Abstract
The estimation of parameters is a key component in statistical modelling and inference. However, parametrization of certain likelihood functions could lead to highly correlated estimates, causing numerical problems, mathematical complexities and difficulty in estimation or erroneous interpretation and subsequently inference. In statistical estimation, the concept of orthogonalization is familiar as a simplifying technique that allows parameters to be estimated independently and thus separates information from each other. We introduce a Fisher scoring iterative process that incorporates the Gram–Schmidt orthogonalization technique for maximum likelihood estimation. A finite mixture model for correlated binary data is used to illustrate the implementation of the method with discussion of application to oesophageal cancer data.
Highlights
The problem of estimating parameters is one of the key stages in fitting a statistical model to a set of data
Maximization algorithms converge rapidly if the initial estimates are good, the likelihood function is well approximated by a quadratic in the neighbourhood of the parameter space and the information matrix is well conditioned, which means that the parameter estimates are not strongly intercorrelated
Parameter orthogonalization is used as an aid in computation, approximation, interpretation and elimination or reduction of the effect of nuisance parameters
Summary
The problem of estimating parameters is one of the key stages in fitting a statistical model to a set of data. Godambe (1991) deals with the problem of nuisance parameters, within a semi-parametric set-up which includes the class of distributions associated with generalized linear models Their technique uses the optimum orthogonal estimating functions of Godambe and Thompson (1989). Since does not enter explicitly into the equations, the solution for j can contain an arbitrary function of as the integrating constant It is noted in the discussion of Cox and Reid that it is sometimes theoretically possible to find a differential equation, simple explicit solutions of the differential equation were not feasible for the some models. Global orthogonalization can not be achieved by this approach
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