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Normal maximum likelihood, weighted least squares, and ridge regression estimates

Withers C. S., Nadarajah S.

Probability and Mathematical Statistics

2012

Vol. 32, Fasc. 1

11--24

There have been many papers published (in almost every statistics related journal) suggesting that normal maximum likelihood is superior or inferior to weighted least squares and other approaches. In this note, we show that the three main estimation methods (normal maximum likelihood, weighted least squares and ridge regression) all have the same asymptotic covariance and that there is no gain in efficiency among them. We also show how the bias of these estimators can be reduced and conduct a simulation study to illustrate the magnitude of bias reduction.

Unbiased estimates for linear regression with roundoff error

Withers C. S., Nadarajah S.

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 2

177--182

We consider the linear regression model, where the residuals have zero mean and an otherwise unspecified distribution F. Suppose that least squares estimates are formed by using rounded values of the dependent variables.We show that these are still unbiased, and that unbiased estimates for the moments and cumulants of F are given by applying Sheppard’s corrections to their estimates.

Charlier and edgeworth expansions for distributions and densities in terms of bell polynomials

Withers C. S., Nadarajah S.

Probability and Mathematical Statistics

2009

Vol. 29, Fasc. 2

271--280

We show that the coefficients of the Charlier differential series for distributions and densities are simply Bell polynomials in the cumulants. The same is true for the Edgeworth expansions of distributions and densities of sample means. We use this to obtain higher order extensions of these well-known series.