Cauchy or not Cauchy? New goodness-of-fit tests for the Cauchy distribution
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Ebner, Bruno; Eid, Lena; Klar, Bernhard Article — Published Version Cauchy or not Cauchy? New goodness-of-fit tests for the Cauchy distribution Statistical Papers Provided in Cooperation with: Springer Nature Suggested Citation: Ebner, Bruno; Eid, Lena; Klar, Bernhard (2022) : Cauchy or not Cauchy? New goodness-of-fit tests for the Cauchy distribution, Statistical Papers, ISSN 1613-9798, Springer, Berlin, Heidelberg, Vol. 65, Iss. 1, pp. 45-78, https://doi.org/10.1007/s00362-022-01382-0 This Version is available at: https://hdl.handle.net/10419/308614 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/
Statistical Papers (2024) 65:45–78 https://doi.org/10.1007/s00362-022-01382-0 REGULAR ARTICLE Cauchy or not Cauchy? New goodness-of-fit tests for the Cauchy distribution Bruno Ebner1 ·Lena Eid1 ·Bernhard Klar1 Received: 13 October 2021 / Revised: 8 November 2022 / Accepted: 1 December 2022 / Published online: 22 December 2022 © The Author(s) 2022 Abstract We introduce a new characterization of the Cauchy distribution and propose a class of goodness-of-fit tests for the Cauchy family. The limit distribution is derived in a Hilbert space framework under the null hypothesis. The new tests are consistent against a large class of alternatives. A comparative Monte Carlo simulation study shows that the test is a good competitor for the state of the art procedures, and we apply the tests to log-returns of cryptocurrencies. Keywords Goodness-of-fit ·Cauchy distribution ·Hilbert-space valued random elements Mathematics Subject Classification Primary 62G10 ·Secondary 62E10 1 Introduction In this article we dedicate our studies to answer the question of whether a data set of univariate real numbers belongs to the famous family of Cauchy distributions. The Cauchy distribution is undoubtedly the standard example for a distribution without existing mean value and was studied in the mathematical world for more than 300 years, having wide applicability in diverse fields ranging from modeling resonances in physics (then often called Lorentz distribution) to cryptocurrencies in finance, see Szczygielski et al. (2020). It is also known as the Breit–Wigner distribution, for an extensive historical overview, see Stigler (1974). To be precise, we write shorthand C(α, β),α∈R,β>0, for the Cauchy distribution with location parameter αand scale parameter β, having density BBernhard Klar [email protected] Bruno Ebner [email protected] 1Institute of Stochastics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany 123
46 B. Ebner et al. f(x,α,β)=1 π β β2+(x−α)2,x∈R. For a detailed discussion on this family, see Johnson et al. (1994), Chapter 16. The Cauchy distribution is a so called heavy tailed distribution and is a member of the stable distributions, see Nolan (2020). Note that X∼C(α, β) if, and only if, (X−α)/β ∼ C(0,1)and hence the Cauchy distribution belongs to the location-scale family of distributions. In the following we denote the family of Cauchy distributions by C:= {C(α, β) :α∈R,β >0}, a family of distributions which is closed under translation and rescaling. We test the composite hypothesis H0:PX∈C(1) against general alternatives on the basis of independent