Hierarchical Maximum Likelihood Parameter Estimation for Cumulative Prospect Theory: Improving the Reliability of Individual Risk Parameter Estimates

2014 ◽  
Author(s):  
Ryan O. Murphy ◽  
Robert H.W. ten Brincke
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Prospect Theory ◽  
Cumulative Prospect Theory ◽  
Individual Risk ◽  
Parameter Estimates ◽  
Risk Parameter

scholarly journals Conditional Maximum Likelihood Estimation in Probability-Branched Multistage Designs

2021 ◽  
Author(s):  
Jan Steinfeld ◽  
Alexander Robitzsch
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Likelihood Estimation ◽  
Parameter Estimate ◽  
Item Parameter ◽  
Parameter Estimates ◽  
Conditional Maximum Likelihood ◽  
Item Parameter Estimates ◽  
Item Parameter Estimation ◽  
Conditional Maximum

This article describes the conditional maximum likelihood-based item parameter estimation in probabilistic multistage designs. In probabilistic multistage designs, the routing is not solely based on a raw score j and a cut score c as well as a rule for routing into a module such as j < c or j ≤ c but is based on a probability p(j) for each raw score j. It can be shown that the use of a conventional conditional maximum likelihood parameter estimate in multistage designs leads to severely biased item parameter estimates. Zwitser and Maris (2013) were able to show that with deterministic routing, the integration of the design into the item parameter estimation leads to unbiased estimates. This article extends this approach to probabilistic routing and, at the same time, represents a generalization. In a simulation study, it is shown that the item parameter estimation in probabilistic designs leads to unbiased item parameter estimates.

scholarly journals Item Parameter Estimation in Multistage Designs: A Comparison of Different Estimation Approaches for the Rasch Model

Psych ◽  
2021 ◽  
Vol 3 (3) ◽  
pp. 279-307
Author(s):  
Jan Steinfeld ◽  
Alexander Robitzsch
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Item Parameter ◽  
Parameter Estimates ◽  
Trait Distribution ◽  
Item Parameter Estimates ◽  
Item Parameter Estimation ◽  
Linear Smoothing ◽  
Log Linear ◽  
Conditional Maximum

There is some debate in the psychometric literature about item parameter estimation in multistage designs. It is occasionally argued that the conditional maximum likelihood (CML) method is superior to the marginal maximum likelihood method (MML) because no assumptions have to be made about the trait distribution. However, CML estimation in its original formulation leads to biased item parameter estimates. Zwitser and Maris (2015, Psychometrika) proposed a modified conditional maximum likelihood estimation method for multistage designs that provides practically unbiased item parameter estimates. In this article, the differences between different estimation approaches for multistage designs were investigated in a simulation study. Four different estimation conditions (CML, CML estimation with the consideration of the respective MST design, MML with the assumption of a normal distribution, and MML with log-linear smoothing) were examined using a simulation study, considering different multistage designs, number of items, sample size, and trait distributions. The results showed that in the case of the substantial violation of the normal distribution, the CML method seemed to be preferable to MML estimation employing a misspecified normal trait distribution, especially if the number of items and sample size increased. However, MML estimation using log-linear smoothing lea to results that were very similar to the CML method with the consideration of the respective MST design.

Skills for Fitting Models to Data

2021 ◽  
pp. 263-280
Author(s):  
Timothy E. Essington
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Numerical Methods ◽  
Maximum Likelihood Estimation ◽  
Direct Method ◽  
Likelihood Estimation ◽  
Worked Examples ◽  
Parameter Estimates ◽  
Survivorship Rate ◽  
A Direct Method

The chapter “Skills for Fitting Models to Data” provides worked examples of the model parameter estimation and model-selection examples presented in Part 2, both in spreadsheets and in R. This chapter presumes that the reader is reasonably comfortable setting up spreadsheets and R code and applying the modeling skills presented in Chapter 15. It begins with maximum likelihood estimation, presenting first a direct method and then numerical methods that yield more precise (usually) estimates of maximum likelihood parameter estimates. It then examines how to estimate parameters that do not appear in probability functions (e.g. a model in which survivorship rate is density dependent). The chapter concludes by discussing likelihood profiles.

Weighted Multicollinearity in Logistic Regression: Diagnostics and Biased Estimation Techniques with an Example from Lake Acidification

1990 ◽  
Vol 47 (6) ◽  
pp. 1128-1135 ◽  
Author(s):  
Brian D. Marx ◽  
Eric P. Smith
Keyword(s):  
Logistic Regression ◽  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Water Chemistry ◽  
Acid Precipitation ◽  
Estimation Methods ◽  
Parameter Estimates ◽  
Regression Diagnostics ◽  
Data Set ◽  
Biased Estimation

