scholarly journals Misclassification Probabilities through Edgeworth-type Expansion for the Distribution of the Maximum Likelihood based Discriminant Function

2021 ◽  
Author(s):  
Emelyne Umunoza Gasana
Keyword(s):  
Maximum Likelihood ◽  
Discriminant Function ◽  
Distribution Of The Maximum ◽  
Edgeworth Type Expansion

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.

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