# Concise Encyclopedia of Statistics by August-Wilhelm Scheer, Wolfram Jost, Helge Heß, Andreas

By August-Wilhelm Scheer, Wolfram Jost, Helge Heß, Andreas Kronz

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Based on this, we define a loss function l(a0, θ ) that accepts This is equivalent to r (1 − θ ) = (n − r) θ , H0 (where the true value of the parameter is and it simplifies to θˆ = r . The estimator mle n θ ) and a loss function l(a1, θ ) that rejects H0. for the maximum likelihood of θ is then simIf the value for the true value obtained by ply the proportion of observed successes. accepting H0 , that is, Now we return to the bayesian method. l(a0 , θ ) dθ , In this case, the Bernoulli distribution, the a priori reference distribution of the parameter θ , is expressed by: is smaller than to the one obtained by reject1 ing H0 , then we can accept H0 .

The interpretation of p-values and the combination of test results. Stat. , 9, 601–614. FURTHER READING Graphical representation Quantitative graph Bayes’ Theorem If we consider the set of the “reasons” that an event occurs, Bayes’ theorem gives a formula for the probability that the event is the Barnard, George A. direct result of a particular reason. Barnard, George Alfred was born in 1915, Therefore, Bayes’ theorem can be interpretin Walthamstow, Essex, England. He gained ed as a formula for the conditional probaa degree in mathematics from Cambridge bility of an event.

Indiv. observed over two years Cases attrib. to smoking (for two-year period) Cases attrib. 5 23 We can calculate the number of cases of breast cancer attributable to tobacco exposure by multiplying the number of individuals observed per year by the attributable risk. By dividing the number of incidents attributable to smoking in the two-year period by two, we obtain the number of cases attributable to smoking per year, and we can then determine the risk attributable to smoking in the population, denoted PAR, as shown in the following example.