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Statistical risk is a quantification of a situation's risk using statistical methods.These methods can be used to estimate a probability distribution for the outcome of a specific variable, or at least one or more key parameters of that distribution, and from that estimated distribution a risk function can be used to obtain a single non-negative number representing a particular conception of ...
An example is shown on the left. The parameter space has just two elements and each point on the graph corresponds to the risk of a decision rule: the x-coordinate is the risk when the parameter is and the y-coordinate is the risk when the parameter is . In this decision problem, the minimax estimator lies on a line segment connecting two ...
The MSE either assesses the quality of a predictor (i.e., a function mapping arbitrary inputs to a sample of values of some random variable), or of an estimator (i.e., a mathematical function mapping a sample of data to an estimate of a parameter of the population from which the data is sampled).
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Risk functions are chosen depending on how one measures the distance between the estimate and the unknown parameter. The MSE is the most common risk function in use, primarily due to its simplicity. However, alternative risk functions are also occasionally used. The following are several examples of such alternatives.
For risk-averse or risk-loving agents, loss is measured as the negative of a utility function, and the objective function to be optimized is the expected value of utility. Other measures of cost are possible, for example mortality or morbidity in the field of public health or safety engineering .
For example, in a study examining the effect of the drug apixaban on the occurrence of thromboembolism, 8.8% of placebo-treated patients experienced the disease, but only 1.7% of patients treated with the drug did, so the relative risk is .19 (1.7/8.8): patients receiving apixaban had 19% the disease risk of patients receiving the placebo. [4]
A decision rule is a function:, where upon observing , we choose to take action (). Also define a loss function L : Θ × A → R {\displaystyle L:\Theta \times {\mathcal {A}}\rightarrow \mathbb {R} } , which specifies the loss we would incur by taking action a ∈ A {\displaystyle a\in {\mathcal {A}}} when the true state of nature is θ ∈ Θ ...