Asymptotic Normality of the Maximum-Likelihood Estimate

1996 ◽  
pp. 119-125
Author(s):  
Thomas S. Ferguson
Keyword(s):  
Maximum Likelihood ◽  
Asymptotic Normality ◽  
Maximum Likelihood Estimate ◽  
Likelihood Estimate

scholarly journals Asymptotic normality of M-estimators in nonhomogeneous hidden Markov models

2011 ◽  
Vol 48 (A) ◽  
pp. 295-306
Author(s):  
Jens Ledet Jensen
Keyword(s):  
Maximum Likelihood ◽  
Asymptotic Normality ◽  
Hidden Markov Models ◽  
Maximum Likelihood Estimate ◽  
Markov Models ◽  
Hidden Markov ◽  
Likelihood Estimate ◽  
Mixing Properties ◽  
Dependent Variables ◽  
Hidden Markov Process

Results on asymptotic normality for the maximum likelihood estimate in hidden Markov models are extended in two directions. The stationarity assumption is relaxed, which allows for a covariate process influencing the hidden Markov process. Furthermore, a class of estimating equations is considered instead of the maximum likelihood estimate. The basic ingredients are mixing properties of the process and a general central limit theorem for weakly dependent variables.

scholarly journals Asymptotic normality of M-estimators in nonhomogeneous hidden Markov models

2011 ◽  
Vol 48 (A) ◽  
pp. 295-306 ◽  
Author(s):  
Jens Ledet Jensen
Keyword(s):  
Maximum Likelihood ◽  
Asymptotic Normality ◽  
Hidden Markov Models ◽  
Maximum Likelihood Estimate ◽  
Markov Models ◽  
Hidden Markov ◽  
Likelihood Estimate ◽  
Mixing Properties ◽  
Dependent Variables ◽  
Hidden Markov Process

Results on asymptotic normality for the maximum likelihood estimate in hidden Markov models are extended in two directions. The stationarity assumption is relaxed, which allows for a covariate process influencing the hidden Markov process. Furthermore, a class of estimating equations is considered instead of the maximum likelihood estimate. The basic ingredients are mixing properties of the process and a general central limit theorem for weakly dependent variables.

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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