行政院國家科學委員會專題研究計畫 成果報告
兩個一致性的模型設定新檢定
研究成果報告(精簡版)
計 畫 類 別 : 個別型
計 畫 編 號 : NSC 98-2410-H-004-057-
執 行 期 間 : 98 年 08 月 01 日至 99 年 07 月 31 日
執 行 單 位 : 國立政治大學經濟學系
計 畫 主 持 人 : 徐士勛
計畫參與人員: 博士班研究生-兼任助理人員:謝子雄
博士班研究生-兼任助理人員:徐兆璿
博士班研究生-兼任助理人員:曾憲政
處 理 方 式 : 本計畫可公開查詢
中 華 民 國 99 年 10 月 28 日
摘要
一般而言, 經濟或計量模型都可以藉由 「條件動差限制式」 來加以定義。 因此, 如何利用條件動差限制式來 檢驗模型設定的正確與否一直是文獻上的重要議題。 其中, 廣為人知的檢定是 Bierens (1982, Journal of Econometrics) 所提出的 「條件動差積分型檢定」 (Integrated Conditional Moment test, 簡稱為 ICM 檢 定)。 然而, 這類的 ICM 檢定有一些已知的缺點: 它需要對應於虛無假設下的一致性參數估計式、 需要運 用數值方法以進行積分運算, 並且其檢定統計量的極限分配會隨著資料的特性而不同。 這些特徵都增加了 ICM 檢定於實際操作時的難度。 準此, 這個計畫提出了兩個方式來改善此類 ICM 檢定。 第一個方法結合 了 ICM 檢定及傅利葉分析, 進而提出了一個容易操作的檢定。 這個檢定統計量和 ICM 檢定具有相同的 極限分配, 但是它不需仰賴數值方法進行積分。 再者, 由於此方法自然而然地將條件動差限制下的參數估 計及模型檢定相結合, 因此我們也不需要額外的一致性參數估計式。 除此之外, 我更擴展了這第一個方法 的精神進而得到了一組無窮多條 「無條件動差限制式」, 並與一般化的實證概似法 (generalized empirical likelihood) 結合, 建立了一個概似比例型態 (Likelihood-Ratio type) 的檢定統計量。 這是我於此計畫中所 提出的第二個檢定方法。 在一些條件假設下, 此統計量經過適當的標準化後, 其極限分配將會是和原始資料 型態無關的標準常態分配。 針對這兩種新的檢定, 我們建立了對應的極限性質, 也提供了模擬的結果。 關鍵詞: 一致性檢定, 條件動差限制式, 條件動差積分型檢定, 傅力葉分析, 一般化的實證概似法
Abstract
Economic and econometric models are usually defined by conditional moment restrictions. Hence, checking the validity of models through these conditional moment restrictions is a central issue in the literature. One of the popular consistent tests is the Integrated Conditional Moment (ICM) test pro-posed by Bierens (1982, Journal of Econometrics). This ICM-type test, however, suffers from some drawbacks: it needs a preliminary consistent estimate of model parameters under the null, the nu-merical method of integrations, and it is not pivotal. These features make the implementation of this ICM-type test cumbersome. This project proposes two approaches to improve the ICM test. The first proposed test statistic is an easy-to-implement version of ICM test by extending the ICM test and the Fourier analysis. It is asymptotic equivalent to the ICM test statistic, but it has an analytic form instead. Moreover, no preliminary consistent estimate is needed because this unified approach relates estimation and diagnostic testing in a rather natural way. Besides, I extend the idea behind the first approach by deriving a new set of infinite many unconditional moment restrictions, and employ gen-eralized empirical likelihood (GEL) method to construct the Likelihood-Ratio type test statistic. After suitable normalization, the asymptotic distribution of the proposed empirical LR test statistic should be standard normal, which is pivotal. For these two proposed tests, we establish the corresponding asymptotics and provide some Monte Carlo simulation results.