identical copies X1,...,Xn of Xdefined on an underlying probability space (, A,P). This testing problem has been considered in the literature: Gürtler and Henze (2000) propose a test procedure based on the empirical characteristic function and Matsui and Takemura (2005) extend this test by considering alternative estimation methods. More recently, Mahdizadeh and Zamanzade (2017) propose to use the likelihood ratio as in Zhang (2002)as well as the Kullback–Leibler (KL) distance, an idea that is extended in Mahdizadeh and Zamanzade (2019). A quantile based method is proposed in Rublik (2001) and compared to the classical omnibus procedures. An empirical power study of goodnessof-fit tests for the Cauchy model based on the empirical distribution function as the Kolmogorov–Smirnov test, the Cramér–von Mises test, the Kuiper test, the Anderson Darling and the Watson test is found in Onen et al. (2001). In Litvinova (2005), two tests for the standard Cauchy distribution based on characterizations are given; however, they are not designed for the composite hypothesis (1). The novel procedure is based on the following new characterization of the standard Cauchy distribution. Theorem 1.1 Let X be a random variable with absolutely continuous density p and E|X| 1+X2<∞. Then X has a Cauchy distribution C(0,1)if and only if Eit −2X 1+X2exp(itX)=0(2) holds for all t ∈R, where i denotes the imaginary unit. Proof For X∼C(0,1)a direct calculation shows the assertion. Let Xbe a random variable with absolutely continuous density function p(x)such that Eit −2X 1+X2exp(itX)=0 123
Cauchy or not Cauchy? New goodness-of-fit tests... 47 holds for all t∈R. Note that since −itE[exp(itX)]is the Fourier-Stieltjes transform of the derivative p(x)of p(x)we have 0=Eit −2X 1+X2exp(itX)=∞ −∞ −p(x)−2x 1+x2p(x)exp(itx)dx for all t∈R. By properties of the Fourier-Stieltjes transform, we hence note that p(x) must satisfy the ordinary differential equation p(x)+2x 1+x2p(x)=0 for almost all x∈R. The only solution satisfying ∞ −∞ p(x)dx=1isp(x)= f(x,0,1),x∈R, and X∼C(0,1)follows. Remark 1.2 The characterization in Theorem 1.1 is related to the spectral representation of the Stein operator of the so called density approach pioneered in Stein et al. (2004). Note that the quotient in (2)is f(x,0,1)/ f(x,0,1)=−2x/(1+x2),x∈R, and the set of so called test functions is given by {exp(itx):t∈R}. For details on this approach see Anastasiou et al. (2022), Sect. 5.4.2. This paper is organized as follows. In Sect. 2, we introduce a family of test statistics, denoted by Tn,a, which is based on the characterization in Theorem 1.1. In Sect. 3,the limit distribution of Tn,ais derived in a Hilbert space framework under the null hypothesis. Furthermore, we derive the limit distribution of the statistic Tn,0=lima→0aTn,a. Consistency of the new tests against a large class of alternatives is shown in Sect. 4.An extensive Monte Carlo simulation study in Sect. 5shows that the test is a good competitor for the state of the art procedures. In Sect. 6, the tests are applied to log-returns of cryptocurrencies. Finally, conclusions are given in Sect. 7. 2 A new class of goodness of fit tests for the Cauchy distribution The testing problem under discussion is invariant with respect to transformations of the kind x→ax +b,x∈R, where a∈Rand b>0. Consequently, a decision in favor or against H0should be the same for X1,...,Xnand aX1+b,...,aXn+b. This goal is achieved if the test statistic Tn, say, is based on the standardized data Yn,1,...,Yn,n, given by Yn,j=Xj−αn βn ,j=1,...,n.(3) Here, αn=αn(X1,...,Xn)and βn= βn(X1,...,Xn)denote consistent estimators of α∈Rand β>0 such that αn(aX1+b,...,aXn+b)=aαn(X1,...,Xn)+b,(4) βn(aX1+b,...,aXn+b)=a βn(X1,...,Xn), (5) 123
