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OTE/SPH
 OTE/SPH
         August 31, 2006
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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.
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