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Data-driven Neyman’s Test for Symmetry
Ximing Wu∗ and Qi Li†
May 21, 2010

Abstract
This study proposes a data driven Neyman’s smooth test for symmetry. For a given sample,
we first construct probability integral transforms with respect to a symmetrization of the sample distribution. Under the null hypothesis of symmetry, the transformed data has a limiting
uniform distribution, reducing testing for symmetry to testing for uniformity. We generalize
Ledwina’s (1994) data driven smooth test to accommodate unknown distributions that do not
necessarily belong to prescribed parametric families. We show that only odd-order orthonormal moments of the transformed data are required in constructing the test statistic. We then
establish consistency and power properties of the proposed test. We also extend the test to dependent data. Our Monte Carlo simulations demonstrate the efficacy of the data driven smooth
test for symmetry. Compared with existing methods, substantial power gains are suggested. An
empirical application on testing symmetry of wage adjustment process is provided.




Texas A&M University, College Station, TX 77843; xwu@ag.tamu.edu.
Texas A&M University, College Station, TX 77843; qi@econmail.tamu.edu.

1

1

Introduction

Many economic variables are known to behave differently under contrasting situations. For instance,
Bacon (1991) made the now famous observation that the retail price of gasoline, relative to its
wholesale price, rises like a rocket but falls like a feather. Whether a quantity of interest exhibits
asymmetric behavior plays an important role in the investigation of many economic questions.
For example, in macroeconomics, people want to know if an economic variable behaves similarly
during expansions and recessions; in finance, investors are interested in knowing whether return
distributions are skewed. Answers to these questions usually provide value input in people’s decision
making.
The knowledge of symmetry is also important in statistical and econometric analyses. For example under symmetry, convergence rate of bootstrap confidence interval coverage probability is of
order n−1 rather than the typical n−1/2 (Hall, 1995). The intercept of a linear regression is identified in adaptive estimation when the error distribution is symmetric (Bickel, 1982). The efficiency
of estimators can be improved by incorporating symmetry (see, e.g., Horowitz and Markatou (1996)
for panel model estimation via deconvolution, and Chen (2000) for binary choice model).
Testing for symmetry has a long history in statistics and econometrics. One can characterize
symmetry tests into two broad categories. The former is based on some characteristics of symmetric
distributions. For instance, there are tests based on moments (see, e.g., Bai and Ng (2005) and
Premaratne and Bera (2005)), tests based on ranks (e.g., H´ajek and Sidak (1967) and Shorack
and Wellner (1986)), and tests based on the empirical characteristic functions (e.g., Cs¨org˝
o and
Heathcote (1987)).
The second group of tests employ a symmetrization of the sample distribution and construct
test statistics as a distance between the sample distribution and its symmetrization. For a random sample {Xi }ni=1 from a continuous distribution, denote the sample distribution Fn and its
symmetrization about a center tendency parameter θ, F˜ . Butler (1969) and Schuster and
n,θ

Barker (1987) designed a Komogorov-Smirnov type test based on supx (Fn (x) − F˜n,θ (x)). Rothman
and Woodroofe (1972) proposed tests based on the integrated squared difference in distributions
R
(Fn (x) − F˜n,θ (x))2 dx. There are also tests based on distance in densities. Let fn and f˜n,θ be
densities, if they exist, associated with Fn and F˜n,θ respectively. Amhad and Li (1997) suggested a
´2

test based on the integrated squared difference in densities
fn (x) − f˜n,θ (x) dx, and Massoumi
µ
¶2
q
R p
˜
and Racine (2009) presented a test based on the Hellinger distance
fn (x) − fn,θ (x) dx.
Following the second approach, this study proposes an alternative test that examines the dis2

