The two-parameter Langevin distribution has been widely used for analyzing directional data. We address the problem of estimating the mean direction in its Cartesian and angular forms. The equivariant point estimation is introduced under different transformation groups. The maximum likelihood...

We investigate the problem of estimating the Cholesky decomposition in a conditional independent normal model with missing data. Explicit expressions for the maximum likelihood estimators and unbiased estimators are derived. By introducing a special group, we obtain the best equivariant estimators.

Independent random samples are taken from two normal populations with means <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\mu _1$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\mu _2$$</EquationSource> </InlineEquation> and a common unknown variance <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\sigma ^2.$$</EquationSource> </InlineEquation> It is known that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\mu _1\le \mu _2.$$</EquationSource> </InlineEquation> In this paper, estimation of the common standard deviation <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\sigma $$</EquationSource> </InlineEquation> is considered with respect to...</equationsource></inlineequation></equationsource></inlineequation></equationsource></inlineequation></equationsource></inlineequation></equationsource></inlineequation>

Estimation of the entropy of several exponential distributions is considered. A general inadmissibility result for the scale equivariant estimators is proved. The results are extended to the case of unequal sample sizes. Risk functions of proposed estimators are compared numerically.