This is an electronic version of an article published as:
Büning, Herbert and Qari, Salmai(2006) 'Power of One-Sample Location Tests Under Distributions with
Equal Lévy Distance', Communications in Statistics - Simulation and Computation, 35: 3, 531 — 545. It is
available online at: http://dx.doi.org/10.1080/03610910600716332
Power of One-Sample Location Tests
under Distributions with Equal Lévy Distance
Herbert Büning and Salmai Qari
Institut für Statistik und Ökonometrie, Freie Universität Berlin,
Boltzmannstr. 20, D-14195 Berlin, Germany
ABSTRACT
In this paper we study the power of one-sample location tests under classical distributions and
two supermodels which include the normal distribution as a special case. The distributions of
the supermodels are chosen in such a way that they have equal distance to the normal as the
logistic, uniform, double exponential and the Cauchy, respectively. As a measure of distance
we use the Lévy metric. The tests considered are two parametric tests, the t-test and a trimmed
t-test and two nonparametric tests, the sign test and the Wilcoxon signed-rank tests. It turns
out, that the power of the tests, first of all, does not depend on the Lévy distance but on the
special chosen supermodel.
Key words: location alternatives, parametric tests, nonparametric tests, nonnormality,
supermodel, Lévy metric, measure of tailweight, Monte Carlo simulation, power comparison
1. INTRODUCTION
Since the famous Princeton study of Andrews et al (1972) Monte Carlo Simulations have
gained more and more attention in studying the efficiency and robustness of statistical tests
and estimates. The efficiency of such procedures strongly depends on the underlying model of
the data mostly assumed to be the normal distribution. Meanwhile, there are an enormous
number of articles investigating the robustness of tests and estimates in the case of departures
from normality by using Monte Carlo simulations. Examples of the models considered are
“classical distributions” like the uniform, logistic, exponential, double exponential, Cauchy
and often so called supermodels including the normal distribution and other classical
distributions as special cases. All the models describe different tailweight and extent of
skewness. Examples of supermodels are the distributions of Box and Tiao (1962), Johnson
and Kotz (1970), the system of Pearson curves, see Kendall and Stuart (1969), the RSTdistributions, see Ramberg and Schmeiser (1972, 1974), and the contaminated normal
distribution (CN), see Büning (1991). The question arises: Does the property of robustness of
a test or an estimate depend on the specially chosen supermodel? Or more precisely: If we
consider two distributions from different two supermodels but with the same “distance” to the
normal distribution would we then obtain (nearly) the same results concerning the power of
the test or the mean square error of the estimate?
In this paper we try to give an answer of this question for some tests in the one-sample
location problem selected for the purpose of illustration. As a measure of distance between
two distribution functions F and G we choose the Lévy metric. First, we select four “classical”
1
symmetric distributions with different tailweight, the uniform, logistic, double exponential
and the Cauchy and then calculate the Lévy distances of this four distribution functions to the
normal. Second, we determine from each of the supermodels, CN and RST, one member
which has the same Lévy distance to the normal as the normal to the uniform, logistic, double
exponential and the Cauchy, respectively. In order to fix such members we have to choose the
parameters in the two supermodels in an appropriate way. To our knowledge there is up to
now no such a robustness study of tests based on the Lévy distance.
We select the parametric t-test and a trimmed version of it as well as two nonparametric tests,
the sign test and the Wilcoxon signed-rank test described in section 2.
In Section 3 we introduce the Lévy metric and calculate the Lévy distances between the
normal distribution and some other distributions considered in our simulation study. In
Section 4 a power study of the four tests for distributions with equal Lévy distance is carried
out via Monte Carlo simulation which will give an answer of the question above.
2. ONE-SAMPLE LOCATION TESTS
2.1 Model and Hypotheses
We consider the one-sample location model: X1 ,..., X n are independent, identically
distributed
random
variables
with
absolutely
continuous
distributions
function
Fθ (x) = F(x − θ) which is symmetric about the location parameter θ ∈ Ω.
