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1
k = Pn
i=1 xk
i log xi
Pni
=1 xk
i −
1
n
nX
i=1
log xi (A.5)
Balakrishnan and Kateri[10] showed that the plots of RHS and LHS of A.5 would
intersect exactly once, at the MLE of shape k. Therefore, A.5 leads us the unique
maximum likelihood estimator of shape k.
Similar with Gamma distribution, the estimator for shapeˆk forWeibull distribution
is calculated by numerical methods.
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A.3.3 Asymptotical Convergence of MLE
(A.W. van der Vaart, Asymptotic Statistics (Cambridge Series in Statistical and
Probabilistic Mathematics) (1998)) In practise, in many cases, a set of independent
identically distributed random variables is used to estimate the distribution
parameters. In such cases, one is interested in the quality of the estimator, that is
the convergence of the estimated parameters to the true parameters as the number
of observations increases to infinity. In literature, this is reffered as asymptotic
behaviour of the estimator. In a certain sense, Maximum Likelihood Estimator is
asymptotically optimal. The behaviour of MLE is listed below:
The MLE is asymptotically unbiased, that is, its bias tends to 0 as the sample size
increases to infinity, where bias of the estimator ˆ is: E(ˆ) − .
The MLE is asymptotically efficient, that is, it achieves the Cramer-Rao lower
bound when the sample size tends to infinity. This means that no asymptotically
unbiased estimator can give better estimator than MLE in the sense of asymptotic
mean squared error.
The estimator is asymptotically normal, that is, as the sample size increases, the
distribution of the MLE tends to Gaussian with mean and the inverse of the Fisher
Information matrix as covariance matrix.
We have to assume some regularity conditions to have the listed behaviours of the
MLE. Those conditions are:
• The first and second derivatives of the log-likelihood function must be defined.
• The Fisher information matrix must not be zero, and must be continuous as
a function of the parameter.
• The maximum likelihood estimator is consistent, i.e, as the sample size goes
to infinity, the estimator converges to the true value of the parameter in probability.
Although the theory does not determine how large sample is required to have good
degree of approximation, in practise the MLE appears to be approximately true if
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the sample size is moderately large.
A.4 CHI-SQUARE GOODNESS OF FIT TEST
Chi-Square goodness of fit test is used to test if the data comes from a particular
distribution. Therefore, the null hypothesis is
H0 : F = F0
and the alternative hypothesis is
H1 : F , F0
where F is the sample distribution and F0 is the particular distribution to be tested
if the sample data comes from.
The test requires that the data first be grouped. The actual number of observations
in each group is compared to the expected number of observations according to F0
and the test statistics is calculated as a function of this difference, namely
2 = Pki
=1 (Oi − Ei)2
Ei
(A.6)
where k is the number of bins the data is divided into, Oi is the actual number of
observations in each bin i, and Ei is the expected number of observations in bin i
with respect to the distribution F0.
A.6 approximately follows the 2 distribution with k − 1 − l degrees of freedom
where l is the number of distribution parameters estimated.
The expected frequency is calculated by
Ei = n(F(Yiu
) − F(Yi
l )) where F is the cumulative distribution function for the
distribution being tested, Yiu
is the upper limit for the class i, Yi
l is the lower limit
of the class i, and n is the sample size.
The Chi-Square goodness of fit test is sensitive to the choice of bins. The test
power will be affected by the number of groups, how they are defined, and the
sample size. There is no optimal choice for the bin width since the optimal bin
width depends on the distribution.
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Appendix B
APPLICATIONS WITH
DIFFERENT TYPES OF KLM
GROUND HANDLING
PROCESS TIMES DATA
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Table of Contents
1 Executive Summary...........................................................................................................................................4
2 Document Management ....................................................................................................................................5
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