Keywords: consistent test, conditional moment restrictions, integrated conditional moment test, Fourier analysis, generalized empirical likelihood method, pivotal
1
Introduction
Economic and econometric models are usually defined by conditional moment restrictions, for exam-ple, the Euler equations in various rational expectation models. How to consistently (and efficiently) estimate parameters of models through these moment restrictions is thus an important issue and has been considered by Chamberlain (1987), Donald et al. (2003), Dom´ınguez and Lobato (2004) and Hsu and Kuan (2010) to mention just a few. On the other hand, because a correct model specifi-cation implies the certain zero conditional moments, checking the validity of models through these conditional moment restrictions is thus another central issue in the literature. A good test should have power approaching one asymptotically for any deviations from the null. Accordingly, the tests should be constructed to against general alternatives, and they are said to be “consistent”. To provide consistentmodel specification tests is the main purpose of this work.
Based on the idea that unconditional moment restrictions can be induced from conditional mo-ment restrictions, many researchers propose the conditional momo-ment tests by testing whether some (finitely) induced unconditional moment are zeros or not, see Newey (1985), Tauchen (1985), among others. It is well known that these tests are in general not consistent because they are “directional”. Other than the specified alternatives, they may not be able to detect all deviations from the null. A way to deliver a consistent model specification test is taking all induced unconditional moment restrictions into account, then any deviations from the null will be revealed by some of these unconditional mo-ments. In this framework, we test the conditional moment restrictions “indirectly” by testing induced unconditional ones. On the other hand, some consistent model specification tests are proposed by using nonparametric methods to measure the “distance” between conditional moment restrictions and zero “directly”, Hong and White (1995), Zheng (1996) and Fan and Li (2000) are a few examples.
In order to achieve consistency, Bierens (1982, 1990) consider infinite many induced uncondi-tional moment restrictions by employing a class of weighting functions indexed by a continuous nuisance parameter. Stinchcombe and White (1998) provide and characterize the features of these weighting functions. A test statistic by integrating these nuisance parameters out is first proposed by Bierens (1982), and Bierens and Ploberger (1997) provide the general asymptotics for this test and name it by the Integrated Conditional Moment (ICM) test. Theoretically, the ICM test has nontrivial local power, and is asymptotically admissible under the normal errors assumption. Boning and Sow-ell (1999) also show that the ICM test proposed by Bierens (1982) and Bierens and Ploberger (1997) is the best ICM test according to the weighted average power criterion considered by Andrews and Ploberger (1994).
This ICM-type test, however, suffers from three drawbacks. First, the preliminary consistent estimate of model parameters under the null is necessary in forming the statistics. A inconsistent
estimate will give the wrong type I error. This point is well known and some examples are provided by Dom´ınguez and Lobato (2006). It means that we may need another estimation method which can deliver a consistent estimate given the conditional moment restrictions under the null before testing. Second, the numerical method of integrations is always needed in computing these test statistics be-cause there are no analytic forms in general. These two features make the implementation of this ICM-type tests cumbersome. Third, this ICM-type of test is not pivotal while the asymptotic distribution and critical values depend on the underlying data generating process. Stinchcombe and White (1998) and Bierens and Ploberger (1997) provide data-independent bounds of the critical values; Whang (2001), Dom´ınguez and Lobato (2006) and Hsu and Kuan (2008) infer based on Bootstrap methods instead.
In order to improve the ICM test, this project proposes two approaches. The first proposed test statistic is to present a global methodology for performing consistent statistical inference on model specification by extending the ICM test and the results in Hsu and Kuan (2010). This proposed test statistic is asymptotic equivalent to the ICM test statistic, but it has an analytic form instead. More-over, because this unified approach relates estimation and diagnostic testing in a rather natural way, we need no preliminary consistent estimate for the parameters of the model under the null. Roughly speaking, I provide an easy-to-implement version of ICM test in this approach. However, because this test is asymptotic equivalent to the ICM test, it is not pivotal, either. Bootstrap methods are employed for inference. Note that the consistent linearity test statistic proposed in Hsu and Kuan (2008) is just a special case of this test.
On the other hand, in order to get a pivotal test, I extend the idea behind the first approach. I de-rive a new set of infinite many unconditional moment restrictions and employ generalized empirical likelihood (GEL) method to construct the Likelihood-Ratio (LR) type test statistic. In the literature, GEL method has been well established and applied to estimation and diagnostic checking for models, see Imbens (2002), Kitamura et al. (2004), and Donald et al. (2003) among others. After suitable nor-malization, the asymptotic distribution of the proposed empirical LR test statistic should be standard normal, which is pivotal. Note that this approach also needs no preliminary consistent estimate for the parameters of the model under the null. To my best knowledge, this is the first attempt to link the GEL method to ICM test for model specification testing.
