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JWBK119-11
Empirical Distribution Function based Approaches 159
Table 11.3 Calculation of absolute deviations.
sup [|F n (x) − F 0 (x)|]
x
Residuals N(x (i) ≤ x) F n (x) F 0 (x) |F n (x i−1 ) − F 0 (x i )| |F n (x i ) − F 0 (x i )|
−0.62 1 0.03 0.039 0.039 0.014
−0.58 2 0.05 0.050 0.025 0.000
−0.57 3 0.08 0.052 0.002 0.023
: : : : : :
−0.19 14 0.35 0.294 0.031 0.056
−0.19 15 0.38 0.294 0.056 0.081D(max)
−0.14 16 0.40 0.345 0.030 0.055
: : : : : :
0.55 39 0.98 0.941 0.009 0.034
0.69 40 1.00 0.975 0.000 0.025

Hence, a band can simply be set up of width ±d α around sample distribution function
F n (x) such that the true distribution function, F(x), lies entirely within this band.
This inversion is allowed because of the measure of deviation and the existence of
the distribution for the Kolmogorov--Smirnov D statistic.
The residuals data from helicopter flight times described in Table 11.1 is used to
demonstrate the Kolmogorov--Smirnov GOF test. Based on the assumptions in linear
regression analysis, the GOF test is for a hypothesized population that is normally
distributed with zero mean and unknown standard deviation, σ.
First, the EDF, F n (x), for each observation is evaluated. The EDF values correspond-
ingtoeachsampleobservationaretabulatedinTable11.3.Giventhatourhypothesized
distribution is normal, theoretical values based on this hypothesized normal CDF can
be evaluated. These are also shown in Table 11.3. The maximum absolute difference
is highlighted in the table, and also shown graphically in Figure 11.1. The band of

1
0.9
0.8
0.7
CDF, F 0 (EDF , F n ) 0.6 D

0.5
0.4
0.3
0.2 α
0.1
0
−1.5 −1 −0.5 0 0.5 1 1.5
Residuals

Figure 11.1 Plot of F n and F 0 against sample observations.