We wish to test
H 0 : θ = θ0 versus H1 : θ > θ0 .
Without loss of generality we can set θ0 = 0.
2.2 t-test and trimmed t-test
At first, we consider the well-known parametric t-test which is the uniformly most powerful
unbiased test under normality, i.e. X i ~ N( θ, σ2 ), i = 1,..., n. The corresponding statistic is
given by
(X − θ) n
t=
,
S
1 n
1 n
where X = ∑ X i and S2 =
∑ (X i − X)2 .
n i=1
n − 1 i =1
Under H 0 : θ = θ0 , the statistic t has a t-distribution with (n-1) degrees of freedom. That
means H 0 has to be rejected at level α if t ≥ t1−α (n − 1) .
Under nonnormal data the statistic t is asymptotically standard normally distributed.
Next, we consider a trimmed version of the t-test, the so called trimmed t-test t γg which was
introduced by Tukey and Mc Laughlin (1963). Let be X (1) ,..., X (n ) the ordered sample of
X1 ,..., X n , γ the fraction of trimming and g = [ γn] , the number of trimmed observations at
both ends of the ordered sample. Furthermore, let be
2
X γg =
1 n −g
∑ X (i) the γ − trimmed mean,
n − 2g i=g+1
X γw =
n−g
1
[gX (g+1) + gX (n −g) + ∑ X (i) ] the γ − winsorized mean of the X-variables and
n
i = g +1
SSD w = g(X (g+1) − X γw )2 + g(X ( n −g) − X γw )2 +
n−g
∑ (X
(i)
− X γw ) 2 the winsorized sum of squared
i = g +1
deviations.
Then the trimmed t-statistic t γg is defined by
t=
(X γg − θ0 )
SSD w / h(h − 1)
with h=n-2g.
Under normality the statistic t γg has approximately a t-distribution with (h-1) degrees of
freedom.
2.3 Sign test and Wilcoxon signed-rank test
The sign test and the Wilcoxon signed-rank test are special cases of linear rank tests. In order
to define linear rank statistics let be
Di = X i − θ0 and D (i) the corresponding ordered absolute differences, i = 1,..., n.
Furthermore, we define indicator variables Vi by
 1, if D (i) belongs to a positive difference D j
Vi = 
0, if D (i) belongs to a negative difference D j.
Then linear rank statistics are of the form
n
L+n = ∑ g(i)Vi with real valued scores g(i) , i = 1,..., n.
i
The statistic
L+ − E(L+n )
Z= n
has a limiting standard normal distributed with
Var(L+n )
1 n
∑ g(i) and
2 i =1
1 n
Var(L+n ) = ∑ (g(i))2 , see e.g. Büning and Trenkler (1994, p. 92).
4 i=1
E(L+n ) =
Thus, critical values c1−α of L+n are given approximately by c1−α = E(L+n ) + Var(L+n )z1−α
where z1−α is the (1 − α) -quantile of the standard normal distribution.
Now, the sign statistic Vn+ is defined by the scores g(i) = 1, i = 1,..., n. That means
n
Vn+ = ∑ Vi .
i =1
The statistic Vn+ is a sum of Bernoulli variables and has a binomial distribution with
parameters n and p=0.5 . Thus we have under H 0 :
3
E(Vn+ ) = n / 2 and Var(Vn+ ) = n / 4.
Critical values of Vn+ can be found in tables of the binomial distribution.
The Wilcoxon signed-rank statistic Wn+ is defined by the scores g(i) = i, i = 1,..., n , i.e.
n
Wn+ = ∑ iVi
i =1
with E(Wn+ ) = n(n + 1) / 4 and Var(Vn+ ) = n(n + 1)(2n + 1) / 24 under the null hypothesis.
For sample sizes n ≤ 20 critical values of Wn+ can be found in Büning and Trenkler (1994).
It should be mentioned that the statistics Vn+ and Wn+ are discrete ones. Thus, in order to have
exact level- α -tests for power comparisons with the t-test and the trimmed t-test t γg a
randomized Vn+ − and Wn+ − test is applied in the simulation study in section 4 for sample sizes
n ≤ 20 , for sample sizes n > 20 we use the normal approximation.
3. LÉVY METRIC
Deviations from the ideal model, e.g. from the normal distribution, may be described by the
Prohorov-, Kolmogorov- or Lévy distance. Here, we consider the Lévy distance in context
with robustness and power studies of the tests from section 2. The Lévy distance d L between
two distribution F, G is defined by
d L (F,G) = inf
{δ
G(x − δ) − δ ≤ F(x) ≤ G(x + δ) + δ, ∀ x ∈ IR}.
Obviously, d L is a metric. The term 2 d L (F, G) is the maximum distance between the graphs
of F and G, measured along a 45o − direction, see the following Figure 1 with δ = d L where F
is within the two dotted lines about G, see Büning (1991) and Huber (1981).
Figure 1 Lévy distance
Now, let us calculate the distance d L between the normal distribution and four other
distributions, the uniform, logistic, double exponential and the Cauchy. Because the Lévy
4
distance is not location- and scale invariant we have to scale all the densities of the
distribution functions considered in order to get a meaningful comparison of the Lévy
distances to the normal distribution function. We do it in the following way: The location
parameter θ of all densities is fixed to θ = 0 and the scale parameter is determined in such a
way that f (0) is equal to 1/ 2 π like the value of the standard normal density for x = 0.
In the following we write N 0 for the normal distribution if θ = 0 ( H 0 ) and N θ if θ ≠ 0 ( H1 ),
analogously for the other distributions.
Table 1 presents the Lévy distances between the normal distribution ( N 0 ) and the logistic
distribution ( L0 ), the double exponential ( D0 ), the uniform ( U 0 ) and the Cauchy ( CA 0 ).
Table 1 Lévy distances d L
L0
D0
0.0767
N 0 0.0164
U0
0.0890
CA 0
0.0932
As already mentioned we consider two supermodels for our power study in section 4, first, the
symmetric contaminated normal distribution, CN (ε, k) , the density of which is given by
1 x −θ
1 x −θ
− (
)2
− (
)2
1
1
e 2 σ + ε⋅
e 2 kσ
2πσ
2π kσ
with σ > 0, ε ∈ [0,1] and k > 1 , see Büning (1991, p.18 ff).
ε
If we set f (0) = 1/ 2 π and θ = 0 we get σ = 1 − ε + . That means that there are various
k
combinations of ε and k to obtain a value of σ for which the condition f (0) = 1/ 2 π is
fulfilled.
Now, we select such values of ε and k so that the Lévy distances between CN 0 (ε, k) and
N 0 are the same as those between N 0 and the logistic L0 (0.0164), double exponential D0
(0.0767), uniform U 0 (0.0890) and Cauchy CA 0 (0.0932), see Table 1. The values of
ε and k are given in Table 2.
f (x) = (1 − ε) ⋅
Table 2 d L (N 0 , CN 0 ( ε, k))
k
3
5
ε
0.105
0.0164
0.3375
0.0767
0.340
0.3665
6
0.0890
0.0932
The second supermodel is the RST-distribution, RST(λ1 , λ 2 , λ 3 , λ 4 ) , which is defined by its
p-quantiles, see Ramberg and Schmeiser (1972, 1974):
pλ3 − (1 − p)λ 4
x p = F (p) = λ1 +
, 0 ≤ p ≤ 1,
λ2
−1
5
where λ1 is a location parameter, λ 2 is a scale parameter and λ 3 , λ 4 are form parameters.
Here, we only consider the symmetric case λ 3 = λ 4 . In the same manner as for CN 0 we
determine values of λ 2 , λ 3 = λ 4 so that RST0 (λ 2 , λ 3 ) := RST(0, λ 2 , λ 3 = λ 4 ) has the same Lévy
distances to N 0 as N 0 to the four classical distributions. Table 3 presents such combinations
of λ 2 and λ 3 = λ 4 .
Table 3 d L (N 0 , RST0 (λ 2 , λ 3 ))
Parameter
dL
λ2
0.3764
0.7400
0.7979
0.8178
λ3
0.2880
0.8170
1.0000
1.0940
0.0164
0.0767
0.0890
0.0932
Alltogether, we have four configurations (Config) each of them with one classical distribution
(classical) and one member of the CN- and RST-distribution for which the Lévy distances are
equal, see Table 4.
Table 4: Four Configurations
Config
classical CN 0
dL
RST0
ε
k
1
L0
0.1050
3
λ2
0.3764
2
D0
0.3375
5
0.7400
0.8170
0.0767
3
U0
0.3400
6
0.7979
1.0000
0.0890
4
CA 0
0.3665
6
0.8178
1.0940
0.0932
λ3
0.2880
0.0164
The simulation study in the next section 4 is based on these four configurations.
Figure 2 presents the densities of the distributions of the four configurations.
6
0.5
0.5
Figure 2: Densities of the four configurations
0.3
0.4
N0
CN0
RST0
D0
0.0
0.1
0.2
densities
0.3
0.2
0.0
0.1
densities
0.4
N0
CN0
RST0
L0
−4
−2
0
2
4
−4
−2
x
2
4
2
4
x
0.5
Configuration 2
0.5
Configuration 1
0.2
0.3
0.4
N0
CN0
RST0
CA0
0.0
0.0
0.1
0.2
densities
0.3
0.4
N0
CN0
RST0
U0
0.1
densities
0
−4
−2
0
2
4
−4
x
−2
0
x
Configuration 3
Configuration 4
From the figures we see that the tailweights of the distributions in the configurations are very
different except for configuration 1 where the Lévy distance between the four distribution and
the normal is very small in contrast to the other configurations. In the Configurations 2, 3 and
4 the RST-distributions have very short tails like the uniform, where the logistic, double
exponential, the Cauchy and the contaminated normal distributions have longer tails than the
normal but with different extent. That means: Distributions with the same Lévy distance to the
normal may have very different tailweight.
4 SIMULATION STUDY
We investigate via Monte Carlo simulation (60 000 runs) the power of the tests from section 2
under the distribution functions in the configurations 1, 2, 3 and 4. Critical values of the t- and
t γg − statistics under nonnormality are found by simulation (100 000 runs). The fraction γ of
trimming is 10%. We consider sample sizes of n=20, 50 and 100, but here we only present
7
results for the case n=20. For the sign- and Wilcoxon signed-rank test we apply
randomization for n=20 in order to achieve the nominal level α = 5%. The location parameter
θ is chosen as the expectation E(X) for all the distributions in the four configurations except,
of course, for the Cauchy where θ is the median. Thus, the power is calculated for the sample
% + θ, i = 1,..., n, where X
% ,X
% ,..., X
% is generated from one of the
X1 , X 2 ,..., X n with X i = X
i
1
2
n
normalized distributions in section 3. We choose θ = 0.3,0.5,0.7 .
Configuration 1:
Table 5 displays the power values of the four tests for selected values of θ under the
distributions from configuration 1 and the normal distribution for comparison. We consider
the case of sample size n=20 and α = 5%.
Table 5: configuration 1 − power values for n=20, α = 5%
θ = 0.3
Nθ
0.3646
Lθ
0.3186
CN θ
0.3146
RSTθ
0.4054
0.3441
0.3273
0.3320
0.3647
V
0.2746
0.2775
0.2759
0.2766
W
θ = 0.5
0.3508
Nθ
0.6942
0.3290
Lθ
0.6154
0.3343
CN θ
0.5840
0.3816
RSTθ
0.7642
0.6656
0.6277
0.6356
0.7015
V
0.5373
0.5318
0.5302
0.5415
W
θ = 0.7
0.6769
Nθ
0.9135
0.6305
Lθ
0.8442
0.6340
CN θ
0.7973
0.7276
RSTθ
0.9557
0.8928
0.8580
0.8665
0.9223
V
0.7782
0.7605
0.7685
0.7881
W
0.9021
0.8592
0.8602
0.9369
t
t γg
+
n
+
n
t
t γg
+
n
+
n
t
t γg
+
n
+
n
Figure 3 shows the power curves separately for all four tests under the distributions from
configuration 1.
8
1.0
0.8
0.6
0.4
power
0.6
0.4
Nθ
CNθ
Lθ
RSTθ
0.0
0.0
0.2
Nθ
CNθ
Lθ
RSTθ
0.2
power
0.8
1.0
Figure 3: Power curves of configuration 1, n=20, α = 5%
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
θ
0.8
1.0
θ
t-test
1.0
0.4
Nθ
CNθ
Lθ
RSTθ
0.0
0.0
0.6
0.8
Nθ
CNθ
Lθ
RSTθ
0.2
0.4
power
0.6
0.8
1.0
t γg -test
0.2
power
0.6
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
θ
0.6
0.8
1.0
θ
Vn+ -test
Wn+ -test
From Table 5 and Figure 3 we get the following results: The power values for each of the four
tests strongly depends on the underlying super model. The reaction of the four tests is very
different under the three distributions with equal Lévy distances to the normal. The power of
the t-test is mostly influenced by the distributions in contrast to the sign test where the power
is nearly the same.
Configuration 2:
Table 6 displays the power values of the four tests under the distributions from configuration
2 and again under the normal distribution for comparison.
Table 6: configuration 2 − power values for n=20, α = 5%
9
θ = 0.3
Nθ
0.3646
Dθ
0.1964
CN θ
0.1790
RSTθ
0.5167
0.3441
0.2213
0.2761
0.4189
V
0.2746
0.2437
0.2729
0.2796
W
θ = 0.5
0.3508
Nθ
0.6942
0.2294
Dθ
0.3649
0.2689
CN θ
0.3059
0.4764
RSTθ
0.8965
0.6656
0.4162
0.4996
0.7916
V
0.5373
0.4345
0.5155
0.5605
W
θ = 0.7
0.6769
Nθ
0.9135
0.4273
Dθ
0.5541
0.4859
CN θ
0.4504
0.8388
RSTθ
0.9948
0.8928
0.6280
0.6827
0.9741
0.7782
0.6225
0.7359
0.8307
t
t γg
+
n
+
n
t
t γg
+
n
+
n
t
t γg
+
n
+
n
V
W
0.9021
0.6320
0.6770
0.9816
Figure 4 shows the power
curves of the four tests under the distributions from configuration 2.
1.0
0.8
0.6
0.4
Nθ
CNθ
Lθ
RSTθ
0.0
0.2
power
0.6
0.4
0.2
Nθ
CNθ
Lθ
RSTθ
0.0
power
0.8
1.0
Figure 4: Power curves of configuration 2, n=20, α = 5%
0.0
0.2
0.4
0.6
0.8
1.0
0.0
θ
0.2
0.4
0.6
θ
t-test
t γg -test
10
0.8
1.0
1.0
0.4
power
0.6
0.8
1.0
0.8
0.6
0.4
Nθ
CNθ
Lθ
RSTθ
0.0
0.0
0.2
power
0.2
Nθ
CNθ
Lθ
RSTθ
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
θ
0.4
0.6
0.8
1.0
θ
Vn+ -test
Wn+ -test
As in configuration 1 we can state that the power values for each of the four tests strongly
depends on the underlying super model except perhaps for the sign test where the power is
again nearly the same for all distributions. The reaction of the four tests is more obvious in
configuration 2 than in configuration 1, especially for the t-test. The tests t γg and Wn+ show
nearly the same reaction. All the tests have smallest power for Dθ , except for the t-test which
has greatest loss in power under CN θ . But now, the sign test has highest power for Dθ
among the four tests.
Configuration 3:
Table 7 displays the power values of the four tests under the distributions from configuration
3 and again under the normal distribution for comparison.
Table 7: configuration 3 − power values for n=20, α = 5%
θ = 0.3
Nθ
0.3646
Uθ
0.5395
CN θ
0.1605
RSTθ
0.5424
0.3441
0.4302
0.2655
0.4347
V
0.2746
0.2802
0.2732
0.2821
W
θ = 0.5
0.3508
Nθ
0.6942
0.4982
Uθ
0.9140
0.2580
CN θ
0.2644
0.5002
RSTθ
0.9145
0.6656
0.8097
0.4705
0.8115
V
0.5373
0.5638
0.5122
0.5668
W
θ = 0.7
0.6769
Nθ
0.9135
0.8552
Uθ
0.9971
0.4611
CN θ
0.3864
0.8557
RSTθ
0.9974
0.8928
0.9787
0.6420
0.9794
V
0.7782
0.8379
0.7293
0.8401
W
0.9021
0.9849
0.6440
0.9853
t
t γg
+
n
+
n
t
t γg
+
n
+
n
t
t γg
+
n
+
n
11
Figure 5 shows the power curves of the four tests under the distributions from configuration 3.
1.0
0.8
0.6
0.4
power
0.6
0.4
Nθ
CNθ
Lθ
RSTθ
0.0
0.0
0.2
Nθ
CNθ
Lθ
RSTθ
0.2
power
0.8
1.0
Figure 5: Power curves of configuration 3, n=20, α = 5%
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
θ
0.8
1.0
θ
t-test
1.0
0.4
Nθ
CNθ
Lθ
RSTθ
0.0
0.0
0.6
0.8
Nθ
CNθ
Lθ
RSTθ
0.2
0.4
power
0.6
0.8
1.0
t γg -test
0.2
power
0.6
0.0
0.2
0.4
0.6
0.8
1.0
0.0
θ
0.2
0.4
0.6
0.8
1.0
θ
Vn+ -test
Wn+ -test
From Table 7 and Figure 5 we get the following results: Under the RST-distribution and the
uniform the differences of power are very small. That is not surprising because we can see
from configuration 3 in Figure 2 that the RST- and the uniform distribution have nearly the
same shape. Again, we see that the power of the tests is heavily influenced by the underlying
supermodel, especially for the t-test.
Configuration 4:
12
Table 8 displays the power values of the four tests under the distributions from configuration
4 and again under the normal distribution for comparison.
Table 8: configuration 4 − power values for n=20, α = 5%
θ = 0.3
Nθ
0.3646
CA θ
0.1352
CN θ
0.1519
RSTθ
0.5498
0.3441
0.2229
0.2526
0.4379
V
0.2746
0.2632
0.2744
0.2792
W
θ = 0.5
0.3508
Nθ
0.6942
0.2269
CA θ
0.2117
0.2519
CN θ
0.2550
0.5064
RSTθ
0.9177
0.6656
0.4056
0.4471
0.8151
V
0.5373
0.4829
0.5129
0.5675
W
θ = 0.7
0.6769
Nθ
0.9135
0.4071
CA θ
0.2954
0.4486
CN θ
0.3746
0.8594
RSTθ
0.9973
0.8928
0.5800
0.6158
0.9811
V
0.7782
0.6779
0.7242
0.8434
W
0.9021
0.5794
0.6246
0.9865
t
t γg
+
n
+
n
t
t γg
+
n
+
n
t
t γg
+
n
+
n
Figure 6 shows the power curves of the tests under the distributions from configuration 4.
