Environmental Decision-Making - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 225

Fig. 3.1 Classical Neyman-
Pearson decision-making
procedure to choose between
the two hypotheses H 0 and H 1

true. In this case, the parameter a of the real density f a can be equal to neither a 0
or a 1 .
Errors of the

first kind, a , and the second kind, b , satisfy the following formulas

respectively:

5 fð E a1 n E a0 nÞð D a0 nÞ 1 = 2

2 n ;

a exp½ 0 : g ð 3 : 2 Þ

b exp½ 0 : 5 fð E a1 n E a0 nÞð D a1 nÞ 1 = 2

2 n ;

g ð 3 : 3 Þ

where

Z 1

E a n ¼ ln f a1 ð x Þ= ½ f a0 ð x Þ f a ð x Þ dx ð 3 : 4 Þ

1

Z 1

2 f a ð x Þ dx E a ðÞ 2

D a n ¼ f ln f a1 ð x Þ= ½ f a0 ð x Þ g ð 3 : 5 Þ

1

Neyman-Pearson theory focuses on testing a statistical hypothesis provided that
the volume of observations allows the calculation of basic statistical characteristics
including the empirical distribution function. In this case, theoretical probability
density function f a (x) is assessed on the base of

2 -criterion. Unfortunately,
empirical results of environmental monitoring usually do not satisfy this condition.
In this case a sequential procedure is used where the sample size is not

ˇ

fixed in
advance. Instead, data are evaluated as they are collected, and further sampling is
stopped in accordance with a pre-de cant results

are observed. Thus, a conclusion may sometimes be reached at a much earlier stage

ned stopping rule as soon as signi