Page 115 - Special Topic Session (STS) - Volume 1

STS410 Abdul Ghapor H. et al.
 5,  10, 15 and 20. For each value of  , the sample size n =20, 30, 50, 70,

130 and 150 are considered for the simulation. With the assumption of   
, the procedures are described below.
Step 1: Generate the values of X variable from the von Mises distribution
of VM  3,2 and for the size of n = 20, 30, 50, 70, 100, 130 and 150; and
  5, 10,15 and 20, respetively. Find Y according to the generated X based
on the model Y    X (mod 2 ) .
Step 2: The variables X and Y are considered with generated random error
terms of where x  X   and y  Y  i for i 1 2 , ,..., n . The error
i
i
i
i
i
terms are  i ~VM , 0 ( )  and  i ~VM , 0 ( )  , respectively where   . 
The variables are fitted to LFRM with parameter estimation as described in
Section 4.2.
Step 3: The values of functional mean circular error cosine (FMCEC) are
calculated for all observations. The estimation of X for this equal error
concentration case is given by 
ˆ
ˆ 
ˆ
X  X ˆ 10  sin x  X 0 i  sin y  X ˆ 0 i .
i
i
ˆ
1 i
ˆ 
cos x  X 0 i  cos y  X ˆ 0 i 
i
i
Step 4: Omit the i observation of the generated data, where i=1, 2, 3, …,n
th
to obtain FMCEC(-i). Repeat this step for all i observations to obtain the set
of values for FMCEC(-i).
Step 5: Calculate the absolute difference between FMCEC and FMCEC(-i).
Then, find value of FDMCEC ( ) i  FMCEC  FMCEC ( ) i  for all i.
Step 6: Repeat steps 1-5 for 500 simulations for each n and  and note
the values 5% upper percentiles of the
FDMCEC  max FMCEC  FMCEC ( ) i  to construct the cut-off equation
based on the significance level of interest. These values of upper percentiles
may be used as the cut-off equations in identifying the outlier for the
unreplicated LFRM for equal error concentration parameters. Table 1 shows
the values of FDMCEC based on 5% upper percentile.

Table 1 The values of 5% percentile of FDMCEC for equal error concentration
n  =5  =10  =15  =20
20 0.0256 0.0103 0.0068 0.0050
30 0.0158 0.0080 0.0050 0.0037
50 0.0198 0.0052 0.0034 0.0024
70 0.0122 0.0036 0.0023 0.0019
100 0.0119 0.0029 0.0019 0.0013
130 0.0091 0.0023 0.0015 0.0011
150 0.0081 0.0018 0.0013 0.0010

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