Maximum Likelihood Theory for Incomplete Data from an Exponential Family

In this paper we study such classes of distribu- tions which are generated from exponential families by loss of information due to the fact that only some function of the exponential family variable is observable. Examples of such classes are mixtures and convolutions of exponential type distributions as well as grouped, censored and folded distribu- tions. Their common structure is analysed. The existence is demonstrated of a n1/2-consistent, asymptotically normally distributed and asymptotically efficient root of the likelihood equation which asymptotically maximizes the likelihood in every compact subset of the parameter space, imposing only the natural requirement that the information matrix is positive definite. It is further shown that even the weaker requirement of local parameter identifiability, which admits of application to non-regular cases, is sufficient for the existence of con- sistent maximum likelihood estimates. Finally the subject of large sample tests based on maximum likelihood estimates is touched upon.

[1]  A. Wald.  Tests of statistical hypotheses concerning several parameters when the number of observations is large , 1943 .

[2]  Calyampudi R. Rao.  Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation , 1948, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  A. Wald.  Note on the Consistency of the Maximum Likelihood Estimate , 1949 .

[4]  ESTIMATION OF PARAMETERS OF MIXED EXPONENTIALLY DISTRIBUTED FAILURE TIME DISTRIBUTIONS FROM CENSORED LIFE TEST DATA , 1958 .

[5]  J. Aitchison,et al.  Maximum-Likelihood Estimation of Parameters Subject to Restraints , 1958 .

[6]  E. Lehmann.  Testing Statistical Hypotheses , 1960 .

[7]  L. S. Nelson,et al.  The Folded Normal Distribution , 1961 .

[8]  A Note on a Problem in Estimation , 1962 .

[9]  Statistical testing of differences in causal behaviour of two morphologically indistinguishable objects. , 1967, Biometrics.

[10]  N. E. Day.  Estimating the components of a mixture of normal distributions , 1969 .

[11]  R. R. Hocking,et al.  The analysis of incomplete data. , 1971 .

[12]  M. Woodbury.  A missing information principle: theory and applications , 1972 .

[13]  R. Sundberg.  Maximum likelihood theory and applications for distributions generated when observing a function of an exponential family variable , 1972 .

[14]  Rolf Sundberg,et al.  On estimation and testing for the folded normal distribution , 1974 .

[15]  K. Valentine,,et al.  Contributions to the theory of care , 1989 .

[16]  D. Wittink,et al.  Estimation and Testing , 2015 .