Necessary conditions for a maximum likelihood estimate to become asymptotically unbiased and attain the Cramer–Rao lower bound. II. Range and depth localization of a sound source in an ocean waveguide

2002 ◽  
Vol 112 (5) ◽  
pp. 1890-1910 ◽  
Author(s):  
Aaron Thode ◽  
Michele Zanolin ◽  
Eran Naftali ◽  
Ian Ingram ◽  
Purnima Ratilal ◽  
...  
Keyword(s):  
Maximum Likelihood ◽  
Lower Bound ◽  
Sound Source ◽  
Maximum Likelihood Estimate ◽  
Necessary Conditions ◽  
Likelihood Estimate ◽  
Cramer Rao Lower Bound ◽  
Depth Localization

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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