Mean Value Estimation Using Low Size Samples Extracted from
Skewed Populations
JOÃO PAULO MARTINS1,3, MIGUEL FELGUEIRAS2,3, RUI SANTOS1,3
1School of Health, P. Porto, Porto, PORTUGAL
2School of Technology and Management, Polytechnic Institute of Leiria, Leiria, PORTUGAL
3CEAUL – Center of Statistics and Applications, Faculdade de Ciências, Universidade de Lisboa,
Lisboa PORTUGAL
Abstract: - The use of the -statistic in statistical inference procedures is usually restricted to normal
populations or to large samples. However, these conditions may not be fulfilled in some situations: the
population can be moderate/highly skewed, or the sample size can be small. In this work, we use the Pearson’s
system of distributions, namely, type IV distributions to model . By some simulation work, it is shown that
this approximation leads to confidence intervals whose coverage is close to the nominal coverage even for low
sample sizes.
Key-Words: - -statistic, Type IV distributions, Pearson’s system, skewness, kurtosis, estimation, confidence
interval, coverage, mean value, simulation
Received: May 20, 2021. Revised: January 23, 2022. Accepted: February 18, 2022. Published: March 17, 2022.
1 Introduction
Let be a random sample with mean
and
standard deviation drawn from a population
with finite mean and standard deviation . The
study of the distribution of the ratio
(1)
under an underlying Normal distribution, presented
by Gosset [1] (under the pseudonym Student) was
one of the seeds of the development of Statistical
Inference. However, the potentialities of using
have not (surprisingly?) been exploited outside the
comfort of the Normal distribution or the scope of
the Central Limit Theorem (CLT).
At the beginning of the last decade, [2] derived
the first four moments of the Student's -Statistic
for any underlying population with finite first four
moments. The derived approximations for all these
four moments depend only on two measures:
skewness and kurtosis. Skewness is defined as
(2)
and kurtosis is given by
. (3)
For a Normal distribution both measures are
equal to zero.
Using Delta method (cf. [3]), [2] derived
expressions for the first four moments of that are
describe in expressions (4) to (7).
(4)
(5)
(6)
(7)
Surprisingly, the first three moments estimates
only depend on .
From equations (4) to (7), clearly, as increases
the importance of skewness decreases. When is
large, Slutsky’s theorem [4] allows the application
of the Student's -statistic to non-normal
populations. However, statistical inference for small
sample sizes may not be possible if the underlying
distribution is not symmetric. This is also clear
when we compute the first-order Edgeworth
expansion of (Edgeworth expansions are a
particular case of the well-known Gram-Charlier
series which allows to write a distribution function
of some variable from a well-known one, usually
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the Normal standard distribution). Let be the
cumulative distribution function (cdf) of . Then,
(8)
where is the third-order derivative of the
cdf of a standard normal distribution, cf [5].
Pearson’s system of distributions is a partition of
the set of all distributions with finite first four
moments cf. [6,7] whose probability/probability
density function satisfies the following differential
equation:
(9)
where and are distribution parameters.
The solutions of equation (9) are divided into
seven groups known as Pearson’s type of
distributions that range from I to VII.
Multiplying equation (9) by (with
and integrating it, it is possible to
derive the relation between the four parameters and
the first four raw moments [8]. [4] provide a more
comprehensive overview on this subject. However,
it should be noted that the expression presented by
[4] for the coefficient contains an inaccurate.
Where it should be , it appears where
, which of course is not equivalent.
[4] performs the partition of the distributions
considering the combination of parameters
(10)
The case corresponds to the type IV
distributions.
Let stand for gamma function and consider
the beta function
(11)
where and .
The probability density function is given by
where and .
None of the most used distributions in Statistics
verifies the density function defined in (12), i.e.,
none is a type IV distribution. Nevertheless, it is
possible to identify in literature examples where
these distributions are used to model real life
problems (cf. [9,10]).
Several researchers have dedicated some of their
attention to this type of distributions. [11]
determined several values of type IV distribution
functions. Later, [12] have constructed an algorithm
for determination of some quantiles that can be
applied to any of the types of distributions of the
Pearson system. [13] analyzed the moments of type
IV distributions. More recently, [14-16] determined
approximate expressions for a type IV distribution
function. Details of packages/macros developed to
allow the use of type IV distributions can be found
in [17] for software and in [18] for SAS
software.
Under broader conditions, [1] showed that is
a type IV distribution if is non-symmetric, i.e.,
. It is clear from equation (8) that as
increases gets close to a normal distribution.
Until recently, there was no closed form expression
for the cdf of a type IV distribution. Moreover, this
type of distribution depends on four parameters.
This is important because if the sample size is high,
it is possible to apply the CLT and if the size is low
it may not be reasonable to estimate all four
parameters. These two issues may help to explain
the little of use of this kind of distributions in
statistical inference. However, it is now possible to
use software to easily compute probabilities or
quantiles. Hence, in the recent years some
applications involving the use of type IV
distributions can be found. For instance, its use in
econometric modelling [19] or in operating room
management [20].