The remainder of this paper is organized as follows. The preliminaries about this issue is given in in Section 2. Section 3 describes two proposed test statistics and their asymptotics. In Section 4, we show some Monte Carlos simulations. Finally, Section 5 concludes.
2
Preliminaries
Assume all random variables are defined on a complete probability space (, F , IP), and denote σ(XXX) ⊂ F the minimal σ-algebra such that XXX : → IRk measurable. In what follows, I consider a
class of modelsM := f(·, θ) : IRm → IR
θ ∈ 2 , where 2 ⊂ IRp, then we say the model is cor-rectly specified if there exists someθosuch that f(XXX, θo) is a version of the conditional expectation of
Y relative toF . As a result, the null and alternative of this model specification test can be represented as
H0: IP(IE [Y |XXX] = f (XXX, θo)) = 1 for some θo∈2 ⊂ IRp, (1)
against
H1: IP(IE [Y |XXX] = f (XXX, θ)) < 1 for all θ ∈ 2. (2)
Obviously, this test is portmanteau since no particular models are specified in the alternative.
Let ZZZ = (Y, XXX) and zzzt = (yt, xxxt) is observable data for t = 1, . . . , T .1 Denote (ZZZ, θ) =
Y − f(XXX, θ) the residual function of the model, then the null hypothesis (1) suggests to test the conditional moment restriction
IE(ZZZ, θo)XXX = 0, with probability one (w.p.1 henceforth). (3) As well known, this conditional moment restriction (3) implies IE [(ZZZ, θo)w(XXX)] = 0, for any
measurable functionw(XXX). Since there are infinite many implied unconditional moment restrictions, intuition suggests that any tests based on an arbitrary finite set of them can not detect all deviations from the null. That’s why the CM tests in Newey (1985) test and Tauchen (1985) are not consistent.
In order to obtain a consistent CM test, one may systematically consider all these unconditional moment restrictions, by using some indexed functions. Let w(XXX, ξξξ) be that function with index ξξξ ∈ 444, where 444 is the nonempty set depended on w(·). A consistent test can then be constructed by testing
H0: IE [(ZZZ, θo)w(XXX, ξξξ)] = 0, ∀τττ ∈ 444, for some θo ∈2, w.p.1.
Given this null hypothesis which involves infinite many unconditional moment restrictions, we may form the tests based on the Lqnorm:
H0: Z 444 IE [(ZZZ, θo)w(XXX, ξξξ)] q dµ(ξξξ) 1/q =0, for some θo∈2, w.p.1, (4) 1Note that xxx
where 1 ≤ q < ∞, and µ is a given probability measure on 444 which is absolutely continuous with respect to Lebesgue measure on444; see Stute (1997), Koul and Stute (1999), Bierens (1982), and Bierens and Ploberger (1997). On the other hand, one may also test the null based on the supremum norm: H0: sup ξξξ∈444 IE [(ZZZ, θo)w(XXX, ξξξ)] =0, for some θo∈2, w.p.1. (5) Bierens (1990) and some Kolmogorov-Smirnov-type tests are based on this null.
3
The Proposed Approaches
Two approaches to testing model specification consistently are proposed and their asymptotics are established in this section. In order to illustrate the idea more easily, we consider the univariate X (and hence a scalarξ) in what follows. Extensions to multivariate XXX is rather straightforward.
3.1 The proposed approach (I)
Given the preliminary consistent estimate ˆθT ofθo under the null, the original ICM test statistic of
Bierens and Ploberger (1997) takes the form ηT( ˆθT) = Z 4 z(ξ , ˆ θT) 2 dµ(ξ) = Z 4 1 √ T T X i =1 (zzzt, ˆθT)w(xt, ξ) 2 dµ(ξ), (6)
the integration here could be cumbersome. Since z(ξ, ˆθT) is a function of ξ given ˆθT, it has its
own Fourier series representation. To be more precise, denote {ψm(·)} the Fourier series which is
orthonormal and complete in the space C(4) of continuous real functions on 4 as well as on the space L2(µ), then z(ξ, ˆθT) = ∞ X m=1 Cm( ˆθT)ψm(ξ),
whereCm( ˆθT) is the corresponding Fourier coefficient
Cm( ˆθT) = Z 4z(ξ, ˆθT)ψm(ξ)dµ(ξ) = Z 4 1 √ T T X t =1 (zzzt, ˆθT)w(xt, ξ)ψm(ξ)dµ(ξ) = √1 T T X t =1 (zzzt, ˆθT) Z 4w(xt, ξ)ψm(ξ)dµ(ξ) := 1 √ T T X t =1 (zzzt, ˆθT)$m(xt),
with $m(·) = R4w(·, ξ)ψm(ξ)dµ(ξ). It shows that each $m(·) can be viewed as a “weighted
av-erage” of allw(·, ξ) for ξ ∈ 4. Any deviations from the null detected by some w(·, ξ) can also be revealed by each$m(·). Besides, the integration in $m(xt) is much easy to compute, and it may
have closed form if we select matched weighting function w and Fourier series, for example, w is exponential function andψm is exponential Fourier series.