An historical data set from the Adirondack region of New York is revisited to study the relationship between water chemistry variables associated with acid precipitation and the presence/absence of brook trout (Salvelinus fontinalis) and lake trout (Salvelinus namaycush). For the trout species data sets, water chemistry variables associated with acid precipitation, for example pH and alkalinity, are highly correlated. Regression models to assess their effects on the probability of the presence of fish species are therefore affected by multicollinearity. Because the appropriate regressions are logistic, correction techniques based on least squares do not work. Maximum likelihood parameter estimation is highly unstable for the trout presence/absence data. Developments in weighted multicollinearity diagnostics are used to evaluate maximum likelihood logistic regression parameter estimates. Further, an application of biased parameter estimation is presented as an option to the traditional maximum likelihood logistic regression. Biased estimation methods, like ridge, principal component, or Stein estimation can substantially reduce the variance of the parameter estimates and prediction variance for certain future observations. In many cases, only a slight modification to the converged maximum likelihood estimator is necessary.

Analysis of Individual Risk Attitude for Risk Management Based on Cumulative Prospect Theory

2009 ◽  
pp. 254-270
Author(s):  
Fei-Chen Hsu ◽  
Hsiao-Fan Wang
Keyword(s):  
Risk Management ◽  
Risk Analysis ◽  
Prospect Theory ◽  
Evaluation Model ◽  
Risk Attitude ◽  
Cumulative Prospect Theory ◽  
Risk Level ◽  
Individual Risk ◽  
Court Case ◽  
Risk Response

In this chapter, we used Cumulative Prospect Theory to propose an individual risk management process (IRM) including a risk analysis stage and a risk response stage. According to an individual’s preferential structure, an individual’s risk level for the confronted risk can be identified from risk analysis. And based on a response evaluation model, the appropriate response strategy is assessed at the risk response stage. The applicability of the proposed model is evaluated by an A-C court case. The results have shown that the proposed method is able to provide more useful and pertinent information than the traditional method of decision tree by using the expected monetary value (EMV).

scholarly journals Item Parameter Estimation With the General Hyperbolic Cosine Ideal Point IRT Model

2018 ◽  
Vol 43 (1) ◽  
pp. 18-33 ◽  
Author(s):  
Seang-Hwane Joo ◽  
Seokjoon Chun ◽  
Stephen Stark ◽  
Olexander S. Chernyshenko
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Sample Size ◽  
Item Response ◽  
Ideal Point ◽  
Point Estimation ◽  
Item Parameter ◽  
Parameter Estimates ◽  
Marginal Maximum Likelihood ◽  
Hyperbolic Cosine

Over the last decade, researchers have come to recognize the benefits of ideal point item response theory (IRT) models for noncognitive measurement. Although most applied studies have utilized the Generalized Graded Unfolding Model (GGUM), many others have been developed. Most notably, David Andrich and colleagues published a series of papers comparing dominance and ideal point measurement perspectives, and they proposed ideal point models for dichotomous and polytomous single-stimulus responses, known as the Hyperbolic Cosine Model (HCM) and the General Hyperbolic Cosine Model (GHCM), respectively. These models have item response functions resembling the GGUM and its more constrained forms, but they are mathematically simpler. Despite the apparent impact of Andrich’s work on ensuing investigations, the HCM and GHCM have been largely overlooked by applied researchers. This may stem from questions about the compatibility of the parameter metric with other ideal point estimation and model-data fit software or seemingly unrealistic parameter estimates sometimes produced by the original joint maximum likelihood (JML) estimation software. Given the growing list of ideal point applications and variations in sample and scale characteristics, the authors believe these HCMs warrant renewed consideration. To address this need and overcome potential JML estimation difficulties, this study developed a marginal maximum likelihood (MML) estimation algorithm for the GHCM and explored parameter estimation requirements in a Monte Carlo study manipulating sample size, scale length, and data types. The authors found a sample size of 400 was adequate for parameter estimation and, in accordance with GGUM studies, estimation was superior in polytomous conditions.

scholarly journals Conditional Maximum Likelihood Estimation in Probability-Branched Multistage Designs

2021 ◽  
Author(s):  
Jan Steinfeld ◽  
Alexander Robitzsch
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Likelihood Estimation ◽  
Parameter Estimate ◽  
Item Parameter ◽  
Parameter Estimates ◽  
Conditional Maximum Likelihood ◽  
Item Parameter Estimates ◽  
Item Parameter Estimation ◽  
Conditional Maximum

This article describes the conditional maximum likelihood-based item parameter estimation in probabilistic multistage designs. In probabilistic multistage designs, the routing is not solely based on a raw score j and a cut score c as well as a rule for routing into a module such as j < c or j ≤ c but is based on a probability p(j) for each raw score j. It can be shown that the use of a conventional conditional maximum likelihood parameter estimate in multistage designs leads to severely biased item parameter estimates. Zwitser and Maris (2013) were able to show that with deterministic routing, the integration of the design into the item parameter estimation leads to unbiased estimates. This article extends this approach to probabilistic routing and, at the same time, represents a generalization. In a simulation study, it is shown that the item parameter estimation in probabilistic designs leads to unbiased item parameter estimates.

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