Performance of some modified test statistics for testing the population
signal to noise ratio: Simulation and Applications
SAMANTHA MENENDEZ, FLORENCE GEORGE, AND B M GOLAM KIBRIA
Department of Mathematics and Statistics
Florida International University
11200 SW 8th St, Miami, FL 33199
UNITED STATES OF AMERICA
Abstract: - This paper considers fifteen existing and proposed test statistics for testing the population SNR. A
theoretical comparison among the test statistics is not possible, a Monte Carlo simulation study has been
conducted. The performance of the test statistics is based on the empirical size and power of the tests considering
a significance level 0.05. The simulation study resulted in some existing and proposed methods performing well
in some conditions. For testing SNR=1, 2 and 5, Method 10 performed the best in all simulation conditions.
However, for testing SNR=0.5 the proposed Method 12 and for testing SNR=10, the proposed Method 15
performed the best. Two real life data sets are analyzed to illustrate the performance of the test statistics.
Key-words: Normal Distribution; Gamma Distribution; Power of the test; Signal to Noise Ratio; Simulation
Study; Size of the test
Received: June 7, 2022. Revised: August 12, 2023. Accepted: September 14, 2023. Published: October 3, 2023.
1 Introduction
Mean is known as an expected value that
represents the central tendency of a probability
distribution which is known as the average of a
data set. When we are interested in the measure of
how dispersed the distribution of data is in relation
to the mean, we refer to the standard deviation.
Standard deviation can vary between high and low
values, when the standard deviation is high the
values fall far from the mean and when standard
deviation is low the values are closer to the mean.
Standard deviation and noise relate vice versa as
variability within a gathered data sample.
Improper filtering where data was measured, and
random errors can be introduced can result in
unexpected noise. Noise has two main sources;
errors introduced by measurement tools and
random errors introduced by processing. Noise can
be any data that has been received and changed in
a manner that it cannot be read which adversely
affects results of any data analysis.
While the mean describes what is being measured,
standard deviation represents noise. Signal to noise
ratio (SNR) is a measure of the strength of the
desired data relative to the undesigned data or
signal known as noise. Population SNR is equal to
the population mean divided by the population
standard deviation (
). In practice, commonly in
digital communications and image processing, a
high SNR means that the signal strength is stronger
in relation to the noise level. Having higher SNR
means there is more useful information than
unwanted noise data in the output. The application
on SNR can be found in John (2007) and Tania
(2008) among others.
SNR is the reciprocal of the commonly used
coefficient of variation (CV) which is unitless and
is defined as the standard deviation divided by the
mean. Since SNR is the reciprocal, when there is
better data there is a higher value for the SNR and
a lower value for the CV. This can also be seen
when, putting SNR into practice, the larger the
standard deviation (noise) the smaller the SNR.
Therefore, it is possible to construct both
confidence interval and hypopaper testing from
CV for SNR. The information on testing the
population SNR is limited. Nevertheless, there are
various methods available for estimating the
confidence interval (CI) for a population CV, such
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
83
Volume 19, 2023
as parametric, nonparametric, modified, and
bootstrapping (Banik and Kibria, 2010). For more
information on the CI for the CV, we refer
Koopmans et al. (1964), Miller (1991), Sharma
and Krishna (1994), McKay (1932), Vangel
(1996), and Curto and Pinto (2009), George and
Kibria (2012), Banik et al. (2012), Albatineh et al.
(2014), and recently Abu-Shaweish et al. (2019)
among others. First Kibria and George (2014)
consider various test statistics based on the
confidence interval of CV for testing the
population SNR.
The objective of this paper is to review and
propose some test statistics for testing the
population SNR based on parametric, and
modified methods for confidence intervals for
SNR. The organization of the paper is as follows:
We will review and propose some test statistics for
testing the SNR in Section 2. A simulation will be
conducted in Section 3. Two real life data are
analyzed in Section. Finally, some concluding
remarks are given in Section 5.
2 Statistical Methodology
Let be an independently and identically
distributed (iid) random sample of size from a
population from a population with finite mean, , and
finite variance, . Let be the sample mean and be
the sample standard deviation. Then
would
be the estimated values of the population SNR (
) and
would be the estimated values of the
population CV (
). The objective of this paper is to
test the population SNR. The null and alternative
hypothesis are defined as follows:
Following Kibria and George (2014), we have
considered the following eleven (11) promising test
statistics, For details about these tests, we refer to
Kibria and George (2014).
2.1 Miller (1991) Method
Method 1. Miller demonstrated that the estimator,
has an approximate normal distribution with
mean
and variance of (1/(n-1))(
[0.5+(
].
Then the approximate upper and lower confidence
limits the population SNR can be constructed.
From the confidence intervals, we will reject the
null hypothesis when is less than the lower
limit or greater than the upper limit. The upper and
lower limits for are given as follows:
(1)
We will reject the null hypothesis when SNR is
more than the above upper limit or less than the
lower limit.
2.2 Sharma and Krishna’s (1994) Method
Method 2. Sharma and Krishna (1994) developed
the asymptotic sampling distribution of the inverse
of the coefficient of variation for making statistical
inferences about population coefficient of
variation (CV). Following Sharma and Krishna
(1994), based on the CI for inverted SNR, we
reject the null hypothesis when is less than
the lower limit or greater than the upper limit,
which are given below:
(2)
2.3 Curto and Pinto’s (2009) Method
Method 3. Using the approximate confidence
interval (CI) for the population inverted SNR, we
will reject the null hypothesis, whe
(3)
2.4 McKay’s (1932) Method
Method 4. Using the approximate confidence
interval (CI) for the population inverted SNR, we
will reject the null hypothesis, when
< [
> [
(4)
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
84
Volume 19, 2023
where
and
are (α/2)th and (1-
α/2)th percentile points for a chi-square distribution
with (n-1) degrees of freedom
2.5 Modified McKay Confidence Interval
(MMcK)
Method 5. Modified McKay confidence interval is
Vangel (1996) modifying McKay’s original 1932
interval. Using the approximate confidence
interval (CI) for the population inverted SNR, we
will reject the null hypothesis, when
< [
> [
(5)
2.6 Panichkitkosolkul’s (2009) Method
Method 6. Panichkitkosolkul’s (2009) modified
the Modified Mckay (in section 2.5) method by
replacing the sample inverted SNR with the
maximum likelihood estimator for a normal
distribution, where
. Using the
approximate confidence interval (CI) for the
population inverted SNR, we will reject the null
hypothesis, when
< [
> [
(6)
2.7 Median Modified Miller
For skewed data the median describes the center of
the distribution better than the mean (Kibria
(2006), and Shi and Kibria (2007)) expressing that
it makes more sense to measure sample variability
in terms of median. The following median
modifications are proposed by Kibria and George
(2014) to improve performance for skewed
distributions and represent both parametric and
nonparametric methods.
Method 7. Null hypothesis will be rejected, when,
(7)
where
is the
modified sample variance.
2.8 Median Modification of McKay
Method 8. Reject the null hypothesis when,
< [
> [
(8)
2.9 Median Modification of Modified
McKay
Method 9. Reject the null hypothesis when,
< [
> [
(9)
2.10 Median Modification of Curto and
Pinto (2009)
Method 10. Reject the null hypothesis when,
(10)
2.11 Kibria and George (2014)
Kibria and George (2014) developed Method 11
based on the normality assumption. First they
developed the confidence interval for inverse of
the population variance and then after some
algebraic simplification they obtained the
confidence interval for population SNR. From the
confidence interval they found the upper and lower
limits for the SNR0, which are given below.
Method 11. Reject the null hypothesis when,
(11)
Now, we want to propose some new test statistics.
Kibria and George (2014) modified Sharma and
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
85
Volume 19, 2023
Krishna’s method using bootstrap technique,
which is computationally expensive and time
consuming and did not improve that much. In this
paper, we will modify Sharma and Krishna’s
method using the modified standard deviation and
MAD. We also modified Miller’s methods and
provided them in the following subsections.
2.12 Modified Kibria and George (2014)
Method 12. Reject the null hypothesis when,
or
(12)
where
and
2.13 Modification of Miller (1991)
Method 13. Reject the null hypothesis when
(13)
where,
This formula is aa modification of Rousseeuw
and Croux (1993).
2.14 Modification of Sharma and
Krishna’s (1994) Method
Method 14. We will reject the null hypothesis
when is less than the lower limit or greater
than the upper limit.
2.15 Median Modification of Sharma and
Krishna’s (1994) Method
Method 15. We reject the null hypothesis when
is less than the lower limit or greater than
the upper limit.
Since a theoretical comparison among the test
statistics is not possible, a Monte Carlo simulation
study using R 4.2.1 to find the empirical size and
power of the tests is conducted in the following
Section.
3. Simulation Study
3.1 Simulation Techniques
In this section, we will compare the performance
of the test statistics that are given in section 2.
Simulation study is done under the symmetric
(normal) and skewed (Gamma) distributional
conditions.
The performance of the test statistics is examined
for both small and large sample sizes (n = 30, 50,
100). Simulation will be run for 5,000. The
simulation results are presented only for α=0.05, a
widely used significance level. The simulation
results of empirical type I error rate and the power
of the tests are presented in Tables 3.1 to 3.30 for
various parametric conditions. In all Tables (3.1 to
3.30), column number 6 represents the empirical
size and the rest of the columns are empirical
power of the tests. For more on simulation studies,
we refer the readers to Shi and Kibria (2007),
Banik and Kibria (2010), Kibria and George
(2014), Andrew et al. (2015) and very recently
Panichkitosolkul (2022) among others.
