Confidence Limits of the Maximum-Likelihood Estimate of the Hominoid three from Mitochondrial-DNA Sequences

Evolution ◽  
1989 ◽  
Vol 43 (3) ◽  
pp. 672 ◽  
Author(s):  
Masami Hasegawa ◽  
Hirohisa Kishino
Keyword(s):  
Mitochondrial Dna ◽  
Maximum Likelihood ◽  
Dna Sequences ◽  
Maximum Likelihood Estimate ◽  
Likelihood Estimate ◽  
Confidence Limits

Box-Cox Transformations

2017 ◽  
Author(s):  
Russell Cheng
Keyword(s):  
Maximum Likelihood ◽  
Normal Form ◽  
Maximum Likelihood Estimate ◽  
Likelihood Estimate ◽  
Grouped Data ◽  
Numerical Example ◽  
Alternative Method ◽  
Modified Likelihood

This chapter examines the well-known Box-Cox method, which transforms a sample of non-normal observations into approximately normal form. Two non-standard aspects are highlighted. First, the likelihood of the transformed sample has an unbounded maximum, so that the maximum likelihood estimate is not consistent. The usually suggested remedy is to assume grouped data so that the sample becomes multinomial. An alternative method is described that uses a modified likelihood similar to the spacings function. This eliminates the infinite likelihood problem. The second problem is that the power transform used in the Box-Cox method is left-bounded so that the transformed observations cannot be exactly normal. This biases estimates of observational probabilities in an uncertain way. Moreover, the distributions fitted to the observations are not necessarily unimodal. A simple remedy is to assume the transformed observations have a left-bounded distribution, like the exponential; this is discussed in detail, and a numerical example given.

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