crepancy between the sample distribution and its symmetrization. Our method is based on the
probability integral transformation ti (θ) = F˜n,θ (Xi ), i = 1, . . . , n. Under the null hypothesis of symmetry, Fn should be close to F˜n,θ and {ti (θ)}ni=1 is distributed according to the standard uniform
distribution asymptotically. Thus testing for symmetry is reduced to testing for uniformity.
There exists a host of uniformity tests in the literature. Among them, Neyman’s smooth test is
particularly popular thanks to its appealing theoretical optimality, ease of construction and good
small sample performance. The finite sample performance of this test, however, can be sensitive
to the number of terms contained in the test statistic. We therefore adopt a data driven version of
Neyman’s test proposed by Ledwina (1994). This test chooses the number of terms according to
an information criterion and is shown to be consistent and adapts to the underlying distribution.
Instead of assuming the underlying distributions belong to known parametric families, we generalize
Ledwina’s (1994) test to a fully nonparametric one which accommodate unspecified alternatives.
Unlike moment-based tests, the proposed smooth test is based on moments of probability integral transforms of data and thus does not require existence of high order moments of raw data. Furthermore, we establish that by construction, even-order orthonormal moments of the transformed
data in the context of symmetry test are constants that only depend on sample sizes. Therefore,
our test uses only odd-order moments, where the number of moments is selected according to some
information criterion. Since the limiting distribution of the test statistic under symmetry depends
on the unknown distribution F when the center parameter is estimated, we use a pre-pivoting procedure to make the test statistic asymptotically pivotal. We derive the large sample distribution
of the proposed test under the null hypothesis of symmetry. For inferences, we use a “symmetric
bootstrap” method to approximate its finite sample distribution. We then generalize the proposed
test to dependent data. Our Monte Carlo simulations demonstrate the efficacy of the proposed
test. In particular, substantial power gains, relative to some existing methods, are reported.
The proposed test offers several advantages. First, it is invariant to monotone transformation
of data due to the probability integral transformation of raw data. Second, it has power against all
asymmetric alternatives and it adapts to the unknown distribution that is not required to belong
to some pre-specified families. Compared with tests based on nonparametrically estimated distributions/densities on raw data, the proposed test is based on an “easier” approximation problem
because the bias of approximating nonparametrically a uniform distribution is zero. Third, the test
statistic has a simple limiting form under the null hypothesis, a Neyman’s test with one moment
condition, and convergence towards this limiting form is speedy. Lastly, the proposed test statistic
can be easily constructed.
3

The rest of the paper is organized as follows. Section 2 briefly reviews the data driven Neyman’s
test. Section 3 introduces a symmetry test that applies the data driven Neyman’s test to the
probability integral transformation of raw data. The asymptotic properties of the proposed test
are discussed. Section 4 extends the proposed test to dependent data. Section 5 provides Monte
Carlo simulations and Section 6 an empirical example on testing symmetry of wage adjustment
process. The last section concludes. Mathematical proofs are gathered in Appendix.

2

Background

Neyman (1937) introduced the smooth goodness-of-fit test for uniformity. Let X = {Xi }ni=1 be an
iid sample from a continuous distribution p0 defined on [0, 1]. The smooth Neyman test rejects null
hypothesis of uniformity for large values of

Nk =

k
X

Ã
n−1/2

j=1

n
X

!2
bj (Xi )

,

(1)

i=1

where k is a positive integer, and b1 , b2 , . . . are normalized Legendre polynomials on [0, 1] constructed as

ª
(j!)−1 dj © 2
j
bj (x) = √
(x

x)
.
2j + 1 dxj

Under the null hypothesis, Nk is distributed asymptotically according to a chi-squared distribution
with k degrees of freedom.
Neyman (1937) constructed his test via a parametric testing problem H0 : γ (k) = 0 versus
H1 : γ (k) 6= 0 for a general exponential distribution
p(x; γ (k) ) = c(γ (k) ) exp


k
X


j=1



γj bj (x) ,


(2)

where k ≥ 1, x ∈ [0, 1], and γ (k) = (γ1 , . . . , γk ) ∈ Rk . The test statistic Nk defined in (1) is derived
as an asymptotically locally optimal solution to this testing problem. Alternatively, Nk can be
interpreted as Rao’s score statistic for H0 in model (2).
Neyman’s test has some appealing theoretical properties and is known for its good finite sample
performance (see, e.g., a review of goodness-of-fit tests by Rayner and Best (1990)). One difficulty
associated with Neyman’s test, however, is that the choice of k can often affect its power. Traditionally, many authors restrict attention to studying the power of Neyman’s test with a small k.
4