3.2 Simulation Results Discussion
3.2.1 When Data is Generated from a Normal
Distribution.
We will consider a test is good test when the test
attains the empirical nominal level at most 0.06
[(05-1.96*sqrt((.95*.05)/5000) =0.04 and
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
86
Volume 19, 2023
05+1.96*sqrt((.95*.05)/5000)=0.06)] for 5% level
of significance test. When we review Tables 3.1 to
3.3, we can see that for testing SNR=0.5, and
sample sizes 30 and 50, none of the tests but
Methods 2 and 15 achieves the empirical nominal
level 0.05. Methods 2 and 15 produce high powers
on both ends. However, when the sample size is
100, Methods 1, 2, 3, 7, 10 and 15 achieved the
nominal level 0.05. When we review Tables 3.4 to
3.6, we can see that for testing SNR=1, and sample
size 30, none of the tests achieves the empirical
nominal level 0.05 and cannot be used for testing
for SNR. However, when the sample sizes are 50
and 100, all methods except methods 2, 11 to 15
achieved the nominal level 0.05. If we review
Tables 3.5 and 3.6, and consider lower and upper
tail powers, it appears that methods 1, 3, 7 and 10
are performing better than Methods 4, 5, 6, 8 and
9. However, method 10 proposed by Kibria and
George (2014) has performed the best in the sense
of attaining the nominal size and high empirical
power. When we review Tables 3.7 to 3.9, we can
see that for testing SNR=2, and for all small
sample sizes, all methods except methods 2, 11 to
15 achieved the nominal level 0.05 and gave high
empirical power. Again, method 10 has performed
the best in the sense of attaining the nominal size
and high empirical power. When we review
Tables 3.10 to 3.12, we can see that for testing
SNR=5, all methods except 2, 13 to 15 achieved
the nominal level 0.05 for sample sizes 50 and 100.
Proposed method 12 performed better than
Method 11 for testing the high values of SNR,
while method 11 for low values. When we review
Tables 3.13 to 3.15, we can see that for testing
SNR=10, all methods except 2, 13 to 15 achieved
the nominal level 0.05 for all sample sizes.
However, the proposed method 12 performed
better than Method 11 for testing the high values
of SNR, while method 11 performed better than
Method 12 for low values of SNR.
3.2.2 When Data is Generated from a Gamma
Distribution.
The simulated empirical sizes and powers are
tabulated in Tables 3.16 to 3.30 when data is
generated from a Gamma distribution. When we
review Table 3.16, we can see that for testing
SNR=0.5, and small sample size 30, Methods 2, 7,
10, and proposed methods 13- 15 attained the
nominal level. Proposed method 15 performed the
best among all methods. Tables 3.17 and 3.18
indicate that the proposed Method 15 performed
the best. When we review Tables 3.19 to 3.21, we
can see that for testing SNR=1, all methods except
Methods 11-14 attained the nominal level.
Proposed method 15 produces the highest power at
the high value of SNR and methods 1 & 2 for the
low values. When we review Tables 3.22-324, we
can see that for testing SNR=2, and n=30, 50 and
100, methods 1, 3, 4 through 10 attained the
nominal level and produced high power for large
sample sizes.
Method 10 performed the best in the sense of
highest power at both ends. When we review
Tables 3.25-3.27, we can see that for testing
SNR=5, Methods 1, 3, 4, 5, 7 through 12 attained
the nominal level for all sample sizes. Proposed
method 15 produces the highest power at the high
value of SNR and method 3 for the low values. All
empirical powers are lower than 55%. When we
review Tables 3.28-3.30, we can see that for
testing SNR=10, Methods 1, 3, 4, 5, 7 through 12
attained the nominal level for all sample sizes.
Proposed method 15 produces the highest power at
the high value of SNR, while Method 3 for the low
values.
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
87
Volume 19, 2023
Table 3.1: Empirical type I error rate and power of tests for Normal (2.5, 25), SNR = 0.5, n = 30
SNR0
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
Method1
1.000
0.973
0.714
0.491
0.354
0.287
0.263
0.256
0.253
Method2
0.593
0.395
0.223
0.122
0.072
0.096
0.200
0.379
0.577
Method3
1.000
0.964
0.707
0.485
0.345
0.277
0.252
0.244
0.241
Method4
1.000
1.000
1.000
1.000
1.000
1.000
0.397
0.122
0.023
Method5
1.000
1.000
1.000
1.000
1.000
0.530
0.177
0.041
0.011
Method6
1.000
1.000
1.000
1.000
1.000
0.569
0.195
0.064
0.013
Method7
1.000
0.972
0.711
0.490
0.354
0.291
0.268
0.260
0.259
Method8
1.000
1.000
1.000
1.000
1.000
1.000
0.354
0.087
0.024
Method9
1.000
1.000
1.000
1.000
1.000
0.531
0.173
0.035
0.012
Method10
1.000
0.964
0.703
0.483
0.346
0.280
0.256
0.249
0.247
Method11
0.989
0.927
0.775
0.593
0.494
0.524
0.626
0.749
0.846
Method12
0.989
0.925
0.769
0.585
0.493
0.528
0.634
0.756
0.852
Method13
1.000
0.972
0.727
0.525
0.397
0.325
0.288
0.273
0.266
Method14
0.589
0.403
0.255
0.153
0.103
0.124
0.228
0.394
0.577
Method15
0.585
0.387
0.216
0.115
0.070
0.096
0.202
0.384
0.587
Table 3.2: Empirical type I error rate and power of tests for Normal (2.5, 25), SNR = 0.5, n = 50
SNR0
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
Method1
1.000
0.901
0.609
0.319
0.152
0.090
0.073
0.069
0.068
Method2
0.798
0.561
0.309
0.129
0.067
0.122
0.301
0.561
0.791
Method3
1.000
0.900
0.609
0.319
0.149
0.087
0.070
0.065
0.065
Method4
1.000
1.000
1.000
1.000
1.000
0.328
0.082
0.011
0.004
Method5
1.000
1.000
1.000
1.000
0.688
0.189
0.044
0.007
0.002
Method6
1.000
1.000
1.000
1.000
0.707
0.192
0.050
0.007
0.002
Method7
1.000
0.898
0.605
0.313
0.149
0.089
0.074
0.070
0.070
Method8
1.000
1.000
1.000
1.000
1.000
0.332
0.079
0.012
0.004
Method9
1.000
1.000
1.000
1.000
0.693
0.185
0.046
0.007
0.002
Method10
1.000
0.898
0.605
0.313
0.146
0.087
0.072
0.067
0.067
Method11
0.999
0.968
0.845
0.626
0.495
0.562
0.723
0.865
0.944
Method12
0.999
0.968
0.841
0.623
0.496
0.567
0.729
0.870
0.947
Method13
1.000
0.891
0.604
0.338
0.189
0.109
0.086
0.078
0.074
Method14
0.789
0.549
0.314
0.162
0.094
0.150
0.318
0.567
0.774
Method15
0.793
0.552
0.300
0.126
0.064
0.124
0.305
0.570
0.795
Table 3.3: Empirical type I error rate and power of tests for Normal (2.5, 25), SNR = 0.5, n = 100
SNR0
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
Method1
1.000
0.959
0.724
0.317
0.066
0.007
0.002
0.002
0.677
Method2
0.980
0.841
0.523
0.187
0.062
0.181
0.507
0.837
0.968
Method3
1.000
0.959
0.725
0.320
0.066
0.007
0.002
0.001
0.691
Method4
1.000
1.000
1.000
1.000
0.230
0.021
0.001
0.000
0.000
Method5
1.000
1.000
1.000
0.829
0.153
0.012
0.001
0.000
0.000
Method6
1.000