48 B. Ebner et al. holds for each a>0 and b∈R.By(4) and (5) it is easy to see that Yn,j,j=1,...,n, do not depend on the location nor on the scale parameter. Hence, Tnhas the property Tn(aX1+b,...,aXn+b)=Tn(X1,...,Xn), and we may and do assume α=0 and β=1 in the following. Motivated by Theorem 1.1, we choose the test statistic Tn=n∞ −∞ 1 n n j=1it −2Yn,j 1+Y2 n,jeitYn,j2 ω(t)dt,(6) which is the weighted L2-distance from (2) to 0. Here, |·|is the complex absolute value and ω:R→(0,∞)denotes a positive weight function that is given by ω(t)=ωa(t)=exp(−a|t|),t∈R. Note that ωis symmetric around the origin, i.e. ω(t)=ω(−t)holds for all t∈R, and that ∞ −∞ t6ω(t)dt<∞(7) holds. We have the integration-free, numerical stable formula Tn=Tn,a=1 n n j,k=18aYn,jYn,k (1+Y2 n,j)(1+Y2 n,k)((Yn,j−Yn,k)2+a2) −16aYn,j(Yn,j−Yn,k) (1+Y2 n,j)((Yn,j−Yn,k)2+a2)2+4a3−12a(Yn,j−Yn,k)2 ((Yn,j−Yn,k)2+a2)3, (8) and hence a whole family of tests depending on the so called tuning parameter a>0. The next result deals with the limit behavior of Tn,afor a→0 and a→∞. Theorem 2.1 For fixed n, we have lim a→0aTn,a−4 a3=8 n n j=1 Y2 n,j (1+Y2 n,j)2= Tn,0,(9) and lim a→∞aTn,a=8 nn j=1 Yn,j 1+Y2 n,j2 .(10) Proof Splitting the sum in (8) in a diagonal and a non-diagonal part results in Tn,a=1 n n j,k=1 Rj,k,a=1 n n j=1 Rj,j,a+1 n j=k Rj,k,a=Td n,a+Tnd n,a, 123
Cauchy or not Cauchy? New goodness-of-fit tests... 49 say. Since Td n,a=8 na n j=1 Y2 n,j/(1+Y2 n,j)2+4 a3 and lima→0Tnd n,a=0, we obtain lim a→0aTn,a−4 a3=lim a→0aTd n,a−4 a3=8 n n j=1 Y2 n,j (1+Y2 n,j)2. By the symmetry of the weight function ωa(·), straightforward calculations show Tn,a=∞ −∞ Z2 n(t)ω a(t)dt, where Zn(t)=1 √n n j=12Yn,j 1+Y2 n,j+tcos(tYn,j)+t−2Yn,j 1+Y2 n,jsin(tYn,j), t∈R. (11) Then, aTn,a=∞ −∞ Z2 n(s/a)exp(−|s|)ds, where Znsatisfies Z2 n(s/a)≤4n(1+|s|)2, for a>1, and lim t→0Zn(t)=1 √n n j=1 2Yn,j 1+Y2 n,j . Now, the second assertion follows from an application of the dominated convergence theorem. 3 Limit distribution under the null hypothesis The asymptotic theory is derived in the Hilbert space Hof measurable, square integrable functions H=L2(R,B,ω(t)dt), where Bis the Borel-σ-field of Rand ω(t)is the weight function defined in the introduction. Notice that Tn=∞ −∞ Z2 n(t)ω(t)dt, where Znis defined in (11), is a real-valued (A⊗B,B)-measurable random element of H. We denote by fH=∞ −∞ f(t)2ω(t)dt1/2 ,f,gH=∞ −∞ f(t)g(t)ω(t)dt 123
50 B. Ebner et al. the usual norm and inner product in H. In the following, we assume that the estimators αnand βnallow linear representations √nαn=1 √n n j=1 ψ1(Xj)+oP(1), (12) √n( βn−1)=1 √n n j=1 ψ2(Xj)+oP(1), (13) where X1,...,Xn∼C(0,1)are independent random variables, oP(1)denotes a term that converges to 0 in probability, and ψ1und ψ2are measurable functions with E[ψ1(X1)]=E[ψ2(X1)]=0,and E[(ψ1(X1), ψ2(X1))(ψ1(X1), ψ2(X1))] =C·I2; here, Cis a positive constant, stands for the transpose of a