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this part of the study, we analyze the relations of 10 tests given in Sect. 3,
focusing the rejection regions with size approximately 0
.
01.
Considering all binary sequences of length
n
= 20 and 30, we formed the
rejection regions
R
T
n
of each test, that is, the set of all
n
bit sequences that
fail
T
i
. The upper and lower acceptable limits for test statistics are calculated
so that
α
0
.
01. If the test statistic is more extreme
than the lower or upper limit given in Table 1, the sequence is assumed to be
non-random.
≈|
R
n
|≈
...
≈|
R
1
n
|≈
Table 1.
Lower limits (LLs) and upper limits (ULs) of the test statistics for 20 bit
sequences and corresponding type I error,
α
Test
n
=20
n
=30
LL UL
α
LL UL
α
Frequency
5
15 0.011818 9
21 0.016125
Overlapping Template
0
8
0.014478 0
10 0.017750
Longest Run of Ones
1
8
0.012691 1
9
0.010727
Runs
5
15 0.011818 9
21 0.016125
RW H eight
1
11 0.014395 1
14 0.010446
RW Ex cu rsion
0
7
0.022461 0
10 0.011818
LC
8
12 0.031250 13 17 0.031250
1-error LC
6
10 0.012996 11 15 0.019407
Maximum Order Complexity 2
10 0.017889 4
12 0.016291
Lempel-Ziv
8
10 0.026367 11 13 0.031378
The (
i, j
)
th
entry of Tables 2 and 3 represents the proportion of sequences
that fail
T
i
and
T
j
to the sequences that fail
T
i
,thatis,
|R
i
n
∩R
n
|
|R
i
n
|
,for
n
=20
,
30.
The expected value of this proportion is 0.01 and tables are expected to be
symmetric for larger
n
values. In the tables, the percentages that significantly
deviate (
>
0
.
10) from expected values are highlighted.
According to Table 2 and 3, frequency, overlapping template (with input tem-
plate 111), longest run of ones, random walk height tests and maximum order
complexity tests are closely related. Also, there is a correlation between the
Table 2.
Relation of tests for all sequences of length
n
=20for
α
=0
.
01
Test
Frequency Overlapping Longest Run Runs
RW
RW
LC
1-error MOC Lempel-Ziv
Template
of Ones
Height Excursion
LC
Frequency
-
0.4334
0.2012
0.04341
1
0
0.1011 0.0657 0.1785
0.3689
Overlapping
0.3538
-
0.4391
0.0516 0.4171
0
0.0491 0.0234 0.1699
0.1632
Template
Longest
0.1874
0.5009
-
0.0634 0.2370
0
0.0385 0.0225 0.1740
0.1040
Run of Ones
Runs
0.04341
0.0632
0.0681
-
0.0618
0.1933
0.0933 0.0531 0.1635
0.1228
RW
0.8210
0.4195
0.2089
0.0507
-
0
0.1050 0.0542 0.1972
0.3108
Height
RW
0
0
0
0.1017
0
-
0.0351 0.0160 0.0369
0.0348
Excursion
LC
0.0383
0.0227
0.0156
0.0353 0.0484
0.0252
-
0.1339 0.1737
0.0424
1-error LC
0.0598
0.0260
0.0220
0.0483 0.0601
0.0277
0.3220
-
0.0961
0.0643
MOC
0.1179
0.1375
0.1235
0.1080 0.1586
0.0464
0.3035 0.0698
-
0.0951
Lempel-Ziv
0.1654
0.0896
0.05006
0.0550 0.1697
0.0296
0.05031 0.0317 0.0645
-