1.000
1.000
0.837
0.153
0.014
0.001
0.000
0.000
Method7
1.000
0.958
0.720
0.312
0.063
0.007
0.002
0.002
0.683
Method8
1.000
1.000
1.000
1.000
0.231
0.020
0.001
0.000
0.000
Method9
1.000
1.000
1.000
0.830
0.153
0.013
0.001
0.000
0.000
Method10
1.000
0.959
0.721
0.314
0.064
0.007
0.002
0.002
0.695
Method11
1.000
0.998
0.943
0.709
0.504
0.629
0.861
0.968
0.996
Method12
1.000
0.998
0.942
0.705
0.503
0.631
0.866
0.969
0.996
Method13
1.000
0.955
0.711
0.333
0.095
0.018
0.003
0.002
0.657
Method14
0.975
0.836
0.522
0.219
0.091
0.199
0.510
0.801
0.947
Method15
0.979
0.837
0.518
0.183
0.060
0.183
0.513
0.840
0.969
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
88
Volume 19, 2023
Table 3.4: Empirical type I error rate and power of tests for Normal (5, 25), SNR = 1, n = 30
SNR0
0.400
0.600
0.700
0.800
1.000
1.200
1.300
1.400
1.500
Method1
0.962
0.705
0.491
0.298
0.082
0.016
0.007
0.003
0.001
Method2
0.882
0.584
0.410
0.258
0.122
0.246
0.398
0.566
0.723
Method3
0.964
0.710
0.498
0.305
0.084
0.017
0.007
0.003
0.001
Method4
1.000
1.000
0.741
0.456
0.128
0.025
0.011
0.004
0.001
Method5
1.000
0.877
0.610
0.366
0.102
0.020
0.008
0.003
0.001
Method6
1.000
0.888
0.629
0.396
0.114
0.023
0.010
0.004
0.001
Method7
0.960
0.694
0.477
0.285
0.076
0.015
0.006
0.002
0.001
Method8
1.000
1.000
0.736
0.445
0.123
0.023
0.010
0.003
0.001
Method9
1.000
0.868
0.596
0.354
0.097
0.019
0.008
0.002
0.001
Method10
0.961
0.700
0.484
0.290
0.079
0.015
0.007
0.002
0.001
Method11
0.992
0.831
0.627
0.415
0.252
0.425
0.557
0.680
0.787
Method12
0.991
0.820
0.613
0.403
0.255
0.441
0.571
0.696
0.799
Method13
0.942
0.678
0.500
0.340
0.134
0.051
0.031
0.019
0.012
Method14
0.850
0.581
0.432
0.311
0.195
0.320
0.439
0.573
0.693
Method15
0.877
0.570
0.396
0.247
0.120
0.251
0.411
0.580
0.736
Table 3.5: Empirical type I error rate and power of tests for Normal (5, 25), SNR = 1, n = 50
SNR0
0.400
0.600
0.700
0.800
1.000
1.200
1.300
1.400
1.500
Method1
0.995
0.850
0.632
0.373
0.064
0.005
0.061
0.217
0.434
Method2
0.981
0.774
0.568
0.341
0.115
0.323
0.543
0.745
0.882
Method3
0.995
0.855
0.635
0.378
0.065
0.008
0.071
0.233
0.453
Method4
1.000
0.882
0.654
0.380
0.068
0.006
0.002
0.001
0.000
Method5
1.000
0.844
0.614
0.352
0.059
0.005
0.002
0.001
0.000
Method6
1.000
0.854
0.636
0.376
0.069
0.006
0.002
0.001
0.000
Method7
0.994
0.842
0.622
0.358
0.059
0.005
0.064
0.226
0.445
Method8
1.000
0.877
0.644
0.366
0.063
0.006
0.002
0.001
0.000
Method9
1.000
0.836
0.601
0.340
0.055
0.005
0.002
0.001
0.000
Method10
0.995
0.846
0.627
0.364
0.060
0.008
0.074
0.241
0.465
Method11
0.999
0.945
0.785
0.535
0.248
0.501
0.678
0.824
0.909
Method12
0.999
0.943
0.777
0.523
0.251
0.514
0.689
0.831
0.913
Method13
0.988
0.808
0.612
0.397
0.112
0.023
0.098
0.256
0.445
Method14
0.964
0.741
0.558
0.378
0.188
0.375
0.541
0.705
0.830
Method15
0.980
0.765
0.557
0.330
0.111
0.331
0.554
0.757
0.889
Table 3.6: Empirical type I error rate and power of tests for Normal (5, 25), SNR = 1, n = 100
SNR0
0.400
0.600
0.700
0.800
1.000
1.200
1.300
1.400
1.500
Method1
1.000
0.979
0.842
0.533
0.055
0.160
0.428
0.728
0.908
Method2
1.000
0.963
0.817
0.528
0.112
0.499
0.791
0.936
0.985
Method3
1.000
0.979
0.845
0.537
0.056
0.164
0.433
0.735
0.908
Method4
1.000
0.964
0.812
0.499
0.050
0.000
0.000
0.132
0.695
Method5
1.000
0.960
0.799
0.481
0.047
0.000
0.000
0.371
0.778
Method6
1.000
0.963
0.810
0.498
0.051
0.000
0.000
0.357
0.767
Method7
1.000
0.978
0.838
0.526
0.053
0.164
0.436
0.738
0.910
Method8
1.000
0.963
0.807
0.489
0.048
0.000
0.000
0.134
0.704
Method9
1.000
0.959
0.795
0.472
0.045
0.000
0.000
0.383
0.785
Method10
1.000
0.978
0.841
0.529
0.054
0.169
0.441
0.743
0.914
Method11
1.000
0.998
0.951
0.743
0.250
0.660
0.864
0.957
0.987
Method12
1.000
0.997
0.949
0.736
0.251
0.666
0.869
0.959
0.988
Method13
1.000
0.957
0.799
0.523
0.116
0.216
0.453
0.677
0.844
Method14
1.000
0.935
0.775
0.519
0.200
0.520
0.740
0.883
0.954
Method15
1.000
0.961
0.812
0.519
0.111
0.508
0.797
0.940
0.985
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
89
Volume 19, 2023
Table 3.7: Empirical type I error rate and power of tests for Normal (10, 25), SNR = 2, n = 30
SNR0
1.200
1.400
1.600
1.800
2.000
2.200
2.400
2.600
2.800
Method1
0.926
0.712
0.413
0.186
0.069
0.028
0.061
0.169
0.349
Method2
0.951
0.813
0.584
0.365
0.269
0.355
0.555
0.745
0.877
Method3
0.929
0.719
0.421
0.191
0.073
0.032
0.069
0.187
0.369
Method4
0.931
0.724
0.426
0.195
0.073
0.022
0.008
0.056
0.218
Method5
0.925
0.715
0.415
0.190
0.072
0.021
0.022
0.109
0.285
Method6
0.939
0.748
0.457
0.220
0.084
0.027
0.020
0.094
0.252
Method7
0.916
0.692
0.394
0.174
0.063
0.028
0.066
0.183
0.367
Method8
0.922
0.708
0.409
0.181
0.068
0.020
0.007
0.062
0.232
Method9
0.915
0.694
0.398
0.176
0.065
0.019
0.025
0.119
0.306
Method10
0.920
0.703
0.403
0.178
0.067
0.033
0.076
0.201
0.386
Method11
0.923
0.678
0.358
0.152
0.107
0.193
0.360
0.557
0.724
Method12
0.914
0.658
0.337
0.145
0.111
0.208
0.379
0.576
0.744
Method13
0.854
0.655
0.442
0.276
0.161
0.117
0.161
0.267
0.402
Method14
0.890
0.740
0.568
0.451
0.431
0.503
0.619
0.721
0.816
Method15
0.945
0.799
0.561
0.346
0.265
0.365
0.570
0.762
0.885
Table 3.8: Empirical type I error rate and power of tests for Normal (10, 25), SNR = 2, n = 50
SNR0
1.200
1.400
1.600
1.800
2.000
2.200
2.400
2.600
2.800
Method1
0.990
0.866
0.532
0.206
0.058
0.050
0.178
0.429
0.688
Method2
0.994
0.934
0.718
0.409
0.256
0.404
0.682
0.882
0.964
Method3
0.990
0.871
0.539
0.210
0.061
0.053
0.184
0.441
0.700
Method4
0.990
0.873
0.542
0.212
0.058
0.025
0.112
0.344
0.626
Method5
0.989
0.862
0.531
0.206
0.057
0.034
0.142
0.382
0.657
Method6
0.991
0.883
0.566
0.231
0.065
0.031
0.121
0.350
0.623
Method7
0.988
0.856
0.513
0.192
0.056
0.054
0.188
0.447
0.703
Method8
0.988
0.863
0.523
0.201
0.056
0.026
0.119
0.362
0.646
Method9
0.988
0.853
0.512
0.194
0.054
0.036
0.152
0.405
0.677
Method10
0.988
0.861
0.520
0.198
0.058
0.056
0.198
0.459
0.713
Method11
0.993
0.879
0.523
0.193
0.108
0.236
0.495
0.737
0.888
Method12
0.992
0.870
0.504
0.182
0.111
0.251
0.514
0.749
0.896
Method13
0.954
0.780
0.530
0.293
0.163
0.164
0.279
0.452
0.634
Method14
0.969
0.858
0.657
0.477
0.430
0.530
0.689
0.817
0.900
Method15
0.994
0.928
0.703
0.395
0.255
0.415
0.699
0.889
0.967
Table 3.9: Empirical type I error rate and power of tests for Normal (10,25), SNR = 2, n = 100
SNR0
1.200
1.400
1.600
1.800
2.000
2.200
2.400
2.600
2.800
Method1
1.000
0.989
0.777
0.301
0.052
0.122
0.466
0.808
0.961
Method2
1.000
0.997
0.903
0.526
0.262
0.523
0.856
0.980
0.998
Method3
1.000
0.989
0.779
0.304
0.053
0.124
0.471
0.810