vector and I2is the 2 ×2identity matrix. See Remark 3.2 for examples of estimation procedures satisfying these assumptions. Theorem 3.1 Let X1,...,Xnbe i.i.d. C(α, β) distributed random variables. Then there exists a centred Gaussian random process Zin Hwith covariance kernel K(s,t)=1 2s2+t2+|s−t|+1e−|s−t| −1 2(t2+|t|+1)e−|t|E 2X1 1+X2 1+scos(sX1)+s−2X1 1+X2 1sin(sX1)ψ1(X1) +1 2t(|t|+1)e−|t|E 2X1 1+X2 1+scos(sX1)+s−2X1 1+X2 1sin(sX1)ψ2(X1) −1 2(s2+|s|+1)e−|s|E 2X1 1+X2 1+tcos(tX1)+t−2X1 1+X2 1sin(tX1)ψ1(X1) +1 2s(|s|+1)e−|s|E 2X1 1+X2 1+tcos(tX1)+t−2X1 1+X2 1sin(tX1)ψ2(X1) +1 4(s2+|s|+1)(t2+|t|+1)e−|s|−|t|E[ψ2 1(X1)]+1 4s(|s|+1)t(|t|+1)e−|s|−|t|E[ψ2 2(X1)] for s,t∈R,such that Tn D −→ Z2 Has n →∞. A proof of Theorem 3.1 is found in Appendix A.1. It is well known that the distribution of Z2 His that of ∞ j=1λjN2 j, where N1,N2,... are i.i.d. standard normal random variables and (λj)is a decreasing sequence of positive eigenvalues of the integral operator Kg(s)=∞ −∞ K(s,t)g(t)ω(t)dt. Due to the complexitiy of the covariance kernel Kas given in Theorem 3.1 it seems hopeless to solve the integral equation Kg(s)=λg(s)and find explicit values of λj, 123
Cauchy or not Cauchy? New goodness-of-fit tests... 51 j≥1. For a numerical approximation method we refer to Subsect. 3.3 in Matsui and Takemura (2005). Note that EZ2 H=∞ −∞ K(t,t)ω(t)dt(14) and VarZ2 H=2∞ −∞ ∞ −∞ K2(s,t)ω(t)ω(s)dtds(15) can be derived for specific estimation procedures. Such results in the theory of goodness-of-fit tests for the Cauchy family are sparse, for some explicit formulae for mean values, see Gürtler and Henze (2000) and Matsui and Takemura (2005). Remark 3.2 The generality of Theorem 3.1 in view of the linear representations of the estimators and hence dependence on the functions ψ1and ψ2leads to explicit covariance kernels for different parameter estimation procedures. To estimate αand βwe choose the following location and scale estimators αnand βn, which all satisfy (4) and (5) respectively. For a compact notation, we write ψ(x)=(ψ1(x), ψ2(x)). Some derivations were partially provided by the computer algebra system Maple, see Maplesoft (2019). 1. Median and interquartile-distance estimators: Let ξp,p∈(0,1), denote the p-quantile of the underlying distribution F, ξp,nthe sample p-quantile, and X(1)≤ ···≤X(n)the order statistics of X1,...,Xn. With ·denoting the floor function, let αn=1 2(X(n 2)+X(n 2+1)), if neven, X(n 2+1),otherwise, (16) be the unbiased empirical median and βn=1 2( ξ3 4,n− ξ1 4,n)(17) the half-interquartile range (iqr) of the sample. Under mild regularity conditions αnand βnare consistent estimators of αand β. Display (3.3) of Gürtler and Henze (2000) then gives the so-called Bahadur representations (see Theorem 2.5.1 in Serfling (1980)) with ψ1(x)=π1 2−1{x≤0},and ψ2(x)=π1 2−1{−1≤x≤1},x∈R. 123
52 B. Ebner et al. It is easy to see that E[ψ1(X1)]=E[ψ2(X1)]=0 and E[ψ(X1)ψ(X1)]=π2 4I2 holds. With these representations, we get the covariance kernel of Zin Theorem 3.1 KMIQ(s,t)=1 2s2+t2+|s−t|+1e−|s−t| +t|t|+12J1(s)−sJ 2(s)−t2+|t|+1s 2J3(s)+J4(s)e−|t| +s|s|+12J1(t)−tJ 2(t)−s2+|s|+1t 2J3(t)+J4(t)e−|s| +π2 16 s2+|s|+1t2+|t|+1+st|s|+1|t|+1e−|s|−|t|, for s,t∈R,where J1(t)=1 0 xsin(tx) (1+x2)2dx,J2(t)=1 0 cos(tx) 1+x2dx, J3(t)=∞ 0 sin(tx) 1+x2dx,J4(t)=∞ 0 xcos(tx) (1+x2)2dx. Direct calculations of integrals lead to EZ2 H=∞ −∞ KMIQ(t,t)ωa(t)dt =(8(a+2)5(1+a)3a3a2+2a+23 )−1π2−8a16 +19 π2−152a15 +173 π2−1368a14 +1003 π2−7720a13 +4126 π2−30192a12 +12594 π2−84304a11 +29128 π2−163520a10 +51460 π2−188832a9+69320 π2−8256a8 +70296 π2+457664a7+52176 π2+1025920a6 +26848 π2+1323264a5+8576 π2+1151488a4 +1280 π2+693248a3+280576 a2+69632 a+8192. 