Small sample properties of the maximum likelihood estimators for an alternative parameterization of the three-parameter lognormal distribution

1987 ◽  
Vol 16 (3) ◽  
pp. 871-884 ◽  
Author(s):  
Jerome F. Eastham ◽  
Vincent N. LaRiccia ◽  
John H. Schenemeyer
Keyword(s):  
Maximum Likelihood ◽  
Lognormal Distribution ◽  
Maximum Likelihood Estimators ◽  
Small Sample ◽  
Small Sample Properties

scholarly journals Estimating logit models with small samples

2021 ◽  
pp. 1-16
Author(s):  
Carlisle Rainey ◽  
Kelly McCaskey
Keyword(s):  
Maximum Likelihood ◽  
American Political Science Review ◽  
Substantial Improvement ◽  
Small Sample ◽  
Small Samples ◽  
Noticeable Improvement ◽  
Small Sample Properties ◽  
Substantial Bias ◽  
Ml Estimator ◽  
American Political Science

Abstract In small samples, maximum likelihood (ML) estimates of logit model coefficients have substantial bias away from zero. As a solution, we remind political scientists of Firth's (1993, Biometrika, 80, 27–38) penalized maximum likelihood (PML) estimator. Prior research has described and used PML, especially in the context of separation, but its small sample properties remain under-appreciated. The PML estimator eliminates most of the bias and, perhaps more importantly, greatly reduces the variance of the usual ML estimator. Thus, researchers do not face a bias-variance tradeoff when choosing between the ML and PML estimators—the PML estimator has a smaller bias and a smaller variance. We use Monte Carlo simulations and a re-analysis of George and Epstein (1992, American Political Science Review, 86, 323–337) to show that the PML estimator offers a substantial improvement in small samples (e.g., 50 observations) and noticeable improvement even in larger samples (e.g., 1000 observations).

scholarly journals On the Existence and Uniqueness of the Maximum Likelihood Estimators of Normal and Lognormal Population Parameters with Grouped Data

2009 ◽  
Vol 2009 ◽  
pp. 1-16 ◽  
Author(s):  
Jin Xia ◽  
Jie Mi ◽  
YanYan Zhou
Keyword(s):  
Maximum Likelihood ◽  
Lognormal Distribution ◽  
Existence And Uniqueness ◽  
Maximum Likelihood Estimators ◽  
Invariance Property ◽  
Type I ◽  
Grouped Data ◽  
Sample Data ◽  
Censored Sample ◽  
Two Parameters

Lognormal distribution has abundant applications in various fields. In literature, most inferences on the two parameters of the lognormal distribution are based on Type-I censored sample data. However, exact measurements are not always attainable especially when the observation is below or above the detection limits, and only the numbers of measurements falling into predetermined intervals can be recorded instead. This is the so-called grouped data. In this paper, we will show the existence and uniqueness of the maximum likelihood estimators of the two parameters of the underlying lognormal distribution with Type-I censored data and grouped data. The proof was first established under the case of normal distribution and extended to the lognormal distribution through invariance property. The results are applied to estimate the median and mean of the lognormal population.

EDA for HLM: Visualization when Probabilistic Inference Fails

2005 ◽  
Vol 13 (4) ◽  
pp. 301-326 ◽  
Author(s):  
Jake Bowers ◽  
Katherine W. Drake
Keyword(s):  
Maximum Likelihood ◽  
Political Science ◽  
Probabilistic Inference ◽  
Asymptotic Properties ◽  
Scientific Progress ◽  
Maximum Likelihood Estimators ◽  
Small Sample ◽  
Good Judgment ◽  
Repeated Sampling ◽  
Level 2

Nearly all hierarchical linear models presented to political science audiences are estimated using maximum likelihood under a repeated sampling interpretation of the results of hypothesis tests. Maximum likelihood estimators have excellent asymptotic properties but less than ideal small sample properties. Multilevel models common in political science have relatively large samples of units like individuals nested within relatively small samples of units like countries. Often these level-2 samples will be so small as to make inference about level-2 effects uninterpretable in the likelihood framework from which they were estimated. When analysts do not have enough data to make a compelling argument for repeated sampling based probabilistic inference, we show how visualization can be a useful way of allowing scientific progress to continue despite lack of fit between research design and asymptotic properties of maximum likelihood estimators.Somewhere along the line in the teaching of statistics in the social sciences, the importance of good judgment got lost amid the minutiae of null hypothesis testing. It is all right, indeed essential, to argue flexibly and in detail for a particular case when you use statistics. Data analysis should not be pointlessly formal. It should make an interesting claim; it should tell a story that an informed audience will care about, and it should do so by intelligent interpretation of appropriate evidence from empirical measurements or observations.—Abelson, 1995, p. 2With neither prior mathematical theory nor intensive prior investigation of the data, throwing half a dozen or more exogenous variables into a regression, probit, or novel maximum-likelihood estimator is pointless. No one knows how they are interrelated, and the high-dimensional parameter space will generate a shimmering pseudo-fit like a bright coat of paint on a boat's rotting hull.—Achen, 1999, p. 26

Sign in / Sign up

Export Citation Format

Share Document