handout4 - ISyE8843A, Brani Vidakovic Handout 4 1 Decision...

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Unformatted text preview: ISyE8843A, Brani Vidakovic Handout 4 1 Decision Theoretic Setup: Loss, Posterior Risk, Bayes Action Let A be action space and a A be an action. For example, in estimation problems, A is the set of real numbers and a is a number, say a = 2 is adopted as an estimator of . In other words, the inference maker took the action a = 2 in estimating . In testing problems, the action space is A = { accept , reject } . The action, as a function of observations is called a decision rule, or simply a rule. An example of a rule is a ( X 1 ,...,X n ) = X . Often, the rules are denoted by ( X ) . No action can be taken without potential losses. Statisticians are pessimistic creatures that replaced nicely coined term utility to a more somber term loss , although, for all practical purposes, the loss is a negative utility. The loss function is denoted by L ( ,a ) and represents the payoff by a decision maker (statistician) if he takes the action a A , and the real state of nature is . The loss function usually satisfies the following properties, L ( a,a ) = 0 and L ( a, ) is nondecreasing function of | a- | . Examples are squared error loss (SEL) L ( ,a ) = ( - a ) 2 , absolute loss , L ( ,a ) = | - a | , the 0-1 loss, L ( ,a ) = 1 ( | a- | > m ) , etc. The most common for estimation problems and mathematically easiest to work with is the SEL. The expected SEL (frequentist risk) is linked with variance and bias of an estimator, E X | ( - ( X )) 2 = V ar ( ( X )) + [ bias ( ( X ))] 2 . where bias ( ( X ) = E X | ( X ))- . One criticism of the SEL is that it grows fast (quadratically) when the error increases, thus severely punishing the errors. Example 1. The LINEX is defined as L ( ,a ) = exp { c ( a- ) } - c ( a- )- 1 , c R. For c > , the loss function L ( ,a ) is quite asymmetric about 0 with overestimation being more costly than under-estimation. As | a- | , the loss L ( ,a ) increases almost exponentially when a- > and almost linearly when a- < . For c < , the linearity-exponentiality phenomenon is reversed. Also, when | a- | is very small, L ( ,a ) is near c ( a- ) 2 / 2 ....
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handout4 - ISyE8843A, Brani Vidakovic Handout 4 1 Decision...

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