2. Maximum likelihood estimators: Matsui and Takemura (2005) show in Lemma A.1 that for the maximum-likelihood estimator αnand βnin the Cauchy family the linear representations are given by ψ1(x)=4x 1+x2,and ψ2(x)=2(x2−1) 1+x2,x∈R. Again, straightforward calculations show E[ψ1(X1)]=E[ψ2(X1)]=0aswellas E[ψ(X1)ψ(X1)]=2I2. Note that there are no closed form expressions for the estimators, such that the log-likelihood equations have to be solved numerically. 123
Cauchy or not Cauchy? New goodness-of-fit tests... 59 Table 2 Percentage of 10,000 MC samples declared significant by various tests for the Cauchy distribution using median and half-IQR estimator (α=0.05,n=20) Tn,1Tn,2Tn,3Tn,4Tn,5Tn,6 Tn,0KL KS C(0,1) 5 5 5 5 5 5 5 5 5 N(0,1) 8 4 1 0 0 0 29 73 5 CN(0.5) 4333339144 CN(0.8) 63111119364 Student(2) 4 2 2 2 2 2 9 22 3 Student(3) 5 2 1 1 1 1 14 34 3 Student(5) 6 3 1 1 1 1 20 49 4 Student(10) 7 3 1 1 1 1 24 61 4 Stable(0.4,0) 55 52 39 26 18 13 79 2 39 Stable(0.7,0) 16 15 12 9 8 7 21 2 12 Stable(1.2,0) 4 3 3 3 3 3 6 10 4 Stable(1.5,0) 5 2 1 1 1 2 14 28 4 Stable(1.8,0) 7 3 1 1 1 1 22 52 5 Stable(0.5,1) 91 97 90 76 61 50 40 65 94 Stable(1.5,1) 10 10 7 6 6 6 19 49 9 Stable(2,1) 8 4 0 0 0 0 28 73 5 Tukey(0.2) 5 2 1 1 1 1 14 36 3 Tukey(0.1) 5 3 1 1 1 1 21 52 4 Tukey(0.05) 7 3 1 0 0 0 24 62 4 Tukey-L(-3) 43423324171264 231 Tukey-L(-2) 21221814111033 116 Tukey-L(-0.5)4322227153 Tukey-L(0.5) 17 11 1 0 0 0 45 94 10 Uniform 32 21 1 1 0 0 56 99 24 Logistic 6 3 1 1 1 1 21 56 4 Laplace 4 2 1 1 1 1 10 32 3 Gumbel 10 9 3 3 2 2 26 70 9 ML(0.25) 100 100 100 97 92 83 93 89 100 ML(0.5) 989993796349558199 ML(0.75) 78 87 69 52 43 39 20 75 85 Exponential 39 44 23 15 13 11 27 92 48 CM AD W D n,1Dn,2Dn,3Dn,4Dn,5Dn,6 C(0,1) 5 5 5 5 5 5 5 5 5 N(0,1) 6 6 15 15 21 26 26 23 19 CN(0.5) 3 3 5 5 5 5 5 4 3 CN(0.8) 448 811131210 8 Student(2) 4 3 5 4 5 5 5 5 3 Student(3) 4 3 7 6 7 9 9 8 6 123
60 B. Ebner et al. Table 2 continued CM AD W D n,1Dn,2Dn,3Dn,4Dn,5Dn,6 Student(5) 4 4 9 8 11 14 14 12 9 Student(10) 5 5 11 11 15 18 19 16 13 Stable(0.4,0) 45 83 58 62 64 66 69 71 74 Stable(0.7,0) 13 23 14 16 18 20 22 24 25 Stable(1.2,0) 434444433 Stable(1.5,0) 436678865 Stable(1.8,0) 5 5 11 10 14 16 16 14 11 Stable(0.5,1) 90 97 88 95 76 57 46 44 46 Stable(1.5,1) 9 9 11 13 11 12 11 10 8 Stable(2,1) 6 6 14 14 20 24 24 22 17 Tukey(0.2) 4 3 7 6 8 10 9 8 7 Tukey(0.1) 4 4 9 8 12 15 15 13 10 Tukey(0.05) 5 5 12 11 15 19 19 17 13 Tukey-L(-3) 357944525862656871 Tukey-L(-2) 174119242832353841 Tukey-L(-0.5) 4 3 5 4 4 4 4 3 3 Tukey-L(0.5) 12 15 31 36 45 50 50 47 40 Uniform 222851606769686559 Logistic 5 4 11 9 13 16 17 15 11 Laplace 3 3 6 5 6 8 8 7 5 Gumbel 8 8 15 17 18 20 20 18 15 ML(0.25) 100 100 100 100 99 98 96 96 96 ML(0.5) 969997998669585557 ML(0.75) 71 79 72 83 52 33 25 23 23 Exponential 31 32 41 49 34 30 28 27 25 Tables 2and 3, the results using the median and interquartile-distance estimators are given, with sample size n=20 and n=50, respectively. Tables 4and 5show the corresponding results for