In practice, the problem of fitting a type IV
distribution to is that it requires finite first four
moments of the underlying distribution and
estimates of its skewness and kurtosis. In the
context of a samples with low sizes this may be a
challenge as previously discussed. Moreover, it is
not clear if it there is any advantage of using a type
IV distribution instead of just using a Normal
distribution even for small sample sizes.
In this work, we address the problem of using the
-statistic in small samples and skewed populations
to perform statistical inference. A Bayesian
approach to this matter can be found in [21].
The outline of this work is as follows. In Section
2, two confidence intervals are presented for the
mean value of a population: one based in the
Normal approximation of and the other based on
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the approximation to a type IV distribution. In
Section 3, simulation work is presented to compare
the two confidence intervals. In two of the studied
cases studied, it also addressed the situation where
the skewness can be written using the mean value.
In the last Section, the obtained results are
discussed.
2 Confidence Intervals for the Mean
Given a random sample drawn from a
population with finite variance, the application of
the CLT allows us to obtain the following
confidence interval for the mean value of :
(13)
where is the quantile of a standard normal
distribution. This confidence interval is widely used.
When the underlying population is normal the
-Student with degrees of freedom should be
used if is not large [4].
In a similar way, it is possible for small sample
sizes to replace the quantiles used in (13) when
dealing with skewed populations. Approximating
by a type IV distribution that verifies the estimates
defined by equations (4) to (7), we get an alternative
confidence interval for :
(14)
where is the quantile of a type IV
distribution. Clearly, this interval is no longer
symmetric. In practice, the application of that
interval requires some knowledge about the
population kurtosis and especially the
population skewness . Thus, its usefulness in
practice needs to be assessed by some simulation
work. It is not clear if there is any advantage in
estimating both skewness and kurtosisin order to
compute estimates for the bounds defined in (14).
The confidence interval defined in (14) only
makes sense when . Otherwise, the
approximation to a type IV distribution is no longer
valid. The distribution would be a type VII
distribution which (with no surprise) corresponds to
the -Student distribution.
To compute the type IV distribution quantiles
involved in (14), package PearsonDS [17] was
used.
3 Simulation Results
To analyse the performance of the confidence
intervals (13) and (14) several underlying
distributions were considered. The first choice was
the exponential distribution , where
because both skewness and kurtosis do not
depend on the parameter . The results concerning
the estimated coverage probability, i.e., the
proportion of intervals that contain for a 95%
confidence level when equation (13) (Normal) and
equation (14) (Type IV) are used are presented in
Table 1 () and 2 (). Samples sizes
ranging from 5 to 200 ( replicas) were
considered.
Table 1. Estimated coverage for
(=6)
Normal
TypeIV
5
81.11
89.44
10
86.95
91.09
15
89.21
91.00
20
90.20
91.09
30
91.79
91.09
50
92.92
91.16
200
94,39
94.01
Table 2. Estimated coverage for
(=6)
Normal
TypeIV
5
81.21
89.63
10
86.81
90.93
15
89.13
91.48
20
90.56
91.90
30
91.79
92.17
50
93.00
92.76
200
94.74
94.15
Clearly, the observed coverages are very similar
when the mean value of the underlying distribution
changes. Comparing the performance of the
confidence intervals (13) and (14), the coverage
improves when a type IV distribution is used for
low sample sizes. For large samples, the coverages
of both intervals are similar with a little advantage
to the Normal distribution.
Both confidence intervals tend to be liberal in the
sense that its coverage is lower than the nominal
probability.
As we are working in a skewed population
setting two more skewed distributions were
considered in the more likely situation of having to
estimate both skewness and kurtosis. To assess the
performance of the confidence intervals with
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discrete and continuous underlying distributions it
was chosen the Poisson and Chi-square with
degrees of freedom
distributions. The usual
confidence levels were used: 0.90, 0.95 and 0.99.
In those two distributions, skewness and kurtosis
can be estimated different ways. It is possible to use:
their observed values as estimates –
strategy TypeIV,o;
the sample mean to estimate both
measures using the relation between
each one and – strategy TypeIV,m.
Recall that for a Poisson distribution :
and (15)
For a Chi-square distribution
:
and (16)
Table 3 to Table 5 present the results of the
estimated coverage probability for both strategies
(compared to the use of the Normal approximation)
at a 95% confidence level for two different Poisson
distributions.
Table 3. Estimated coverage for (
=1)
Normal
TypeIV,o
Type IV,m
10
91.04
92.76
93.23
15
91.45
92.51
92.85
20
92.67
93.15
93.05
30
93.26
93.84
93.73
50
93.96
93.96
93.79
200
94.62
94.63
94.75
Table 4. Estimated coverage for (
=0.3333)
Normal
TypeIV,o
Type IV,m
10
91.41
93.89
93.58
15
92.56
94.17
93.87
20
93.12
94.20
94.22
30
93.93
94.53
94.39
50
94.21
94.61
94.48
200
94.95
95.03
94.82
Clearly, using type IV distributions shortens the gap
between the nominal and real confidence. For small
sample sizes the gap is roughly half of what we
would get using the Normal approximation.