Ledwina (1994) proposed a data driven Neyman’s test. Instead of a single exponential model (2),
she considers a nested family of exponential models p(·; γ (k) ), 1 ≤ k ≤ K with K fixed, and chooses
in that family the density that fits the data best according to Schwarz’s Bayesian Information
Criteria (BIC).
Formally consider, for 1 ≤ s ≤ K,
Ls = log

n
Y

p(Xi ; γˆ (s) )

(3)

i=1

where γˆ (s) is the MLE solution to the model. We then choose the density fitting the data best
according to the following rule
½
¾
1
1
S = s : Ls − s log n > Lj − j log n, 1 ≤ j, s ≤ K .
2
2

(4)

When there are ties, the model with the smallest dimension is selected. Having chosen a model of
dimension S, we use NS as the data-driven Neyman’s test.
Assuming that the unknown distribution takes the form p(·; γ (s) ) for some s = 1, . . . , K, Ledwina
(1994) establishes the consistency and power properties of the above data driven test. In this study,
we generalize it to the case where the unknown distribution does not necessarily come from the
exponential family p(·; γ), but can be approximated arbitrarily well by this family.
Note that (4) can be interpreted as an approximation of the selection rule based on a minimum
description-length criterion by Barron and Cover (1991). They consider the density estimation
problem

(
qn = arg min C(q) − log
q∈Γ

n
Y

)
q(Xi ) ,

i=1

where Γ is the candidate set of sequence of parametric densities and C(q) measures the complexity of
q. The solution qn , termed the minimum complexity estimator, achieves a balance between accuracy
and simplicity. Under the condition C(q)/n → 0 and some regularity conditions on the underlying
distribution p0 , they establish the convergence rate of qn in terms of squared Hellinger distance.
When p ∈ Γ, the convergence rate is of order n−1 . When p0 belongs to an s dimensional parametric
family, the convergence rate is of order log n/n. More generally, when p0 does not necessarily come
from a finite parametric family but has degree of smoothness r, its approximation by a sequence of
parametric families converges at rate O((log n/n)2r/(2r+1) ). Therefore, with minimum complexity
estimation, we converge at a rate within a logarithmic factor of the rate obtainable with knowledge
5

of the smoothness class of the density. Below we establish the consistency and power properties of
the data driven smooth test under the condition that p0 is a smooth density not necessarily from
the prescribed regular exponential family.
Theorem 1 Let pS , 1 ≤ S ≤ K be the model selected according to (4) based on an iid sample
{Xi }ni=1 from distribution with density p0 defined on [0, 1]. Suppose that
Z
0

1

(Dr log p0 (x))2 dx < ∞,

where r is a positive integer, S → ∞ and S log n/n → 0 as n → ∞. When p0 is uniform, S → 1
and γˆ (1) → 0 in probability; when p0 is non-uniform, γˆ (S) converges to a non-zero constant, and
the power of NS converges to one as n → ∞.
Under H0 for any given s, Ns has an asymptotical central chi-squared distribution with s
degrees of freedom. Furthermore, under H0 , S → 1 as n → ∞. So the distribution of NS can be
approximated by the central chi-squared distribution with one degree of freedom. In practice, one
often replace the maximized log-likelihood Ls in (3) by a local approximation

