The data that are analysed are from a monitoring survey which was carried out in 1994 in the forests of Baden-Württemberg, a federal state in the south-western region of Germany. The survey is part of a large monitoring scheme that has been carried out since the 1980s at different spatial and...

The distributional assumption for a generalized linear model is often checked by plotting the ordered deviance residuals against the quantiles of a standard normal distribution. Such plots can be difficult to interpret, because even when the model is correct, the plot often deviates...

Suppose X and Y are binary exposure and outcome variables, and we have full knowledge of the distribution of Y, given application of X. We are interested in assessing whether an outcome in some case is due to the exposure. This "probability of causation" is of interest in comparative historical...

A scoring rule <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$S(x; q)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> provides a way of judging the quality of a quoted probability density <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$q$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>q</mi> </math> </EquationSource> </InlineEquation> for a random variable <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$X$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>X</mi> </math> </EquationSource> </InlineEquation> in the light of its outcome <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$x$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>x</mi> </math> </EquationSource> </InlineEquation>. It is called proper if honesty is your best policy, i.e., when you believe <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$X$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>X</mi>...</math></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation>