ABSTRACT
Statistics is the science of decision making in the face of uncertainties about some characteristic properties of a population on the basis of sample observations X 1 , … , X n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2679.tif"/> from it. These uncertainties are measured in numerical quantities that could be a scalar or a vector or a matrix. We shall denote them by θ. The space of all possible values of θ is called the state space and is denoted by Ω. The job of a statistician is to take an action among the set of all possible actions available for the problem at hand on the basis of decision rules. These decision rules depend only on the sample observations X 1 , … , X n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2680.tif"/> . The set of all actions will be denoted by a https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2681.tif"/> . A decision rule is denoted by δ( X 1 , … , X n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2682.tif"/> ). If we observe x 1 , … , x n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2683.tif"/> and use the decision δ, we take action δ( x 1 , … , x n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2684.tif"/> ). The space of all values ( x 1 , … , x n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2685.tif"/> ) is denoted by Χ, the sample space. The set of all decision rules will be denoted by P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2686.tif"/> .