0.962
Method4
1.000
0.989
0.778
0.304
0.049
0.087
0.419
0.779
0.956
Method5
1.000
0.988
0.771
0.298
0.049
0.097
0.436
0.793
0.960
Method6
1.000
0.990
0.790
0.314
0.053
0.089
0.415
0.773
0.954
Method7
1.000
0.987
0.767
0.292
0.050
0.129
0.479
0.814
0.963
Method8
1.000
0.987
0.768
0.295
0.047
0.092
0.431
0.790
0.960
Method9
1.000
0.986
0.763
0.289
0.046
0.104
0.447
0.802
0.961
Method10
1.000
0.988
0.769
0.295
0.051
0.132
0.483
0.818
0.964
Method11
1.000
0.994
0.811
0.319
0.105
0.336
0.708
0.929
0.989
Method12
1.000
0.993
0.802
0.312
0.105
0.348
0.718
0.933
0.990
Method13
0.999
0.939
0.708
0.365
0.169
0.237
0.478
0.725
0.889
Method14
0.999
0.967
0.817
0.556
0.439
0.576
0.789
0.926
0.978
Method15
1.000
0.996
0.897
0.512
0.262
0.534
0.863
0.981
0.998
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
90
Volume 19, 2023
Table 3.10: Empirical type I error rate and power of tests for Normal (25, 25), SNR = 5, n = 30
SNR0
4.000
4.400
4.600
4.800
5.000
5.200
5.400
5.600
6.400
Method1
0.489
0.246
0.165
0.105
0.069
0.048
0.047
0.066
0.297
Method2
0.869
0.707
0.643
0.610
0.611
0.631
0.680
0.729
0.916
Method3
0.498
0.254
0.172
0.109
0.072
0.050
0.052
0.072
0.313
Method4
0.495
0.252
0.170
0.109
0.070
0.048
0.044
0.063
0.290
Method5
0.493
0.250
0.169
0.108
0.070
0.049
0.048
0.067
0.299
Method6
0.544
0.292
0.197
0.127
0.084
0.056
0.045
0.058
0.257
Method7
0.467
0.227
0.153
0.098
0.064
0.046
0.052
0.072
0.319
Method8
0.472
0.231
0.157
0.100
0.065
0.047
0.049
0.067
0.313
Method9
0.470
0.230
0.156
0.099
0.066
0.047
0.051
0.072
0.321
Method10
0.473
0.235
0.158
0.102
0.067
0.051
0.056
0.079
0.336
Method11
0.329
0.139
0.090
0.062
0.062
0.083
0.122
0.184
0.528
Method12
0.312
0.131
0.083
0.060
0.065
0.087
0.136
0.205
0.548
Method13
0.503
0.337
0.281
0.239
0.214
0.198
0.200
0.212
0.377
Method14
0.818
0.750
0.730
0.726
0.737
0.745
0.769
0.798
0.888
Method15
0.854
0.691
0.631
0.612
0.617
0.637
0.691
0.741
0.922
Table 3.11: Empirical type I error rate and power of tests for Normal (25,25), SNR = 5, n = 50
SNR0
4.000
4.400
4.600
4.800
5.000
5.200
5.400
5.600
6.400
Method1
0.659
0.312
0.181
0.100
0.057
0.044
0.067
0.109
0.541
Method2
0.944
0.785
0.689
0.614
0.598
0.623
0.696
0.774
0.970
Method3
0.667
0.320
0.186
0.104
0.059
0.047
0.070
0.114
0.548
Method4
0.667
0.318
0.186
0.103
0.058
0.044
0.064
0.107
0.536
Method5
0.665
0.315
0.185
0.102
0.058
0.045
0.066
0.109
0.541
Method6
0.701
0.351
0.207
0.118
0.064
0.045
0.058
0.095
0.501
Method7
0.640
0.297
0.170
0.094
0.053
0.046
0.073
0.119
0.560
Method8
0.645
0.302
0.174
0.096
0.053
0.045
0.071
0.117
0.555
Method9
0.644
0.300
0.173
0.095
0.054
0.046
0.072
0.119
0.560
Method10
0.646
0.302
0.175
0.097
0.056
0.047
0.076
0.124
0.568
Method11
0.536
0.206
0.115
0.064
0.057
0.085
0.146
0.240
0.714
Method12
0.516
0.193
0.108
0.062
0.061
0.092
0.158
0.257
0.729
Method13
0.605
0.389
0.300
0.236
0.204
0.197
0.216
0.256
0.530
Method14
0.880
0.774
0.737
0.731
0.735
0.756
0.775
0.804
0.928
Method15
0.937
0.771
0.672
0.605
0.596
0.634
0.707
0.782
0.975
Table 3.12: Empirical type I error rate and power of tests for Normal (25, 25), SNR = 5, n = 100
SNR0
4.000
4.400
4.600
4.800
5.000
5.200
5.400
5.600
6.400
Method1
0.892
0.480
0.263
0.130
0.064
0.059
0.126
0.263
0.867
Method2
0.993
0.892
0.761
0.634
0.598
0.667
0.777
0.862
0.997
Method3
0.894
0.483
0.265
0.132
0.065
0.061
0.129
0.268
0.868
Method4
0.895
0.483
0.265
0.131
0.064
0.058
0.124
0.259
0.866
Method5
0.894
0.482
0.263
0.130
0.064
0.059
0.126
0.262
0.867
Method6
0.906
0.506
0.284
0.145
0.071
0.053
0.115
0.241
0.852
Method7
0.885
0.467
0.255
0.126
0.061
0.061
0.134
0.276
0.872
Method8
0.887
0.470
0.256
0.127
0.061
0.059
0.131
0.273
0.871
Method9
0.886
0.469
0.255
0.127
0.061
0.061
0.133
0.275
0.871
Method10
0.886
0.470
0.256
0.128
0.062
0.064
0.136
0.281
0.873
Method11
0.852
0.403
0.212
0.104
0.061
0.106
0.226
0.397
0.917
Method12
0.843
0.390
0.200
0.100
0.060
0.113
0.238
0.411
0.923
Method13
0.786
0.496
0.365
0.276
0.225
0.225
0.289
0.366
0.757
Method14
0.951
0.833
0.775
0.751
0.737
0.765
0.810
0.849
0.966
Method15
0.992
0.885
0.751
0.628
0.598
0.672
0.785
0.866
0.998
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
91
Volume 19, 2023
Table 3.13: Empirical type I error rate and power of tests for Normal (250, 25), SNR = 10, n = 30
SNR0
6.000
7.000
8.000
9.000
10.000
11.000
12.000
13.000
14.000
Method1
0.991
0.869
0.503
0.202
0.067
0.059
0.146
0.359
0.572
Method2
1.000
0.995
0.946
0.825
0.794
0.843
0.921
0.970
0.989
Method3
0.992
0.873
0.513
0.208
0.070
0.065
0.159
0.376
0.588
Method4
0.991
0.871
0.508
0.205
0.068
0.060
0.147
0.362
0.574
Method5
0.991
0.871
0.508
0.205
0.068
0.062
0.149
0.364
0.575
Method6
0.995
0.897
0.555
0.240
0.081
0.054
0.123
0.315
0.530
Method7
0.989
0.850
0.482
0.185
0.061
0.064
0.162
0.385
0.596
Method8
0.989
0.854
0.488
0.189
0.063
0.065
0.163
0.389
0.598
Method9
0.989
0.853
0.487
0.189
0.063
0.065
0.165
0.389
0.601
Method10
0.990
0.856
0.491
0.192
0.065
0.068
0.176
0.401
0.611
Method11
0.963
0.707
0.330
0.104
0.053
0.128
0.343
0.570
0.765
Method12
0.957
0.682
0.311
0.096
0.057
0.142
0.369
0.593
0.779
Method13
0.938
0.752
0.515
0.329
0.219
0.209
0.301
0.423
0.548
Method14
0.996
0.965
0.905
0.867
0.868
0.897
0.914
0.947
0.969
Method15
1.000
0.994
0.942
0.813
0.796
0.849
0.928
0.974
0.990
Table 3.14: Empirical type I error rate and power of tests for Normal (250, 25), SNR = 10, n = 50
SNR0
6.000
7.000
8.000
9.000
10.000
11.000
12.000
13.000
14.000
Method1
0.999
0.969
0.693
0.262
0.065
0.088
0.308
0.616
0.838
Method2
1.000
0.999
0.983
0.875
0.776
0.867
0.953
0.989
0.998
Method3
0.999
0.971
0.697
0.268
0.068
0.093
0.318
0.623
0.844
Method4
0.999
0.970
0.696
0.265
0.067
0.088
0.308
0.616
0.838
Method5
0.999
0.970
0.695
0.265
0.066
0.089
0.310
0.617
0.839
Method6
0.999
0.977
0.729
0.296
0.075
0.080
0.276
0.583
0.811
Method7
0.999
0.965
0.671
0.245
0.062
0.096
0.329
0.632
0.847
Method8
0.999
0.965
0.676
0.248
0.063
0.097
0.330
0.633
0.848
Method9
0.999
0.965
0.674
0.248
0.063
0.098
0.331
0.634
0.849
Method10
0.999
0.965
0.678
0.252
0.065
0.102
0.338
0.640
0.852
Method11
0.999
0.931
0.546
0.161
0.056
0.174
0.476
0.754
0.914
Method12
0.999
0.923
0.525
0.153
0.058
0.187
0.498
0.772
0.921
Method13
0.989
0.883
0.632
0.360
0.220
0.251
0.398
0.576
0.730
Method14
0.999
0.991
0.940
0.885
0.861
0.892
0.939
0.967
0.983
Method15
1.000
0.999
0.981
0.868
0.781
0.874
0.957
0.990
0.998
Table 3.15: Empirical type I error rate and power of tests for Normal (250, 25), SNR = 10, n = 100
SNR0
6.000
7.000
8.000
9.000
10.000
11.000
12.000
13.000
14.000
Method1
