A-37 
where <T> is the normal cumulative density function. In this greatly simplified 
scenario, Pj would be the outcome of N independent Bernoulli trials. The ideal CFD 
would be the cumulative distribution function of M outcomes of a binomial random 
variable with N trials. If we allow ^ to have positive off diagonal elements, then the 
Bernoulli trials become dependent (i.e. adjacent cells are more likely to either both 
exceed or both meet the standard than distant cells). This should make the distribu¬ 
tion of the Pj more variable than under the independent binomial model, but the 
expectation of Pj would be constant over time. If we relax the assumption that 
is 0, then the expectation of the Pj would vary over time which would increase the 
variability of the Pj even more. 
Under the simplifying assumptions of independence, constant mean, and constant 
variance, it is possible to obtain an analytical formulation for the CFD based on the 
parameters of Eqn 5.1.1.1, However, when the more realistic time dependent, space 
dependent model with seasonal nonstationarity is considered, an analytical formula¬ 
tion is not tractable. The lack of an analytical formulation for this estimator under 
realistic dependence assumptions, e.g. non-trivial and points toward com¬ 
puter intensive simulation techniques to develop statistical inference procedures for 
this problem. None-the-less, it is interesting to consider the behavior of the CFD 
under the simplified model. 
5.3 CFD BEHAVIOR UNDER A SIMPLIFIED MODEL 
In what follows, the behavior of the CFD under various parameter formulations for 
Equation 5.1.1.1 are presented in graphical form. There are four parameters involved: 
p the population mean, <7 t the temporal variance, a s the spatial variance, and C the 
criterion threshold. In the examples that follow, three of these parameters are held 
constant and the fourth is varied to illustrate the effect of the varied parameter. 
In this exercise, the parameters of Equation 5.1.1.1 are simplified as follows: = o t 
I and ^ = <7 S I, where I is the identity matrix. Thus in both the temporal and spatial 
dimensions, independence and constant variance is assumed. 
Example 1. Example 1 considers the effect of changing the population mean on the 
shape of the CFD. 
Table 5.1. Parameter values and key for the family of curves shown in Figure 5.1. 
F 
<T S 
c 
color 
curve 
number 
5 
i 
1 
5 
Red 
1 
4 
i 
1 
5 
Orange 
2 
3 
i 
1 
5 
Brown 
3 
2 
i 
1 
5 
Green 
4 
1 
i 
1 
5 
Blue 
5 
appendix a • The Cumulative Frequency Diagram Method for Determining Water Quality Attainment 