After invoking Paserval’s Theorem, the ICM test statisticηT( ˆθT), which is the L2norm ofz(ξ, ˆθT),
can then be expressed as the summation of the magnitudes of the corresponding Fourier coefficients: ηT( ˆθT) = Z 4 z(ξ , ˆ θT) 2 dµ(ξ) = ∞ X m=1 Cm( ˆθT) 2 .
In this step, we simply the construction ofηT( ˆθT) from integration to summation. In accordance with
Bessel’s inequality, |Cm|is close to zero when m is large enough and hence is not helpful to detect
the deviations form the null. Therefore, given ˆθT, we can consider
η∗ T( ˆθT, mT) = mT X m=1 Cm( ˆθT) 2 ,
where mT is some positive integer and needs to grow with the sample size T to ensure this test to
be consistent. Compared with the original ICM test statistic ηT( ˆθT) in (6), the computation of this
statistic is rather easy in practice. Besides, Paserval’s Theorem ensures thatη∗
T( ˆθT, ∞) = ηT( ˆθT). It
means thatη∗T( ˆθT, mT) shares the same asymptotics with the ICM statistic ηT( ˆθT) theoretically, if mT
is allowed to increase to infinity. Notice that the consistent linearity test statistic proposed in Hsu and Kuan (2008) is just a special case ofη∗
T( ˆθT, mT).
Recall that a preliminary consistent estimate ˆθT is crucial in above analysis. In this part of this
project, I propose a new test statistic for model specification without any preliminary consistent pa-rameter estimates instead. That is
JT1(mT) = min θ∈2η ∗ T(θ, mT) = min θ∈2 mT X m=1 |Cm(θ)|2. (7)
The asymptotics of this J1
T(mT) test statistic can be easily established based on the results of
Bierens(1982, 1990), Bierens and Ploberger (1997), Stinchcombe and White (1998) and Hsu and Kuan (2010). To see this, we impose the following conditions.
[A1] The observed data zzzt =(yt, xt)0, t = 1, . . . , T, are independent realizations of ZZZ = (Y, X)0.
[A2] For eachθ ∈ 2, (·, θ) is measurable, and for each zzz, (ZZZ, ·) is continuous on 2, where 2 is a compact subset in IRp. Also,θoin2 is the unique solution to IE[(ZZZ, θ)|X] = 0 under H0.
[A3] IE[supθ∈2|(ZZZ, θ)|4|X] < ∞; (ZZZ, θ) is twice continuously differentiable in a neighborhood ofθo, the corresponding first and second derivatives are bounded, and the second moment of
the first derivative is nonsingular.
[A4] The functionw(·) is generically comprehensive revealing.
Given the local alternative of the form H1L : IE [Y |X ] = f(XXX, θo) +
g(XXX) √
T for someθo
∈2, w.p.1, (8)
where function g is measurable with respect toF . Notice that when g is a zero function, the local alternative degenerates to the null of interest. Define
φt(ξξξ) = w(xxxt, ξξξ) − B(θo, ξξξ)0A(θo)−1∇θ0 f(xxxt, θo) with A(θo) = plimT →∞T1 PT t =1[∇θ0f(xxxt, θo)] [∇θ0 f(xxxt, θo)]0and B(θo, ξξξ) = plimT →∞1 T PT t =1∇θ0 f(xxxt, θo)w(xxxt, ξξξ).