1.000
0.999
0.917
0.369
0.055
0.194
0.654
0.926
0.993
Method2
1.000
1.000
0.999
0.927
0.776
0.914
0.989
1.000
1.000
Method3
1.000
0.999
0.920
0.374
0.057
0.197
0.659
0.926
0.993
Method4
1.000
0.999
0.919
0.372
0.056
0.194
0.653
0.926
0.993
Method5
1.000
0.999
0.919
0.372
0.056
0.195
0.655
0.926
0.993
Method6
1.000
0.999
0.930
0.398
0.059
0.175
0.629
0.919
0.991
Method7
1.000
0.999
0.910
0.358
0.055
0.205
0.667
0.929
0.993
Method8
1.000
0.999
0.911
0.359
0.056
0.205
0.667
0.929
0.993
Method9
1.000
0.999
0.911
0.359
0.056
0.206
0.668
0.929
0.993
Method10
1.000
0.999
0.911
0.361
0.057
0.209
0.672
0.930
0.993
Method11
1.000
0.999
0.862
0.279
0.056
0.295
0.756
0.957
0.996
Method12
1.000
0.999
0.854
0.265
0.056
0.312
0.765
0.959
0.996
Method13
1.000
0.983
0.808
0.438
0.219
0.330
0.592
0.815
0.931
Method14
1.000
0.999
0.980
0.905
0.867
0.902
0.966
0.988
0.997
Method15
1.000
1.000
0.999
0.922
0.780
0.920
0.989
1.000
1.000
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
92
Volume 19, 2023
Table 3.16: Empirical type I error rate and power of tests for Gamma (0.25,2), SNR = 0.5, n = 30
SNR0
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
Method1
1.000
0.988
0.689
0.268
0.073
0.039
0.036
0.036
0.036
Method2
0.842
0.527
0.203
0.044
0.006
0.001
0.018
0.122
0.421
Method3
1.000
0.986
0.695
0.273
0.072
0.036
0.032
0.032
0.032
Method4
1.000
1.000
1.000
1.000
1.000
1.000
0.125
0.000
0.000
Method5
1.000
1.000
1.000
1.000
1.000
0.278
0.028
0.000
0.000
Method6
1.000
1.000
1.000
1.000
1.000
0.340
0.021
0.000
0.000
Method7
1.000
0.975
0.512
0.139
0.066
0.058
0.057
0.057
0.057
Method8
1.000
1.000
1.000
1.000
1.000
1.000
0.000
0.000
0.000
Method9
1.000
1.000
1.000
1.000
1.000
0.500
0.000
0.000
0.000
Method10
1.000
0.973
0.516
0.138
0.060
0.051
0.050
0.050
0.050
Method11
1.000
0.999
0.939
0.605
0.220
0.223
0.472
0.737
0.915
Method12
1.000
0.999
0.898
0.401
0.148
0.353
0.690
0.901
0.975
Method13
1.000
1.000
1.000
1.000
0.086
0.008
0.001
0.001
0.000
Method14
1.000
1.000
1.000
0.116
0.015
0.003
0.001
0.000
0.000
Method15
0.740
0.309
0.070
0.012
0.002
0.001
0.026
0.209
0.631
Table 3.17: Empirical type I error rate and power of tests for Gamma (0.25,2), SNR = 0.5, n = 50
SNR0
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
Method1
1.000
0.985
0.746
0.260
0.026
0.002
0.001
0.001
0.001
Method2
0.974
0.768
0.350
0.067
0.004
0.005
0.084
0.409
0.813
Method3
1.000
0.985
0.752
0.265
0.027
0.001
0.001
0.001
0.001
Method4
1.000
1.000
1.000
1.000
1.000
0.048
0.000
0.000
0.000
Method5
1.000
1.000
1.000
1.000
0.393
0.015
0.000
0.000
0.000
Method6
1.000
1.000
1.000
1.000
0.447
0.012
0.000
0.000
0.000
Method7
1.000
0.971
0.568
0.080
0.005
0.001
0.001
0.001
0.001
Method8
1.000
1.000
1.000
1.000
1.000
0.000
0.000
0.000
0.000
Method9
1.000
1.000
1.000
1.000
0.409
0.000
0.000
0.000
0.000
Method10
1.000
0.973
0.574
0.083
0.005
0.001
0.001
0.001
0.001
Method11
1.000
1.000
0.978
0.703
0.260
0.327
0.684
0.921
0.990
Method12
1.000
1.000
0.955
0.494
0.192
0.535
0.884
0.985
0.999
Method13
1.000
1.000
1.000
1.000
0.266
0.005
0.000
0.000
0.000
Method14
1.000
1.000
1.000
0.992
0.029
0.002
0.000
0.000
0.000
Method15
0.951
0.603
0.137
0.012
0.001
0.010
0.157
0.649
0.949
Table 3.18: Empirical type I error rate and power of tests for Gamma (0.25,2), SNR = 0.5, n = 100
SNR0
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
Method1
1.000
0.999
0.891
0.326
0.018
0.000
0.000
0.000
0.706
Method2
0.999
0.969
0.663
0.134
0.007
0.038
0.385
0.890
0.996
Method3
1.000
0.999
0.892
0.329
0.019
0.000
0.000
0.000
0.724
Method4
1.000
1.000
1.000
1.000
0.060
0.000
0.000
0.000
0.000
Method5
1.000
1.000
1.000
0.765
0.031
0.000
0.000
0.000
0.000
Method6
1.000
1.000
1.000
0.763
0.030
0.000
0.000
0.000
0.000
Method7
1.000
0.997
0.775
0.095
0.001
0.000
0.000
0.000
0.921
Method8
1.000
1.000
1.000
1.000
0.011
0.000
0.000
0.000
0.000
Method9
1.000
1.000
1.000
0.608
0.004
0.000
0.000
0.000
0.000
Method10
1.000
0.997
0.778
0.097
0.001
0.000
0.000
0.000
0.927
Method11
1.000
1.000
0.996
0.817
0.292
0.528
0.916
0.996
1.000
Method12
1.000
1.000
0.989
0.629
0.259
0.805
0.989
1.000
1.000
Method13
1.000
1.000
1.000
1.000
1.000
0.004
0.000
0.000
0.000
Method14
1.000
1.000
1.000
1.000
0.642
0.001
0.000
0.000
0.575
Method15
0.999
0.933
0.385
0.021
0.001
0.086
0.670
0.984
1.000
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
93
Volume 19, 2023
Table 3.19: Empirical type I error rate and power of tests for Gamma (1,2), SNR = 1, n = 30
SNR0
0.400
0.600
0.700
0.800
1.000
1.200
1.300
1.400
1.500
Method1
0.998
0.847
0.589
0.314
0.041
0.002
0.001
0.000
0.000
Method2
0.972
0.718
0.476
0.260
0.048
0.096
0.248
0.491
0.711
Method3
0.998
0.852
0.598
0.323
0.044
0.003
0.001
0.000
0.000
Method4
1.000
1.000
0.725
0.393
0.055
0.004
0.001
0.000
0.000
Method5
1.000
0.910
0.613
0.325
0.044
0.002
0.001
0.000
0.000
Method6
1.000
0.916
0.647
0.358
0.053
0.004
0.001
0.000
0.000
Method7
0.995
0.751
0.471
0.235
0.032
0.002
0.001
0.000
0.000
Method8
1.000
1.000
0.676
0.344
0.051
0.003
0.001
0.000
0.000
Method9
1.000
0.857
0.527
0.265
0.037
0.002
0.001
0.000
0.000
Method10
0.995
0.761
0.480
0.242
0.035
0.002
0.001
0.000
0.000
Method11
1.000
0.944
0.756
0.435
0.102
0.280
0.474
0.657
0.803
Method12
1.000
0.895
0.631
0.333
0.133
0.411
0.590
0.744
0.858
Method13
1.000
0.914
0.656
0.380
0.085
0.018
0.008
0.004
0.002
Method14
1.000
0.793
0.537
0.332
0.092
0.057
0.190
0.433
0.645
Method15
0.943
0.586
0.365
0.193
0.041
0.164
0.373
0.608
0.783
Table 3.20: Empirical type I error rate and power of tests for Gamma (1,2), SNR = 1, n = 50
SNR0
0.400
0.600
0.700
0.800
1.000
1.200
1.300
1.400
1.500
Method1
0.999
0.941
0.747
0.397
0.032
0.000
0.012
0.098
0.337
Method2
0.998
0.894
0.664
0.360
0.047
0.192
0.478
0.749
0.916
Method3
0.999
0.943
0.753
0.404
0.033
0.001
0.017
0.112
0.363
Method4
1.000
0.941
0.733
0.388
0.032
0.001
0.000
0.000
0.000
Method5
1.000
0.926
0.700
0.361
0.029
0.000
0.000
0.000
0.000
Method6
1.000
0.931
0.728
0.388
0.034
0.001
0.000
0.000
0.000
Method7
0.999
0.887
0.593
0.285
0.022
0.000
0.027
0.193
0.491
Method8
1.000
0.887
0.584
0.280
0.022
0.000
0.000
0.000
0.000
Method9
1.000
0.853
0.549
0.256
0.020
0.000
0.000
0.000
0.000
Method10
0.999
0.890
0.599
0.289
0.023
0.001
0.037
0.213
0.516
Method11
1.000
0.989
0.901
0.605
0.122
0.415
0.662
0.844
0.946
Method12
1.000
0.975
0.807
0.460
0.154
0.562
0.763
0.897
0.964
Method13
1.000
0.993
0.852
0.510
0.084
0.009
0.003
0.027
0.222
Method14
1.000
0.970
0.779
0.469
0.100
0.108
0.367
0.658
0.842
Method15
0.994
0.795
0.511
0.255
0.042
0.323
0.618
0.827
0.945
