scholarly journals Asymptotics of Maximum Likelihood without the LLN or CLT or Sample Size Going to Infinity

2013 ◽  
pp. 1-24 ◽  
Author(s):  
Charles J. Geyer
Keyword(s):  
Maximum Likelihood ◽  
Sample Size

A comparative review of methods for comparing means using partially paired data

2015 ◽  
Vol 26 (3) ◽  
pp. 1323-1340 ◽  
Author(s):  
Beibei Guo ◽  
Ying Yuan
Keyword(s):  
Missing Data ◽  
Maximum Likelihood ◽  
Sample Size ◽  
T Test ◽  
Type I ◽  
Paired Data ◽  
Modified Maximum Likelihood ◽  
Comparative Review ◽  
Paired T Test ◽  
Normally Distributed

In medical experiments with the objective of testing the equality of two means, data are often partially paired by design or because of missing data. The partially paired data represent a combination of paired and unpaired observations. In this article, we review and compare nine methods for analyzing partially paired data, including the two-sample t-test, paired t-test, corrected z-test, weighted t-test, pooled t-test, optimal pooled t-test, multiple imputation method, mixed model approach, and the test based on a modified maximum likelihood estimate. We compare the performance of these methods through extensive simulation studies that cover a wide range of scenarios with different effect sizes, sample sizes, and correlations between the paired variables, as well as true underlying distributions. The simulation results suggest that when the sample size is moderate, the test based on the modified maximum likelihood estimator is generally superior to the other approaches when the data is normally distributed and the optimal pooled t-test performs the best when the data is not normally distributed, with well-controlled type I error rates and high statistical power; when the sample size is small, the optimal pooled t-test is to be recommended when both variables have missing data and the paired t-test is to be recommended when only one variable has missing data.

Finite Sample Properties of Several Predictors From an Autoregressive Model

1987 ◽  
Vol 3 (3) ◽  
pp. 359-370 ◽  
Author(s):  
Koichi Maekawa
Keyword(s):  
Maximum Likelihood ◽  
Sample Size ◽  
Asymptotic Expansions ◽  
Autoregressive Model ◽  
Mean Squared Error ◽  
The Other ◽  
Finite Sample ◽  
Squared Error ◽  
Distributional Properties ◽  
Finite Sample Properties

We compare the distributional properties of the four predictors commonly used in practice. They are based on the maximum likelihood, two types of the least squared, and the Yule-Walker estimators. The asymptotic expansions of the distribution, bias, and mean-squared error for the four predictors are derived up to O(T−1), where T is the sample size. Examining the formulas of the asymptotic expansions, we find that except for the Yule-Walker type predictor, the other three predictors have the same distributional properties up to O(T−1).

scholarly journals Two-Stage Adaptive Optimal Design with Fixed First-Stage Sample Size

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Adam Lane ◽  
Nancy Flournoy
Keyword(s):  
Maximum Likelihood ◽  
Optimal Design ◽  
Sample Size ◽  
Asymptotic Distribution ◽  
Fixed Number ◽  
Maximum Likelihood Estimates ◽  
Small Pilot Study ◽  
Distribution Of The Maximum ◽  
Optimal Procedures ◽  
Mean Function

In adaptive optimal procedures, the design at each stage is an estimate of the optimal design based on all previous data. Asymptotics for regular models with fixed number of stages are straightforward if one assumes the sample size of each stage goes to infinity with the overall sample size. However, it is not uncommon for a small pilot study of fixed size to be followed by a much larger experiment. We study the large sample behavior of such studies. For simplicity, we assume a nonlinear regression model with normal errors. We show that the distribution of the maximum likelihood estimates converges to a scale mixture family of normal random variables. Then, for a one parameter exponential mean function we derive the asymptotic distribution of the maximum likelihood estimate explicitly and present a simulation to compare the characteristics of this asymptotic distribution with some commonly used alternatives.

OPTIMAL INVARIANT INFERENCE WHEN THE NUMBER OF INSTRUMENTS IS LARGE

2009 ◽  
Vol 25 (3) ◽  
pp. 793-805 ◽  
Author(s):  
Laura Chioda ◽  
Michael Jansson
Keyword(s):  
Asymptotic Behavior ◽  
Maximum Likelihood ◽  
Sample Size ◽  
Maximum Likelihood Estimator ◽  
Instrumental Variables ◽  
Asymptotic Efficiency ◽  
Rotation Invariance ◽  
Limited Information ◽  
Likelihood Estimator ◽  
Rotation Invariant