Under H1L in (8), a Taylor expansion ofz(ξξξ, ˆθT) around θoand laws of large numbers yield
z(ξξξ, ˆθT) = 1 √ T T X t =1 (ZZZ, θo)φt(ξξξ) + 1 T T X t =1 g(xxxt)φt(ξξξ) + oIP(1) :=ZT(ξξξ) + Z g T(ξξξ) + oIP(1).
Under either the null or (local) alternative, the asymptotics of test statistic JT1are established based on the limiting processes ofZT(ξξξ) and Z
g T(ξξξ).
Theorem 3.1 (Asymptotics of J1 T test)
Given conditions [A1]–[A4], and If mT =o(T1/2) as T → ∞, we have
(a) For allθ ∈ 2, η∗ T(θ, mT) P −→ Z 4 IE[(ZZZ, θ)w(X, ξ)] 2 dµ(ξ). (b) argminθ∈2η∗ T(θ, mT) P −→θounder H0in (1). (c) under H0in (1), JT1(mT) d −→ Z 444 |Z(ξξξ)|2dξξξ,
whereZ(ξξξ) the limiting function of ZT(ξξξ), is a Gaussian process with zero mean and
covari-ance function0(ξξξ1, ξξξ2) = plimT →∞
PT
(d) under the alternative H1in (2), JT1(mT) diverges.
(e) under the local alternative HL 1 in (8), JT1(mT) d −→ Z 444 Z(ξξξ) + Zg(ξξξ) 2 dξξξ,
whereZ(ξξξ) is defined in (c) and Zg(ξξξ) is the limiting function of Zg T(ξξξ).
There are some remarks. First, from Theorem 3.1(b), we know that theθ which minimizes the η∗
T(θ, mT) is the consistent estimate of θo under the null if mT grows with T at the rate o(T1/2).
Second, even though the JT1 statistic shares the same asymptotics with the original ICM test, the proposed JT1statistic improves the original ICM test in two directions: one is that the construction of proposed statistic J1
T is more easier because no numerical integration is needed; the other is that the JT1
statistic does not depend on the preliminary consistent parameter estimates. This is a unified approach which links the estimation and diagnostic testing in a natural way by extending the ICM test and the work in Hsu and Kuan (2010). Besides, because the asymptotic properties of J1
T is equivalent to the
original ICM test, it is not pivotal, the bootstrap or simulation methods should be further imposed to obtain critical values.2
3.2 The proposed approach (II)
In this part, we propose a pivotal test by extending the proposed approach (I). We rewrite the null hypothesis in the ICM test approach as
H0: Z 4 IE [(ZZZ, θo)w(X, ξ)] 2 dµ(ξ) 1/2 =0, for some θo∈2, w.p.1.
By having Fourier representation of IE [(ZZZ, θo)w(X, ξ)] with respect to an orthonomal Fourier series
{ψm(·)}, under the null, we have Z 4 IE [(ZZZ, θo)w(X, ξ)] 2 dµ(ξ) 1/2 = ∞ X m=1 IE[(ZZZ, θo)$m(X)] 2!1/2 =0,
where $m(X) is defined above. This result immediately suggests that we can test a set of infinite
many unconditional moment restrictions instead. That is,
H0: IE[(ZZZ, θ)$m(X)] = 0, m = 1, 2, . . . , mT, mT → ∞. (9) 2It the simulations below, wild bootstrap method is used for obtaining the corresponding critical values.
The(ZZZ, θ)$m(X), m = 1, 2, . . . , mT are stacked into an mT ×1 vectorρ(ZZZ, θ, mT). Given this set
of unconditional moment restrictions and the sample counterparts, the second test statistic based on Donald et al. (2003) is constructed by
JT2(mT) = 2 ( min θ maxλ T X t =1 8 λ0ρ(zzz t, θ, mT) − T 8(0) ) , (10)
where8 is C2in a neighborhood of 0, concave on an open interval of the real line containing 0, andλ is a mT×1 vector of Lagrange multipliers. JT2(mT) in (10) is nothing but objective function of the GEL
estimation method while having mT unconditional moment restrictions. Similar to the conventional
likelihood-ratio test statistic, the asymptotic distribution of JT2(mT) under H0 is approximated by
χ2(m
T − p) which has mean (mT − p) and variance 2(mT − p); more discussions may refer to
Donald et al. (2003) for example. The asymptotics of JT2(mT) follows.