Table 3.21: Empirical type I error rate and power of tests for Gamma (1,2), SNR = 1, n = 100
SNR0
0.400
0.600
0.700
0.800
1.000
1.200
1.300
1.400
1.500
Method1
1.000
0.995
0.921
0.605
0.022
0.080
0.352
0.736
0.946
Method2
1.000
0.989
0.900
0.595
0.049
0.437
0.810
0.971
0.998
Method3
1.000
0.995
0.922
0.608
0.022
0.084
0.360
0.741
0.948
Method4
1.000
0.989
0.894
0.559
0.019
0.000
0.000
0.082
0.701
Method5
1.000
0.988
0.885
0.541
0.017
0.000
0.000
0.289
0.798
Method6
1.000
0.989
0.893
0.558
0.020
0.000
0.000
0.275
0.780
Method7
1.000
0.983
0.822
0.413
0.009
0.180
0.540
0.850
0.973
Method8
1.000
0.976
0.778
0.363
0.007
0.000
0.000
0.168
0.825
Method9
1.000
0.972
0.761
0.343
0.006
0.000
0.000
0.481
0.896
Method10
1.000
0.983
0.824
0.416
0.010
0.187
0.548
0.856
0.975
Method11
1.000
1.000
0.984
0.827
0.138
0.646
0.899
0.985
0.999
Method12
1.000
0.999
0.958
0.677
0.193
0.786
0.951
0.995
0.999
Method13
1.000
1.000
0.992
0.803
0.076
0.010
0.161
0.551
0.852
Method14
1.000
1.000
0.987
0.795
0.101
0.234
0.652
0.911
0.985
Method15
1.000
0.974
0.789
0.405
0.047
0.631
0.905
0.989
0.999
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
94
Volume 19, 2023
Table 3.22: Empirical type I error rate and power of tests for Gamma (4,2), SNR = 2, n = 30
SNR0
1.200
1.400
1.600
1.800
2.000
2.200
2.400
2.600
2.800
Method1
0.955
0.749
0.421
0.167
0.051
0.018
0.032
0.128
0.314
Method2
0.974
0.856
0.614
0.350
0.213
0.296
0.525
0.755
0.898
Method3
0.957
0.759
0.433
0.174
0.054
0.020
0.040
0.142
0.337
Method4
0.959
0.763
0.438
0.178
0.054
0.015
0.004
0.029
0.173
Method5
0.955
0.750
0.426
0.172
0.052
0.015
0.011
0.071
0.249
Method6
0.964
0.792
0.472
0.206
0.066
0.019
0.009
0.058
0.208
Method7
0.930
0.697
0.380
0.144
0.043
0.020
0.053
0.169
0.365
Method8
0.936
0.712
0.397
0.154
0.047
0.014
0.003
0.050
0.223
Method9
0.929
0.699
0.385
0.148
0.045
0.014
0.017
0.105
0.300
Method10
0.934
0.707
0.391
0.151
0.047
0.024
0.061
0.187
0.388
Method11
0.954
0.712
0.352
0.123
0.069
0.146
0.328
0.543
0.737
Method12
0.929
0.662
0.316
0.116
0.087
0.190
0.376
0.586
0.768
Method13
0.917
0.715
0.475
0.267
0.142
0.083
0.097
0.193
0.340
Method14
0.942
0.805
0.610
0.436
0.360
0.426
0.558
0.707
0.824
Method15
0.954
0.811
0.562
0.320
0.225
0.340
0.570
0.782
0.910
Table 3.23: Empirical type I error rate and power of tests for Gamma (4,2), SNR = 2, n = 50
SNR0
1.200
1.400
1.600
1.800
2.000
2.200
2.400
2.600
2.800
Method1
0.995
0.913
0.567
0.198
0.043
0.026
0.128
0.389
0.678
Method2
0.997
0.959
0.761
0.423
0.208
0.348
0.665
0.892
0.978
Method3
0.995
0.916
0.574
0.201
0.044
0.029
0.138
0.404
0.690
Method4
0.995
0.917
0.578
0.204
0.044
0.013
0.074
0.307
0.611
Method5
0.994
0.911
0.567
0.199
0.042
0.017
0.091
0.344
0.645
Method6
0.995
0.923
0.605
0.225
0.053
0.015
0.079
0.312
0.608
Method7
0.990
0.873
0.512
0.172
0.038
0.041
0.175
0.450
0.715
Method8
0.990
0.878
0.522
0.178
0.037
0.018
0.106
0.364
0.659
Method9
0.989
0.869
0.511
0.173
0.036
0.027
0.134
0.406
0.687
Method10
0.990
0.877
0.518
0.175
0.040
0.045
0.186
0.461
0.728
Method11
0.996
0.922
0.559
0.181
0.066
0.183
0.461
0.738
0.900
Method12
0.992
0.883
0.503
0.162
0.086
0.238
0.517
0.771
0.913
Method13
0.986
0.852
0.571
0.300
0.134
0.103
0.207
0.405
0.606
Method14
0.994
0.917
0.717
0.475
0.373
0.454
0.645
0.808
0.912
Method15
0.993
0.934
0.705
0.378
0.220
0.401
0.705
0.906
0.980
Table 3.24: Empirical type I error rate and power of tests for Gamma (4,2), SNR = 2, n = 100
SNR0
1.200
1.400
1.600
1.800
2.000
2.200
2.400
2.600
2.800
Method1
1.000
0.993
0.804
0.289
0.035
0.095
0.438
0.826
0.976
Method2
1.000
0.998
0.925
0.550
0.221
0.488
0.873
0.988
0.999
Method3
1.000
0.993
0.807
0.293
0.035
0.098
0.443
0.829
0.976
Method4
1.000
0.993
0.806
0.294
0.033
0.066
0.381
0.797
0.971
Method5
1.000
0.992
0.799
0.286
0.032
0.075
0.400
0.808
0.972
Method6
1.000
0.994
0.816
0.309
0.038
0.067
0.378
0.792
0.969
Method7
1.000
0.984
0.741
0.242
0.035
0.143
0.513
0.856
0.981
Method8
1.000
0.983
0.741
0.246
0.029
0.108
0.457
0.832
0.977
Method9
1.000
0.982
0.737
0.239
0.030
0.119
0.477
0.842
0.979
Method10
1.000
0.984
0.743
0.245
0.036
0.145
0.518
0.859
0.982
Method11
1.000
0.996
0.840
0.314
0.076
0.304
0.725
0.946
0.993
Method12
1.000
0.990
0.781
0.266
0.100
0.374
0.768
0.957
0.994
Method13
1.000
0.980
0.769
0.387
0.129
0.162
0.426
0.716
0.896
Method14
1.000
0.995
0.882
0.577
0.373
0.525
0.785
0.931
0.979
Method15
1.000
0.996
0.879
0.477
0.244
0.560
0.899
0.989
0.999
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
95
Volume 19, 2023
Table 3.25: Empirical type I error rate and power of tests for Gamma (25,2), SNR = 5, n = 30
SNR0
4.000
4.400
4.600
4.800
5.000
5.200
5.400
5.600
6.400
Method1
0.492
0.249
0.158
0.095
0.062
0.043
0.042
0.058
0.275
Method2
0.882
0.729
0.647
0.609
0.600
0.617
0.671
0.712
0.921
Method3
0.502
0.258
0.164
0.101
0.065
0.046
0.046
0.063
0.289
Method4
0.500
0.256
0.162
0.098
0.064
0.042
0.039
0.056
0.269
Method5
0.497
0.254
0.161
0.098
0.064
0.043
0.042
0.059
0.276
Method6
0.552
0.297
0.194
0.120
0.075
0.049
0.040
0.049
0.236
Method7
0.467
0.231
0.147
0.088
0.057
0.045
0.048
0.066
0.302
Method8
0.474
0.236
0.149
0.090
0.058
0.043
0.046
0.063
0.296
Method9
0.472
0.235
0.149
0.090
0.058
0.045
0.048
0.067
0.304
Method10
0.475
0.239
0.150
0.092
0.061
0.047
0.054
0.073
0.317
Method11
0.336
0.133
0.081
0.055
0.054
0.073
0.111
0.170
0.521
Method12
0.315
0.122
0.076
0.056
0.062
0.084
0.129
0.193
0.546
Method13
0.501
0.336
0.279
0.235
0.199
0.186
0.182
0.195
0.364
Method14
0.823
0.748
0.719
0.713
0.726
0.738
0.771
0.800
0.882
Method15
0.861
0.708
0.636
0.610
0.604
0.630
0.680
0.727
0.927
Table 3.26: Empirical type I error rate and power of tests for Gamma (25,2), SNR = 5, n = 50
SNR0
4.000
4.400
4.600
4.800
5.000
5.200
5.400
5.600
6.400
Method1
0.668
0.307
0.183
0.097
0.053
0.042
0.057
0.101
0.540
Method2
0.954
0.792
0.684
0.607
0.577
0.621
0.700
0.783
0.972
Method3
0.672
0.312
0.186
0.099
0.055
0.044
0.060
0.106
0.552
Method4
0.672
0.311
0.186
0.098
0.054
0.042
0.055
0.097
0.534
Method5
0.670
0.309
0.184
0.097
0.054
0.042
0.056
0.100
0.540
Method6
0.707
0.340
0.211
0.116
0.064
0.042
0.048
0.088
0.498
Method7
0.638
0.286
0.167
0.088
0.051
0.044
0.066
0.119
0.568
Method8
0.645
0.290
0.170
0.090
0.051
0.045
0.064
0.115
0.564
Method9
0.642
0.288
0.170
0.089
0.051
0.045
0.066
0.118
0.568
Method10
0.646
0.291
0.171
0.091
0.054
0.048
0.069
0.126
0.580
Method11
0.544
0.210
0.111
0.062
0.052
0.075
0.138
0.231
0.721
Method12
0.514
0.193
0.104
0.060
0.057
0.088
0.157
0.255
0.740
Method13
0.614
0.387
0.299
0.233
0.200
0.189
0.208
0.247
0.523
Method14