This paper studies the asymptotic behavior of a Gaussian linear instrumental variables model in which the number of instruments diverges with the sample size. Asymptotic efficiency bounds are obtained for rotation invariant inference procedures and are shown to be attainable by procedures based on the limited information maximum likelihood estimator. The bounds are obtained by characterizing the limiting experiment associated with the model induced by the rotation invariance restriction.

scholarly journals Weighted Maximum Likelihood Correlation Coefficient to Handle Missing Values and Outliers in Data Set

2021 ◽  
Vol 20 ◽  
pp. 415-430
Author(s):  
Juthaphorn Sinsomboonthong ◽  
Saichon Sinsomboonthong
Keyword(s):  
Missing Data ◽  
Maximum Likelihood ◽  
Sample Size ◽  
Correlation Coefficient ◽  
Missing Values ◽  
Mean Squared Error ◽  
Correlation Coefficients ◽  
Median Percentage ◽  
Data Set ◽  
Median Correlation

The proposed estimator, namely weighted maximum likelihood (WML) correlation coefficient, for measuring the relationship between two variables to concern about missing values and outliers in the dataset is presented. This estimator is proven by applying the conditional probability function to take care of some missing values and pay more attention to values near the center. However, outliers in the dataset are assigned a slight weight. These using techniques will give the robust proposed method when the preliminary assumptions are not met data analysis. To inspect about the quality of the proposed estimator, the six methods—WML, Pearson, median, percentage bend, biweight mid, and composite correlation coefficients—are compared the properties in two criteria, i.e. the bias and mean squared error, via the simulation study. The results of generated data are illustrated that the WML estimator seems to have the best performance to withstand the missing values and outliers in dataset, especially for the tiny sample size and large percentage of outliers regardless of missing data levels. However, for the massive sample size, the median correlation coefficient seems to have the good estimator when linear relationship levels between two variables are approximately over 0.4 irrespective of outliers and missing data levels

scholarly journals A Comparative Study of the Bias Correction Methods for Differential Item Functioning Analysis in Logistic Regression with Rare Events Data

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Marjan Faghih ◽  
Zahra Bagheri ◽  
Dejan Stevanovic ◽  
Seyyed Mohhamad Taghi Ayatollahi ◽  
Peyman Jafari
Keyword(s):  
Logistic Regression ◽  
Maximum Likelihood ◽  
Sample Size ◽  
Differential Item Functioning ◽  
Bias Correction ◽  
Rare Events ◽  
Estimation Methods ◽  
Type I ◽  
Item Functioning ◽  
Events Data

The logistic regression (LR) model for assessing differential item functioning (DIF) is highly dependent on the asymptotic sampling distributions. However, for rare events data, the maximum likelihood estimation method may be biased and the asymptotic distributions may not be reliable. In this study, the performance of the regular maximum likelihood (ML) estimation is compared with two bias correction methods including weighted logistic regression (WLR) and Firth's penalized maximum likelihood (PML) to assess DIF for imbalanced or rare events data. The power and type I error rate of the LR model for detecting DIF were investigated under different combinations of sample size, moderate and severe magnitudes of uniform DIF (DIF = 0.4 and 0.8), sample size ratio, number of items, and the imbalanced degree (τ). Indeed, as compared with WLR and for severe imbalanced degree (τ = 0.069), there were reductions of approximately 30% and 24% under DIF = 0.4 and 27% and 23% under DIF = 0.8 in the power of the PML and ML, respectively. The present study revealed that the WLR outperforms both the ML and PML estimation methods when logistic regression is used to evaluate DIF for imbalanced or rare events data.

scholarly journals Skewness of Maximum Likelihood Estimators in the Weibull Censored Data

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1351 ◽  
Author(s):  
Tiago M. Magalhães ◽  
Diego I. Gallardo ◽  
Héctor W. Gómez
Keyword(s):  
Monte Carlo ◽  
Maximum Likelihood ◽  
Sample Size ◽  
Censored Data ◽  
Maximum Likelihood Estimators ◽  
Monte Carlo Study ◽  
Parameter Distribution ◽  
Regression Parameter ◽  
Skewness Coefficient ◽  
Matrix Formula

In this paper, we obtain a matrix formula of order n − 1 / 2 , where n is the sample size, for the skewness coefficient of the distribution of the maximum likelihood estimators in the Weibull censored data. The present result is a nice approach to verify if the assumption of the normality of the regression parameter distribution is satisfied. Also, the expression derived is simple, as one only has to define a few matrices. We conduct an extensive Monte Carlo study to illustrate the behavior of the skewness coefficient and we apply it in two real datasets.

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