1
2 Ns ,

where Ns is

given by (1) with k replaced by s.
Under uniformity, γˆ (s) → 0 for any given s. Theorem 1 entails that in this case, only one
moment condition is desired. Although for all s, Ns is consistent and has unitary asymptotic
power, its finite sample power can be affected by s. More precisely, under the null, NS=1 is the
most powerful test since generally the bigger is the number of ‘nuisance’ parameter involved in the
test, the lower is its power.
Since when S is data driven there is a non-zero probability that S > 1 under the null hypothesis,
it complicates the finite sample distribution of NS . Ledwina (1994) proposed to use Monte Carlo
methods to approximate the finite sample distribution of NS . The distribution of NS quickly
converges to its asymptotic distribution. More importantly, the distribution is shown to stabilize
under reasonably large K. Thus the test is not sensitive to the choice of K.
The properties of the data driven smooth test can be readily understood via simple Monte Carlo
examples. Table 1 reports the frequency of S = s for s = 1, . . . , K for K = 10 under uniformity
for small to moderate sample sizes. Each experiment is repeated 10,000 times. As predicted by
Lemma 1, S converges to 1 quickly as sample size increases. Even for sample size as small as 25,
the proportion of S = 1 is larger than 90%.
Table 2 tabulates simulated 5% critical values for Ns for various K. One can see that even
6

Table 1: Counts of {S = s} under uniformity
s
n
1
2
3
4
5
6
7
25 9021 621 199 73 32 22 16
50 9416 423 108 31 7
7
3
100 9608 332 42 13 4
0
1
200 9770 197 24
9
0
0
0

with K = 10
8
7
4
0
0

9
4
1
0
0

10
5
0
0
0

relatively rare cases with S > 1 have a substantial influence on the speed in which simulated
critical values of NS approach the limiting critical value. The suggested finite sample deviation of
the distribution of NS from its limiting distribution supports use of simulated critical values when
S is data driven. On the other hand, it is seen that the critical values stabilize as K increases,
indicating that the tests are not sensitive to the choice of K, provided it is reasonably large.

n
25
50
100
200

Table 2: 5% critical values of Ns simulated
K
1
2
3
4
5
6
3.793 5.486 6.562 6.951 7.135 7.257
3.841 5.327 5.748 5.891 5.998 6.008
3.777 5.208 5.336 5.361 5.367 5.367
3.765 4.550 4.632 4.647 4.647 4.647

for various values of K
7
7.322
6.028
5.367
4.647

8
7.339
6.035
5.367
4.647

9
7.350
6.035
5.367
4.647

10
7.356
6.035
5.367
4.647

The data-driven smooth test has been generalized in several directions. Kallenberg and Ledwina
(1997) extended the test to more general cases wherein the hypotheses are composite. Inglot and
Ledwina (2006) suggested that a host of criterion can be employed to construct consistent data
driven smooth tests. For instance, they showed that both AIC and BIC penalties lead to consistent
data-driven smooth tests. They further proposed a test whose penalty term is data-driven such
that the selection criteria is chosen automatically between AIC and BIC.

3

A data-driven Neyman’s test for symmetry

In this section, we present a symmetry test based on the data driven Neyman’s test discussed
P
above. Let Fn (x) = n1 ni=1 I (Xi ≤ x) be the empirical CDF of an iid sample {Xi }ni=1 from a
continuous distribution F . Let θˆ be an estimated center tendency parameter of Fn , and Fn,θˆ (x) =
³
´
1 Pn
ˆ − x the empirical distribution obtained by rotating Fn about θ.
ˆ Further define
I
X

2
θ
i
i=1
n

7

n
o
Y = X1 , . . . , Xn , 2θˆ − X1 , . . . , 2θˆ − Xn . The symmetrization of Fn about θˆ is then defined as
( 2n
)
X
1
1
F˜n,θˆ (x) =
I(x ≤ Yi ) −
2n
2
i=1
´
1

Fn (x) + Fn,θˆ (x) −
=
2
4n
³
´´

1
=
Fn (x) + 1 − Fn 2θˆ − x −
.
2
4n

(5)

Since our test is based on F˜n,θˆ, we shall first investigate its large sample properties. The
consistency of the empirical process F˜n,θˆ is established by Arcones and Gin´e (1991). They show
that for n−1/2 -bootstrap consistent location parameter θ, the empirical process F˜ ˆ converges to F
n,θ

almost surely provided that F is symmetric and absolutely continuous with a uniformly integrable
density f . Note that the consistence requirement on the location parameter θ is slightly stronger
than usual in the sense that we must impose joint convergence of the parameter and the empirical
process. On the other hand, this requirement is automatically satisfied if θ is differentiable in any
reasonable sense (see theorem 2.3 of Acrones and Gin´e (1991) for details).
Acrones and Gin´e (1991) investigate theoretically the Komogorov-Smirnov type test τn =
− F˜n,θˆ(x)) in Schuster and Barker (1987). Because the limiting distribution of

n1/2 supx (Fn (x)