0.881
0.777
0.738
0.725
0.728
0.748
0.773
0.808
0.924
Method15
0.944
0.773
0.668
0.596
0.583
0.635
0.715
0.793
0.975
Table 3.27: Empirical type I error rate and power of tests for Gamma (25,2), SNR = 5, n = 100
SNR0
4.000
4.400
4.600
4.800
5.000
5.200
5.400
5.600
6.400
Method1
0.904
0.480
0.264
0.115
0.053
0.052
0.119
0.257
0.880
Method2
0.995
0.902
0.764
0.636
0.597
0.649
0.759
0.869
0.997
Method3
0.905
0.484
0.266
0.118
0.054
0.053
0.122
0.262
0.883
Method4
0.905
0.485
0.266
0.118
0.054
0.052
0.117
0.253
0.880
Method5
0.905
0.483
0.266
0.117
0.053
0.052
0.118
0.256
0.880
Method6
0.917
0.511
0.286
0.128
0.055
0.048
0.103
0.233
0.867
Method7
0.888
0.458
0.246
0.108
0.052
0.060
0.134
0.285
0.889
Method8
0.891
0.463
0.248
0.111
0.053
0.059
0.132
0.281
0.887
Method9
0.889
0.461
0.247
0.110
0.052
0.060
0.134
0.284
0.889
Method10
0.890
0.462
0.248
0.111
0.053
0.062
0.137
0.289
0.890
Method11
0.865
0.403
0.202
0.083
0.054
0.095
0.217
0.392
0.934
Method12
0.848
0.383
0.187
0.080
0.060
0.109
0.242
0.415
0.940
Method13
0.798
0.494
0.353
0.257
0.211
0.212
0.267
0.354
0.769
Method14
0.951
0.845
0.780
0.736
0.729
0.759
0.801
0.850
0.972
Method15
0.993
0.887
0.744
0.627
0.604
0.660
0.774
0.878
0.997
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
96
Volume 19, 2023
Table 3.28: Empirical type I error rate and power of tests for Gamma (100,2), SNR = 10, n = 30
SNR0
6.000
7.000
8.000
9.000
10.000
11.000
12.000
13.000
14.000
Method1
0.991
0.861
0.531
0.207
0.069
0.058
0.152
0.337
0.557
Method2
1.000
0.995
0.949
0.847
0.788
0.840
0.924
0.967
0.988
Method3
0.992
0.867
0.541
0.215
0.071
0.061
0.161
0.353
0.575
Method4
0.991
0.864
0.537
0.210
0.069
0.058
0.153
0.339
0.561
Method5
0.991
0.863
0.536
0.210
0.070
0.058
0.154
0.341
0.562
Method6
0.994
0.890
0.586
0.250
0.086
0.052
0.128
0.292
0.509
Method7
0.987
0.844
0.502
0.192
0.064
0.063
0.168
0.360
0.582
Method8
0.987
0.848
0.507
0.195
0.065
0.064
0.169
0.362
0.585
Method9
0.987
0.848
0.507
0.195
0.065
0.065
0.171
0.364
0.586
Method10
0.988
0.851
0.513
0.200
0.067
0.069
0.178
0.377
0.600
Method11
0.966
0.725
0.334
0.107
0.052
0.133
0.320
0.554
0.756
Method12
0.958
0.702
0.313
0.098
0.059
0.146
0.342
0.579
0.775
Method13
0.944
0.764
0.520
0.326
0.216
0.213
0.292
0.413
0.546
Method14
0.995
0.969
0.908
0.860
0.859
0.895
0.917
0.948
0.967
Method15
1.000
0.993
0.943
0.839
0.786
0.844
0.927
0.971
0.989
Table 3.29: Empirical type I error rate and power of tests for Gamma (100,2), SNR = 10, n = 50
SNR0
6.000
7.000
8.000
9.000
10.000
11.000
12.000
13.000
14.000
Method1
1.000
0.968
0.683
0.250
0.060
0.090
0.317
0.625
0.845
Method2
1.000
1.000
0.981
0.866
0.783
0.868
0.963
0.987
0.998
Method3
1.000
0.970
0.689
0.255
0.063
0.094
0.326
0.634
0.851
Method4
1.000
0.970
0.686
0.253
0.061
0.090
0.318
0.627
0.845
Method5
1.000
0.970
0.686
0.252
0.061
0.091
0.320
0.627
0.846
Method6
1.000
0.975
0.718
0.285
0.067
0.082
0.288
0.588
0.823
Method7
1.000
0.962
0.664
0.233
0.059
0.100
0.337
0.646
0.855
Method8
1.000
0.964
0.667
0.236
0.060
0.101
0.337
0.646
0.855
Method9
1.000
0.964
0.667
0.235
0.060
0.102
0.338
0.648
0.856
Method10
1.000
0.965
0.668
0.237
0.061
0.104
0.344
0.656
0.860
Method11
0.999
0.929
0.547
0.155
0.054
0.178
0.475
0.767
0.923
Method12
0.999
0.920
0.525
0.145
0.057
0.198
0.501
0.782
0.930
Method13
0.990
0.889
0.629
0.351
0.222
0.247
0.396
0.586
0.730
Method14
1.000
0.991
0.942
0.871
0.859
0.906
0.937
0.971
0.982
Method15
1.000
1.000
0.975
0.858
0.784
0.877
0.965
0.989
0.998
Table 3.30: Empirical type I error rate and power of tests for Gamma (100,2), SNR = 10, n = 100
SNR0
6.000
7.000
8.000
9.000
10.000
11.000
12.000
13.000
14.000
Method1
1.000
0.999
0.919
0.377
0.052
0.192
0.643
0.930
0.990
Method2
1.000
1.000
0.999
0.926
0.777
0.914
0.991
0.999
1.000
Method3
1.000
0.999
0.921
0.380
0.052
0.197
0.646
0.931
0.990
Method4
1.000
0.999
0.920
0.379
0.052
0.191
0.643
0.930
0.990
Method5
1.000
0.999
0.920
0.378
0.052
0.192
0.643
0.930
0.990
Method6
1.000
1.000
0.928
0.404
0.055
0.174
0.622
0.918
0.989
Method7
1.000
0.999
0.911
0.364
0.050
0.207
0.658
0.934
0.990
Method8
1.000
0.999
0.913
0.366
0.051
0.206
0.657
0.934
0.990
Method9
1.000
0.999
0.913
0.366
0.051
0.207
0.658
0.934
0.990
Method10
1.000
0.999
0.914
0.366
0.051
0.210
0.662
0.935
0.990
Method11
1.000
0.999
0.868
0.285
0.048
0.291
0.752
0.959
0.994
Method12
1.000
0.999
0.857
0.272
0.049
0.305
0.762
0.962
0.996
Method13
1.000
0.984
0.806
0.425
0.222
0.337
0.601
0.817
0.931
Method14
1.000
1.000
0.979
0.906
0.870
0.916
0.966
0.989
0.998
Method15
1.000
1.000
0.998
0.921
0.775
0.921
0.992
1.000
1.000
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
97
Volume 19, 2023
4. Applications
In this section we will consider and analyze
two (2) real life data sets to illustrate the
performance of the test statistics.
4.1 Chemotherapy Treatment
In this section, we will consider the
chemotherapy data, which was given on a
group of individuals who had chemotherapy
treatment exclusively for 45 years (Bekker et
al. 2000). The observations are as follows:
0.047, 0.115, 0.121, 0.132, 0.164, 0.197,
0.203, 0.260, 0.282, 0.296, 0.334, 0.395,
0.458, 0.466, 0.501, 0.507, 0.529, 0.534,
0.540, 0.641, 0.644, 0.696, 0.841, 0.863,
1.099, 1.219, 1.271, 1.326, 1.447, 1.485,
1.553, 1.581, 1.589, 2.178, 2.343, 2.416,
2.444, 2.825, 2.830, 3.578, 3.658, 3.743,
3.978, 4.003, 4.033.
The histogram of the failure time data is
presented in Figure 4.1, which looks like a
gamma distribution. The skewness and
kurtosis are obtained as 0.97 and 2.66
respectively.
Using R 4.2.2, we obtained the estimated
parameters of the gamma distribution as 1.1
and 0.82. Now using the KS test, we found the
P-value as 0.6029 and concluded that the
chemotherapy data follow a gamma
distribution with parameters 1.1 and 0.82. The
histogram, density plot, Q-Q plot, empirical
and theoretical cdf plot and P-P plot are
provided in Figure 4.2, which also supported
the gamma distribution. That means the
population signal to ratio will be
Then it is logical to test the true signal
to ratio as, H0: SNR=1.05. The lower and
upper critical values for all tests are presented
in Table 4.1.
Figure 4.1: Histogram of chemotherapy data
Figure 4.2: Empirical and theoretical
density, QQ plot, CDF and P-P plots of
chemotherap data
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
98
Volume 19, 2023
Table 4.1: Lower and upper critical values of the
chemotherapy treatment data
From Table 4.1, we observed that all methods
including our proposed methods 12 to 15 have
accepted the null hypothesis H0: SNR=1.05.
