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with the bin-width = 3.15.
The R function ”hist” is used for this purpose. The number of observations in
each bin is:
17
Figure 2.1: Gamma Q-Q Plot for shifted GHP Times
1 1 0 1 0 0 10 25 145 469 674 524 304 220 90 66 50 29 22 22
17 10 3 3 7 6 6 3 1 4 3 3 0 0 1 2 0 0 0 1 0 0 0 1 0 1
As one observes, there are many empty bins as well as the bins which include less
than 5 observations. We rearrange the bins as follows:
(0, 24], (24, 27], (27, 30], (30, 33], (33, 36], (36, 39], (39, 42], (42, 48], (48, 56],
(56, 71], (71, 90], (90, 102], (102, 150]
so that the observations, xi’s, in each bin are:
13 25 145 469 674 524 304 281 151 94 28 11 6
Figure 2.2 and Figure 2.3 show the histograms of the former and the latter cases.
The test statistics is, 2 = Pki
=1 (Oi−Ei)2
Ei
= 105370.4, which gives a p-value very
18
Figure 2.2: Histogram of Shifted GHP Data with bin-width=3.15
close to 0 with 13 − 1 − 2 = 10 degrees of freedom. Therefore one concludes
that we cannot accept the null-hypothesis and Gamma distribution with estimated
parameters does not fit to our sample.
Because for this particular study we are interested in predicting the quantiles between
0.1 and 0.9, one may want to see how our data behaves between 10th and
90th percentiles. In order to find out this, first we determine the sample quantiles.
2.2 QUANTILES
Any real-valued random variable X can be characterized by its (right continuous)
distribution function
F(x) = P(X x)
where the -quantile for any 0 < < 1 is:
F−1( ) = inf{x : F(x) }
19
Figure 2.3: Histogram of Shifted GHP Data with Rearranged Bins
A simple optimization problem is used to find the quantiles as follows:
Under the piecewise linear loss function, illustrated in Figure 2.4, (u) = u( −
1{u < 0}) for some 2 (0, 1), a point estimate is required for a random variable,
X, with (posterior) distribution function F. One would like to minimize the expectation
of the above loss function, , given X. Fox and Rubin (1964) studied the
admissibility of the quantile estimator under this loss function. We would like to
minimize the expectation
E (X − ˆx) = ( − 1) Z ˆx
−1
(x − ˆx)dF(x) + Z 1
ˆx
(x − ˆx)dF(x)
Differentiating with respect to ˆx, we obtain
0 = (1 − ) Z ˆx
−1
dF(x) − Z 1
ˆx
dF(x) = F( ˆx) −
Any element of {x : F(x) = } minimizes the expected loss because F, being a distribution
function, is monotone. The result is either unique (in that case ˆx = F−1( ))
20
or an interval of quantiles (from which the smallest to be chosen for convention
that the empirical distribution function is left continuous).
Figure 2.4: Quantile Regression function
If we replace F by the empirical distribution function Fn, where Fn = n−1 Pni
=1 1{Xi x},
we may still choose ˆx to minimize the expected loss:
Z (x − ˆx)dFn(x) = n−1
nX
i=1
(Xi − ˆx)
The resulting ˆx would give us the -sample quantile. Thus, denoting the sample
quantile by q( ), one can formulize finding the quantile as follows:
q( ) = argminˆx = n−1
nX
i=1
(Xi − ˆx)( − 1{Xi − ˆx < 0})
We now have expressed the -sample quantile as a linear optimization problem
which can be solved with well known methods such as interior point and simplex
algorithms [9]Koenker [2005].
21
Royal Dutch Airlines (KLM) is interested in the quantiles of the ground handling
process times in order to test and improve the quality of the schedules. Historical
data are available for 3 years, since 2006, each year divided into 4 operation plan
(OP) periods.We would like to know more about the quantile for the next OP
period based on the historical quantiles.
2.3 ESTIMATION OF THE DISTRIBUTION : AN
OPTIMIZATION APPROACH
In this section, we would like to use an optimization approach in order to determine
the distribution of the data. In the previous section, we used Maximum Likelhood
Estimator, however the estimated Gamma distribution did not fit to our whole sample.
Unlike the previous approach, here, we try to see if a Gamma distribution
can be fit to our data when we cencor the observations between 10th and 90th percentiles.
In other words, let a and b denote the 0.1 and 0.9 quantiles, respectively.
We use the data Xi, where a Xi b to see how good an estimated Gamma
distribution fits to the cencored part of the sample. We construct a minimization
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