1.0
0.8
0.6
0.4
Nθ
CNθ
Lθ
RSTθ
0.0
0.2
power
0.6
0.4
0.2
Nθ
CNθ
Lθ
RSTθ
0.0
power
0.8
1.0
Figure 6: Power curves of configuration 4, n=20, α = 5%
0.0
0.2
0.4
0.6
0.8
1.0
0.0
θ
0.2
0.4
0.6
θ
t-test
t γg -test
13
0.8
1.0
1.0
0.4
power
0.6
0.8
1.0
0.8
0.6
0.4
Nθ
CNθ
Lθ
RSTθ
0.0
0.0
0.2
power
0.2
Nθ
CNθ
Lθ
RSTθ
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
θ
0.6
0.8
1.0
θ
Vn+ -test
Wn+ -test
From Table 8 and the Figure 6 we can state as in the configurations 1, 2 and 3 that the power
strongly depends on the assumed supermodel but now all the tests have smallest power under
the Cauchy, a distribution with very long tails. Again, the t-test reveals highest and the
Vn+ − test smallest variations under the four distributions.
1.0
0.8
0.6
0.4
Nθ
CNθ
Lθ
RSTθ
0.0
0.2
power
0.6
0.4
0.2
Nθ
CNθ
Lθ
RSTθ
0.0
power
0.8
1.0
Now, let us have a special look at the t-test. From Table 5, 6, 7, and 8 we see that the t-test
reacts differently on changes of the supermodels, most in configuration 4 and least of all in
configuration 1 which is illustrated by Figure 7. The reason for that may be the increasing
Lévy distances between the normal and the distributions from configuration 1 up to
configuration 4.
Figure 7: Power of the t-test under the four configurations
0.0
0.2
0.4
0.6
0.8
1.0
0.0
θ
0.2
0.4
0.6
θ
Configuration 1
Configuration 2
14
0.8
1.0
1.0
0.4
Nθ
CNθ
Lθ
RSTθ
0.0
0.2
power
0.0
0.6
0.8
1.0
0.8
0.6
0.4
power
0.2
Nθ
CNθ
Lθ
RSTθ
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
θ
0.6
0.8
1.0
θ
Configuration 3
Configuration 4
At the end of this section it should be noted that similar results are true for n=50 and n=100,
although the influence of the supermodel on the power of the tests becomes smaller with
increasing sample sizes, a fact which is not surprising.
5. CONCLUDING REMARKS
As an answer of our question in the Introduction we may say that the power of tests strongly
depends on the assumed supermodel. We have seen that distributions with the same distance
to the normal distribution can produce very different power results. A better criterion for
robustness as the Lévy distance may be the tailweight or skewness of a distribution in
comparison to the normal. The distributions in our simulation study have different tailweight
as already seen from Figure 2 in section 3. But how can we measure tailweight or skewness of
a distribution? A comprehensive study of such so called selector statistics is carried out by
Hüsler (1988). Because in this paper only symmetric distributions are considered we restrict
our attention to a measure of tailweight, e.g., the measure M T which is defined as follows
x
− x 0.025
M T = 0.975
where x p is the p-quantile of the distribution, see Büning (1991, p. 262).
x 0.875 − x 0.125
Obviously, the measure M T is location and scale invariant.
For the normal distribution we have M T = 1.704 . Values of M T for all the distributions in
our four configurations are given in Table 9.
Table 9 M T for some distributions
Config1 L
CN
1.883
2.012
MT
Config2 D
CN
2.161
3.424
MT
Config3 U
CN
1.267
1.841
MT
Config4 CA
CN
5.263
3.446
MT
RST
1.567
RST
1.304
RST
1.267
RST
1.254
15
From Table 9 we can see that the values of M T in each of the four configurations are very
different although the distributions have the same Lévy distance to the normal. Otherwise,
distributions may have the same tailweight but very different distances to the normal, see e.g.
the RST-distributions in the configurations 2, 3 and 4. For these three RST-distributions the
power of the four tests is nearly the same, see Tables 6, 7 and 8. On the other hand, the
distribution L in configuration 1 and CN in configuration 3 have nearly the same tailweight
and different distances to the normal, but the power under these two distributions is more or
less different for each of the tests. The question arises: Is the tailweight of distributions a
better concept for power comparisons than the Lévy distance? A convincing answer might be
of great interest.
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Gruyter.
Huber, P.J. (1981). Robust statistics. New York: Wiley.
Hüsler, J. (1988). On the asymptotic behaviour of selector statistics, Communications in
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