, a sample estimate of probability, empFi, is obtained with the plotting position of Cunnane [48]:…”

Section: Flood Hazard Mappingmentioning

confidence: 99%

Flood Hazard Mapping Combining Hydrodynamic Modeling and Multi Annual Remote Sensing data

et al. 2015

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This paper explores a method to combine the time and space continuity of a large-scale inundation model with discontinuous satellite microwave observations, for high-resolution flood hazard mapping. The assumption behind this approach is that hydraulic variables computed from continuous spatially-distributed hydrodynamic modeling and observed as discrete satellite-derived flood extents are correlated in time, so that probabilities can be transferred from the model series to the observations. A prerequisite is, therefore, the existence of a significant correlation between a modeled variable (i.e., flood extent or volume) and the synchronously-observed flood extent. If this is the case, the availability of model simulations over a long time period allows for a robust estimate of non-exceedance probabilities that can be attributed to corresponding synchronously-available satellite observations. The generated flood hazard map has a spatial resolution equal to that of the satellite images, which is higher than that of currently available large scale inundation models. The method was applied on the Severn River (UK), using the outputs of a global inundation model provided by the European Centre for Medium-range Weather Forecasts OPEN ACCESSRemote Sens. 2015, 7 14201and a large collection of ENVISAT ASAR imagery. A comparison between the hazard map obtained with the proposed method and with a more traditional numerical modeling approach supports the hypothesis that combining model results and satellite observations could provide advantages for high-resolution flood hazard mapping, provided that a sufficient number of remote sensing images is available and that a time correlation is present between variables derived from a global model and obtained from satellite observations.

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Estimating the Mean and Variance of Normal Populations from Singly Truncated and Doubly Truncated Samples

Cited by 216 publications

References 2 publications

Youden Index and Optimal Cut‐Point Estimated from Observations Affected by a Lower Limit of Detection

Youden Index and Optimal Cut‐Point Estimated from Observations Affected by a Lower Limit of Detection

The singly truncated normal distribution: A non-steep exponential family

Flood Hazard Mapping Combining Hydrodynamic Modeling and Multi Annual Remote Sensing data

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