Theorem 3.2 (Asymptotics of JT2test)
Given conditions [A1]–[A4], and If mT =o(T1/3) as T → ∞, under the null, we have
JT2(mT) − (mT −p)
√
2(mT − p) d
−→ N(0, 1).
There are some remarks. First, the base stone of the proposed test J2
T(mT), $m(·), is quite different
from the some particular basis functions used in Donald et al. (2003). Each$m(·) can be viewed as
a “weighted average” of all basis functions w(·, ξ) for ξ ∈ 4. Second, unlike the ICM-type tests (including J1
T(mT) test), this test is pivotal instead. Last, but not least, as well as JT1(mT), this test
based on JT2(mT) needs no preliminary consistent estimate of θo.
4
Monte Carlo Simulations
In this section, we consider two experiments to evaluating the performance of the Cramer-von Mises type test statistic: C MT in (6), the Kolmogorov-Smirnov type test statistic: K ST in (5) and the
proposed statistics JT1(mT) and JT2(mT). The null hypothesis of both experiments is that the model is
linear, and the nominal size is 5% for all cases.
In the first experiment, like what in Hsu and Kuan (2008), we specify the model as yt =xt +
axt
1 + exp(−xt)
+t,
where xt follows a standard normal distribution. Various values of a, a = −1.5, −1, −0.5, 0, 0.5,
Table 1: Rejection probabilities. T =100 T =200 a a Test 0 0.5 1 1.5 0 0.5 1 1.5 JT1(1) 0.052 0.201 0.566 0.870 0.063 0.362 0.869 0.995 JT1(2) 0.050 0.199 0.574 0.873 0.059 0.365 0.874 0.995 JT1(3) 0.051 0.203 0.577 0.877 0.064 0.369 0.874 0.995 JT2(1) 0.258 0.543 0.889 0.993 0.254 0.694 0.983 1.000 JT2(2) 0.402 0.628 0.902 0.991 0.388 0.736 0.986 1.000 JT2(3) 0.573 0.741 0.935 0.995 0.535 0.819 0.989 1.000 C MT 0.049 0.134 0.409 0.744 0.062 0.250 0.716 0.972 K ST 0.049 0.144 0.439 0.769 0.060 0.266 0.744 0.976
otherwise, we compare the powers. In this setting, the number of Monte Carlo replications and of bootstrap replications are 3000 and 3000, respectively. The results are reported in Table 1. Roughly speaking, the test JT1and JT2have better power performances than C MT and K ST. However, the test
JT2suffers serious size-distortion problem when we focus on the case with a = 0. This shortcoming of test JT2needs further investigation in the future work.
In the second experiment, we consider another nonlinear model as
yt = 1 +t if yt −1> 0; 0 if yt −1=0; −1 +t if yt −1< 0.
Four sample size are considered, T = 50, 100, 200, 500. The results are reported in Table 2. Based on these results, the power performance of J1
T, C MT and K ST are quite similar in all cases.
Table 2: Rejection probabilities.
T Test 50 100 200 500 JT1(1) 0.848 0.994 1.000 1.000 JT1(2) 0.839 0.994 1.000 1.000 JT1(3) 0.850 0.994 1.000 1.000 JT1(4) 0.840 0.992 1.000 1.000 JT1(5) 0.848 0.994 1.000 1.000 C MT 0.845 0.990 1.000 1.000 K ST 0.876 0.995 1.000 1.000
5
Concluding Remarks
In this project, I construct two consistent model specification tests. The first proposed approach is the unified approach which links the estimation and diagnostic testing in a natural way by extending the ICM test of Bierens (1982) and Bierens and Ploberger (1997) and the work in Hsu and Kuan (2010). This test improves the original ICM test in two directions: it is more easier to implement and it does not depend on the preliminary consistent parameter estimates. The second proposed approach fur-ther extend the first proposed approach by using GEL methods. For this test, it needs no preliminary consistent parameter estimates, its implementation is not hard, and it is pivotal. To my best knowl-edge, this is the first attempt to link the GEL method to ICM tests for model specifications. The corresponding asymptotics of these two tests are established, and some interesting Monte Carlo sim-ulations are considered. Based on limited experiments, the finite sample performance of JT1are not worse than the common used Cramer-von Mises type and Kolmogorov-Smirnov type tests; the J2
T test
has good power performance but with serious size-distortion problem. The size-distortion problem of this pivotal test is interesting and is worth further investigations.
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