Comparing TypeIV,o to TypeIV,m there is no clear
winner between the two strategies of estimating
skewness.
Table 5. Estimated coverage for (
=0.2)
Normal
TypeIV,o
Type IV,m
10
91.76
92.96
94.78
15
92.89
94.16
94.29
20
93.25
94.48
94.09
30
93.98
94.44
94.52
50
94.47
94.69
94.70
200
94.89
94.88
94.87
The script of the simulations performed with an
underlying Poisson distribution, due to discrete
nature of the distribution, must take into account
two issues than turn impossible the estimation of the
skewness directly from the sample (strategy
TypeIV,o). A finite value for the sample skewness
cannot be computed if all values are equal because
there is no possibility of estimating the sample
variance. Another issue arises, if the sample is
symmetric as the distribution from Pearson’s system
that is going to fit data is no longer a type IV
distribution (it would be a type VII distribution).
Every replica/sample that met one of those two
criteria were excluded from the simulation and
replaced by other (simulated) sample. Thus, in
Tables 3 to 5 the sample size was not
considered due to the high number of samples with
all values equal or symmetric.
Table 6 to Table 8 are like the previous ones but
consider underlying Chi-squared distributions
Table 6. Estimated coverage for
(
=2)
Normal
TypeIV,o
Type IV,m
5
85.49
89.85
91.75
10
90.11
92.3
92.53
15
91.59
92.95
92.74
20
92.41
93.31
93.25
30
93.38
93.78
93.51
50
93.90
94.06
93.76
200
94.81
94.72
94.56
All confidence intervals are once again liberal.
For a given , the real coverage tends to be closer to
the nominal coverage when and/or are
close to zero.
As expected, the strategies that use a type IV
distribution overcome the approximation to the
Normal distribution. However, for a large , the
performances are similar. When using a type IV
distribution, it is not clear what is the best strategy
to follow: TypeIV,o or TypeIV,m.
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Table 7. Estimated coverage for
(
=1)
Normal
TypeIV,o
Type IV,m
5
86.51
91.14
91.51
10
91.13
93.48
93.06
15
92.30
93.74
93.40
20
92.98
94.00
93.73
30
93.74
94.36
94.00
50
94.26
94.55
94.28
200
94.57
94.73
94.64
Table 8. Estimated coverage for
(
=0.6667)
Normal
TypeIV,o
Type IV,m
5
89.92
91.63
91.38
10
91.18
93.70
93.46
15
92,54
94.10
93.82
20
92.97
94.11
94.13
30
93.78
94.45
94.29
50
94,27
94.68
94.47
200
94.85
94.90
94.92
All conclusions are similar for other levels of
coverage: 0.90 and 0.99 (results not shown).
4 Discussion
More than one hundred years after Gosset’s work,
under the pseudonym of Student, the -ratio
potentiality has not been totally exploited yet. The
-Student statistic is in the genesis of what we now
call Statistical Inference.
Trying to use very well-known methods to
situations where assumptions are violated is
common and, above all, a need in the way data is
often messier than desired [22]. In the literature,
several works regarding sample size calculation for
skewed populations can be found (cf. [23,24]).
In [25], using simulation, it is described that the
normative value of 50 for the sample size is not
enough when the population is skewed.
However, when it is not possible to get a sample
whose size is not equal or higher to the desired one
inferential statistics may also be performed even if
with some constraints.
This work showed how the -Student statistics can
be used outside the CLT assumptions. Population
skewness should be considered and even in samples
with only 5 individuals it is possible to improve the
coverage of the confidence intervals (comparing to
the straightforward application of the CLT).
However, there was no clear winner between the
two analyzed strategies that used type IV
distributions. This is, at some extent, surprising
since the sample mean has some optimal proprieties
as an estimator of the population mean value.
Therefore, we would expect a better performance of
strategy TypeIV,m.
When both skewness and kurtosis
do not exceed 1, a coverage of about 94% is
observed for samples sizes as low as 15 (Tables 4, 5
and 8). As those measures increase and are closer to
1, the required sample sizes for that coverage is
about 20 (Table 7) or 50 (Table 3 and 6). Clearly,
when skewness or kurtosis is very high, a large
sample is required to achieve that level of coverage
as seen when the underlying distribution was
Exponential (Tables 1 and 2).
5 Conclusion
Clearly, the approximations to a type IV distribution
is only a plus when the sample size is low and/or
skewness is at least moderate. For instance, for
, the performances were, in general, quite
similar. In the future, it can be studied, in more
detail, conditions where it makes sense to use the
type IV distribution instead of the Normal
distribution.
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