τn under the null hypothesis of symmetry depends on the unknown distribution F , Schuster and
Barker (1987) proposed a “symmetric bootstrap”method for inferences. Let Fn∗ be the empirical CDF obtained by resampling from F˜ ˆ and θˆ∗ its center parameter. Denote by F˜ ∗ the
corresponding symmetrization of

Fn∗

n,θ
about θˆ∗ .

n,θˆ∗

Schuster and Barker (1987) used the bootstrap dis-

tribution of τn∗ = n1/2 supx (Fn∗ (x) − F˜n,θˆ∗ (x)) to approximate the distribution of τn . Arcones and
Gin´e (1991) established that τn and τn∗ have the same limiting distribution when F is symmetric
whereas τn → ∞ and τn∗ still converges weakly almost surely if F is not symmetric.
In this study we consider a test based on the probability integral transformation of the raw
ˆ = F˜ ˆ(Xi ), i = 1, . . . , n. For a given r = 1, . . . , K, we first
data. Denote the transformed data ti (θ)
n,θ

calculate the sample orthonormal moments
n
³
´
X
ˆ ,
ˆbr (θ)
ˆ = 1
br ti (θ)
n
i=1

where br ’s are normalized Legendre polynomials defined on [0, 1]. Note that tests based on moments
of raw data X require the existence of those moments. In addition, higher order sample moments

8

can be sensitive to outliers and require a large sample for precise estimations (see, e.g., Bai and
Ng (2005) on the estimation of sample kurtosis). In contrast thanks to the probability integral
transformation, our test does not require existence of moments of X and thus is relatively robust
to outliers.
Generally, Neyman’s test can then be constructed as Ns =

³√
´2
ˆbr (θ)
ˆ , where s is either
n
r=1

Ps

pre-specified or obtained in some data driven manner. However in the current context, we show
that only odd orthonormal moments are required because even orthonormal moments depend only
on sample size according to the following proposition.
Proposition 1 Let F˜n,θ be the symmetrization of a random sample X from a continuous distribution and θ be an arbitrary constant. Then for all even integers r, ˆbr (θ) = ar,n , where ar,n is a
constant that depends on only r and n.
In other words, the transformed data {ti (θ)}ni=1 are “symmetric” by construction in the sense
P
that its rth order arithmetic moment µ
ˆr (θ) = 1/n ni=1 tri (θ), where r is an even integer, does not
contain additional information given {ˆ
µ1 (θ), . . . , µ
ˆr−1 (θ)}. We note that the above results hold for
not only the location parameter, but also an arbitrary θ.
Because of the lack of independent information of even order moments, we construct the data
driven smooth test of symmetry based on the first s odd orthonormal moments of transformed
data. Define, for s ≥ 1,
Ns =

s ³
´2
X

ˆ
nˆb2r−1 (θ)
.

(6)

r=1

The maximized log-likelihood of (2), which is in fact the log-likelihood ratio statistic, is locally
equivalent to 12 Ns . Following Ledwina (1994), we choose the number of terms in Ns according to
the following information criterion
S = min {s : Ns − s log n ≥ Nj − j log n, 1 ≤ j, s ≤ K} .

(7)

Upon choosing an optimal S, we next consider distribution of NS . According to Theorem 1,
under the null hypothesis S → 1 as n → ∞. If the center parameter θ is known, one can easily
show that N1 has a limiting χ2 distribution with one degree of freedom. However, since θ has to
be estimated, the limiting distribution of N1 generally depends on the unknown distribution F .
Below we present the limiting distribution of N1 where the unknown center parameter is estimated
by the sample average. Results for other alternative estimators of the center parameter, such as
9


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