Distance and decision rules

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Distance between distribution functions of ten becomes a useful and convenient concept in statistics. In a class of distributions, distance can be defined in various ways. We, therefore, have to choose an adequate distance for a particular problem. Fur the r , to t r ea t the problem efficiently, we are required, or at least i t is ve ry desirable, to control all errors tha t may be committed in making decision or inference. For this purpose we have to formulate, or reformulate if necessary, the problem suitably. A proper formulation of the problem is very important for ge t t ing command over possible errors. For instance, suppose tha t we wan t to know whether or not the random variable under consideration can be considered to have mean zero, when it is given that the random variable has the Gaussian dist r ibut ion N(O, 1). In this case, if we take the problem as the one inquiring jus t whether or not the mean of the random variable is zero, we can not control possible errors. For we can find a distribution which has mean not equal to 0, bu t which is as near to the distribution N(0, 1) as desired, and this makes it impossible to control all possible errors in inference or decision making based on a finite number of observations. In terms of hypothesis testing, we can not make the first and second kinds of errors simultaneously as small as desired (i.e., while we can make the first kind of error smaller than any given (positive) value, this is not the case wi th the second kind of error). One way to avoid such inconvenience is to formulate the problem as follows. That is, introducing an adequate distance d ( . , . ) i n the space of distributions concerned, we set the problem as making decision whether the random variable under consideration has F0=N(0 , 1) or some F with d(F, Fo)>~ (>0) , where ~ is a constant which is to be predetermined from the actual situation of the problem. For the problem thus formulated we can control the errors (see Matusi ta [1], Matusita, Akaike [2]). So far, the author has t rea ted various problems with the same idea, the idea of controlling possible errors (see Matusi ta [1], [3], [5], [7], [8],