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3.5.3 Improvement of the Model Paramters
In this section, we try to find a parameter, namely
k, depending on the iteration
step for the model
ˆ
k+1 = ˆk +
kS (Yk,ˆk)
with the initial value ˆ0 = 1, for any 0 <  < 1.
We combine the method of ARRSES in section 3.3.2. There is a significant improvement
in the estimations. (Results will be included later).
3.6 EVALUATION OF FORECAST RESULTS
3.7 COMPARISON AND COMBINATION OF FORECAST
METHODS
57
Bibliography
[1] van der Vaart A.W., Asymptotic Statistics, Cambridge Series in Statistical and
Probabilistic Mathematics, (1998).
[2] Williams D., Pobability with Martingales, Cambridge Mathematical Textbooks,
(1991).
[3] Bowman K.O., Shenton L.R., Properties of Estimators for the Gamma Distribution,
Union Carbide Corporation Nuclear Divison, Oak Ridge, Tennessee
(1968).
[4] Geyer C.J., Maximum Likelihood in R, (2003).
[5] Belitser E., van de Geer S., On Robust Recursive Nonparametric Curve Estimation,
High Dimensional Probability II(2000), pp. 391-404.
[6] Rice J.A., Mathematical Statistics and Data Analysis, Wadsworth and Brooks,
Pacific Grove, California (1988).
[7] Makridakis S., Wheelright S., McGee V., Forecasting: Methods and Applications,
John Wiley and Sons, Inc., 2nd edition, (1983).
[8] Granger C.W.J., Newbold P., Forecasting Economic Time Series, Academic
Press, Inc., (1977).
[9] Koenker R., Quantile Regression, Cambridge University Press, New York,
(2005).
58
[10] Balakrishnan N., Kateri M., On the Maximum Likelihood Estimation of Parameters
of Weibull Distribution Based on Complete and Censored Data, Elsevier
Statistics and Probability Letters 78 (2008), pp. 2971-2975.
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59
Appendix A
BACKGROUND
A.1 GAMMA DISTRIBUTION
The Gamma Distribution is a two-parameter continuous distribution which is widely
used in statistical applications, such as life testing procedures, process time examinations
and whether modification experiments. It has a scale parameter  (which
is equivalent to a rate parameter 1
 ) and a shape parameter k.
The probabilty density function of Gamma distribution with shape k and scale  is
given by:
f (x) = xk−1 e(−x/)
􀀀(k)
−k, for 0  x < 1 and k,  > 0.
where 􀀀(z) = R1
0 tz−1e−tdt.
If k is an integer, the distribution represents the sum of k independent exponentially
distributed random variables with mean .
The cumulative distriubtion function of Gamma distribution with parameters k and
 is:
F(x) = Z x
0
f (u)du =

(k, x/)
􀀀(k)
where
 is the incomplete gamma function,
(s, t) = Rt
0 us−1e−udu.
The mean and the variance of a Gamma(k, ) distributed random variable are
μ = k and 2 = k2,
60
respectively.
A.2 WEIBULL DISTRIBUTION (2-parameter)
Like the Gamma distribution, 2-parameter Weibull distribution is a continuous
probability distribution. It has a shape parameter k and a scale parameter . It
is widely used to analyse life data. The shape parameter k is related with the failure
rate as follows:
If the failure rate decreases by time then k < 1, if it increases, k > 1. If the failure
rate is constant over time k = 1.
The probability density function of a Weibull random variable with shape k and
scale  is given by
f (x) =
k
 x
k−1
e−(x/)k
, x  0, k > 0,  > 0
The cumulative distribution function is
F(x) = 1 − e−(x/)k
for x  0 with k > 0 shape and  > 0 scale parameters.
The mean of a random variable, X, distributed according to Weibull distribution
with shape k and scale  is:
μ = 􀀀  1 +
1
k !
where 􀀀(z) is the previously defined gamma function. Similarly, the variance of X
is given by
2 = 2􀀀  1 +
2
k !− μ2
A.3 MAXIMUM LIKELIHOOD ESTIMATION
Maximum Likelihood Estimation (MLE) is a statistical method which is used to fit
a statistical model to the given data providing model’s parameters.
61
Suppose that random variables X1, X2, . . . , Xn have a joint density or frequency
function f (x1, x2, . . . , xn|). Given observed values Xi = xi, i = 1, 2, . . . , n, the
likelihood of  as a function of x1, x2, . . . , xnis defined as
　
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