These results are consistent with the
simulation results. It is noted that the
proposed method 15 performed very well in
the simulation results and it is consistent with
the real data.
4.2 Mathematics Grades
The below datasets contain the 2013
mathematics grades for 48 students enrolled
in the slow-paced program. From Linhart and
Zucchini (1986), the following observations
were obtained: 29, 25, 50, 15, 13, 27, 15, 18,
7, 7, 8, 19, 12, 18, 5, 21, 15, 86, 21, 15, 14,
39, 15, 14, 70, 44, 6, 2,58, 19, 50, 23, 11, 6,
34, 18, 28, 34, 12, 37, 4, 60, 20, 23, 40, 65,
19, 31.
The histogram of the mathematics
grade data is presented in Figure-4.5, which
looks like a gamma distribution. The
skewness and kurtosis are obtained as 1.29
and 4.22 respectively.
Figure 4.5: Histogram of mathematics grade
data
The estimated parameters of the
gamma distribution are obtained as 1.97 and
0.08. Now using the KS test, we found the P-
value as 0.958 and concluded that the
mathematics grade data follow a gamma
distribution with parameters 1.97 and 0.82.
The histogram, density plot, Q-Q plot,
empirical and theoretical cdf plot and P-P plot
are provided in Figure 4.6, which also
supported the gamma distribution. That
means the population signal to ratio will be
Then it is logical to test the
true signal to ratio as, H0: SNR=1.40. The
lower and upper critical values for all tests are
presented in Table 4.3.
From Table 4.3, we observed that all methods
including our proposed methods 12 to 15 have
accepted the null hypothesis H0: SNR=1.40.
These results are consistent with the
simulation results.
Method
Lower
Limit
Upper
Limit
1
0.800
1.643
2
0784
1.368
3
0.802
1.633
4
0.796
1.990
5
0.788
1.871
6
0.799
1.874
7
0.804
1.313
8
0.726
2.047
9
0.717
1.878
10
0.806
1.306
11
0.852
1.300
12
0.789
1.204
13
0.696
1.525
14
0.664
1.248
15
0.705
1.289
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
99
Volume 19, 2023
Figure 4.6: Empirical and theoretical
density, QQ plot, CDF and P-P plots of
mathematics grade data
Table 4.3: Lower and upper critical values of
the mathematics grades data
Method
Lower
Limit
Upper
Limit
1
1.041
1.904
2
1.063
1.629
3
1.043
1.896
4
1.041
2.053
5
1.034
1.999
6
1.046
2.012
7
1.001
1.744
8
0.976
1.999
9
0.969
1.938
10
0.888
2.243
11
1.074
1.616
12
1.016
1.528
13
1.173
2.075
14
1.216
1.781
15
0.989
1.555
5 Conclusion
In this paper we consider fifteen different test
statistics (Methods 1 to 15) for testing the
population SNR. We consider some existing
test statistics, which were developed from the
confidence interval of SNR and from the work
of Miller (1991), Sharma and Krishna (1994),
Curto and Pinto (2009), McKay’s (1932),
Panichkitkosolkul (2009) and Kibria and
George (2014) among others. We also
propose a few new test statistics based on the
robust estimator of population standard
deviation σ. Since a theoretical comparison
among the test statistics is not possible, a
Monte Carlo simulation study has been
conducted under both symmetric and skewed
distributions to compare the performance of
the test statistics. The performance of the test
statistics is determined based on the empirical
size and power of the tests. We have
considered the most popular and widely used
significance level 0.05 for finding the size and
power of the test. To see the impact of sample
size on the test statistics, we considered n=30,
50 & 100. From simulation study it appears
that Methods 1, 3, 4, 5, 6, 7, 8, 9,10 and
proposed Methods 12 and 15 are promising
and performed well in some conditions.
However, Method 10 proposed by Kibria and
George (2014) performed the best when
testing for SNR=1, 2 and 5 in all simulation
conditions both at the lower and upper end of
the alternative hypotheses. However, for
testing SNR=0.5 the proposed Method 12 and
for testing SNR=10, the proposed Method 15
performed the best. Two real life data on
chemotherapy treatment and mathematics
grades are analyzed to illustrate the
performance of the test statistics. It appears
that the simulation results and applications are
consistent to some extent. The conclusions of
this paper are restricted to the given
simulation conditions of this paper. For a
definite statement one might need more
simulation conditions and more sample sizes
and do a simulation under various
distributional conditions. Hope the findings of
the paper will be an asset for the practitioners.
This study was done for two-tailed tests.
However, it would be interesting to compare
these methods for one tail test. Additionally,
it would be interesting to use t statistic instead
of z-statistic to compare the performance of
the test statistics under small sample sizes.
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
100
Volume 19, 2023
Acknowledgements
Authors are grateful to three anonymous
reviewers for their constructive comments
and suggestions, which certainly helped to
improve the quality and presentation of the
paper.
References:
[1] Abu-Shaweish, M O A, Akyz, H. E. and
Kibria, B. M. G. (2019). Performance of
Some Confidence Intervals for Estimating
the Population Coefficient of Variation
Under both Symmetric and Skewed
Distributions. Statistics, Optimization and
Information Computing, 7, 277-290.
[2] Albatineh, A, N., Kibria, B. M. G., Wilcox,
M. L. and Zogheib, B. (2014). Confidence
interval estimation for the population
coefficient of variation using ranked set
sampling: a simulation study, Journal of
Applied Statistics, 41(4), 733-751.
[3] Andrew, H., George, F. and Kibria, B. M. G.
(2015). Methods for Identifying
Differentially Expressed Genes: An
Empirical Comparison. Journal of
Biometrics and Biostatistics. 6:5, 1-6.
[4] Banik, S., Kibria, B M G. (2010).
Comparison of some parametric and
nonparametric type one sample confidence
intervals for estimating the mean of a
positively skewed distribution.
Communications in Statistics-Simulation and
Computation, 39: 361-389.
[5] Banik, S., Kibria, B. M. G. and D. Sharma
(2012). Testing the Population Coefficient of
Variation. Journal of Modern Applied
Statistical Methods. 11(2), 325 – 335.
[6] Bekker, A., J. J. J. Roux and P. J. Mosteit.
2000. A generalization of the compound
Rayleigh distribution: using a Bayesian
method on cancer survival times. Commun.
Stat. Theory Methods. 29: 1419–1433.
[7] Curto, J. D., Pinto, J. C. (2009). The
coefficient of variation asymptotic
distribution in the case of non-iid random
variables. Journal of Applied Statistics,
36(1), 21-32.
[8] George, F. and Kibria, B. M. G. (2012).
Confidence Intervals for estimating the
population signal-to-noise ratio: a simulation
study. Journal of Applied Statistics
39(6):1225–1240.
[9] John, C. R. (2007). The image processing
handbook. CRC Press, Boca Raton, Florida.
[10] Kibria, B. M. G. (2006). Modified
confidence intervals for the mean of the
asymmetric distribution. Pakistan Journal
of Statistics, 22 (2), 109-120.
[11] Kibria, B. M. G. and George, F. (2014)
Methods for Testing Population Signal-to-
Noise Ratio, Communications in Statistics -
Simulation and Computation, 43:3, 443-461,
DOI: 10.1080/03610918.2012.70454
[12] Koopmans, L. H., Owen, D. B.,
Rosenblatt, J. I. (1964) Confidence intervals
for the coefficient of variation for the normal
and log normal distributions. Biometrika
Trust , 51(1/2), 25-32.
[13] Linhart, H. and W. Zucchini. 1986.
Model Selection. Wiley, New York, USA
[14] McGibney, G., Smith, M.R. (1993). An
unbiased signal-to-noise ratio measure for
magnetic resonance images. Medical
Physics, 20(4), 1077-1079.
[15] McKay, A.T.(1932). Distribution of the
coefficient of variation and the extended $t$
distribution. Journal of Royal Statistical
Society, 95, 695-698.
[16] Miller, E.G.(1991). Asymptotic test
statistics for coefficient of variation.
Communications in Statistics - Theory and
Methods, 20, 3351-3363.
[17] Panichkitkosolkul, W. (2009). Improved
confidence intervals for a coefficient of
variation of a normal distribution. Thailand
Statistician, 7(2), 193-199.
[18] Panichkitkosolkul, W. and Tulyanitikul,
B. (2022). Performance of statistical methods
for testing the signal-to-noise ratio of a log-
normal distribution. 2020 IEEE 7th
International Conference on Industrial
Engineering and Applications, 656-661.
[19] Rousseeuw, P.J.,and Croux,C. (1993).
Alternatives to the median absolute
deviation, Journal of the American Statistical
Association,88(424), 1273-1283.
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
101
Volume 19, 2023
[20] Sharma, K. K., Krishna, H. (1994).
Asymptotic sampling distribution of inverse
coefficient-of-variation and its applications.
IEEE Transactions on Reliability, 43(4), 630-
633.
[21] Shi, W. and Kibria, B. M. G. (2007). On
some confidence intervals for estimating the
mean of a skewed population. International
Journal of Mathematical Education in
Science and Technology. 38 (3), 412-421.
[22] Tania, S. (2008). Image fusion:
algorithms and applications. Academic Press.
[23] Vangel, M. G. (1996). Confidence
intervals for a normal coefficient of variation.
The American Statistician, 15(1), 21-26.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.10
Samantha Menendez,
Florence George, B. M. Golam Kibria
E-ISSN: 2224-3488
102
Volume 19, 2023