the maximum likelihood estimator. The main conclusions that can be drawn from the simulation results are the following: – As always in similar situations, there exists no uniformly most powerful test, which is in accordance to the results in Janssen (2000). – As expected, the power of nearly all test statistics increases for increasing sample size for all alternative distributions. An exception is the KL distance based test. Its power decreases for certain alternatives, in particular for the Mittag-Leffler distribution. This behavior has been reported previously, see Tables 3 to 6 in Mahdizadeh and Zamanzade (2019), and may be due to the choice of the window size. 123
Cauchy or not Cauchy? New goodness-of-fit tests... 61 Table 3 Percentage of 10,000 MC samples declared significant by various tests for the Cauchy distribution using median and half-IQR estimator (α=0.05,n=50) Tn,1Tn,2Tn,3Tn,4Tn,5Tn,6 Tn,0KL KS C(0,1) 554555555 N(0,1) 24 24 4 1 0 0 72 100 26 CN(0.5) 85333323147 CN(0.8) 16132 1 1 1 514614 Student(2) 6 4 2 2 2 2 23 52 6 Student(3) 10 7 2 2 1 1 38 82 9 Student(5) 14 12 2 1 1 1 52 96 13 Student(10) 18 18 3 1 1 0 64 100 18 Stable(0.4,0) 91 90 76 47 25 15 99 0 80 Stable(0.7,0) 23 23 14 10 8 7 46 0 16 Stable(1.2,0) 6 4 3 3 3 3 12 19 6 Stable(1.5,0) 11 8 3 2 2 2 38 53 9 Stable(1.8,0) 18 16 3 1 1 1 62 88 18 Stable(0.5,1) 100 100 100 100 97 87 71 13 100 Stable(1.5,1) 28 41 26 18 15 13 52 81 56 Stable(2,1) 23 23 4 1 1 0 72 100 26 Tukey(0.2) 10 7 2 1 1 1 40 87 9 Tukey(0.1) 15 13 2 1 1 1 56 98 15 Tukey(0.05) 19 17 3 1 1 1 64 100 18 Tukey-L(-3)707457352013960 64 Tukey-L(-2)333524161210640 26 Tukey-L(-0.5) 5 4 3 3 3 3 13 37 5 Tukey-L(0.5) 56 61 11 1 0 0 90 100 68 Uniform 87 87 18 1 0 0 97 100 95 Logistic 15 14 3 1 1 1 59 100 15 Laplace 6 4 2 1 1 1 21 94 6 Gumbel 33 43 18 8 6 5 68 100 58 ML(0.25) 100 100 100 100 100 100 100 4 100 ML(0.5) 100 100 100 100 98 91 86 15 100 ML(0.75) 100 100 100 93 78 62 30 45 100 Exponential 94 96 75 44 32 26 64 100 100 CM AD W D n,1Dn,2Dn,3Dn,4Dn,5Dn,6 C(0,1) 5 5 5 5 5 5 5 5 5 N(0,1) 29 55 67 65 83 89 92 93 94 CN(0.5)6 713131516161513 CN(0.8)142338374954555554 Student(2) 6 8 14 10 15 19 21 23 23 123
62 B. Ebner et al. Table 3 continued CM AD W D n,1Dn,2Dn,3Dn,4Dn,5Dn,6 Student(3) 101627213240444749 Student(5) 152841365262677072 Student(10) 20 41 54 49 70 78 83 85 86 Stable(0.4,0) 91 99 98 97 96 97 97 98 98 Stable(0.7,0) 17 39 26 29 34 37 41 45 48 Stable(1.2,0)558678888 Stable(1.5,0) 9 13 24 20 29 34 36 37 37 Stable(1.8,0) 19 35 50 46 63 71 74 77 77 Stable(0.5,1) 100 100 100 100 100 100 100 99 99 Stable(1.5,1) 38 53 60 61 71 72 70 69 68 Stable(2,1) 28 55 67 65 84 89 92 94 94 Tukey(0.2) 9 16 27 22 34 42 47 50 52 Tukey(0.1) 16 32 44 39 57 67 72 75 77 Tukey(0.05) 21 41 54 50 70 78 83 85 87 Tukey-L(-3)739991909495969798 Tukey-L(-2)297247485864687275 Tukey-L(-0.5) 5 5 8 7 8 10 11 12 12 Tukey-L(0.5) 63 89 92 96 99 100 100 100 100 Uniform 86 98 99 100 100 100 100 100 100 Logistic 17 35 48 43 63 72 77 80 82 Laplace 7 12 18 13 24 32 37 40 42 Gumbel 42 64 73 74 86 89 90 91 91 ML(0.25) 100 100 100 100 100 100 100 100 100 ML(0.5) 100 100 100 100 100 100 100 100 100 ML(0.75) 100 100 100 100 100 100 98 94 91 Exponential 92 98 99 100 100 99 98 97 97 – The new tests Tn,a,a=1,...,6, perform better using the maximum likelihood estimator than with median and half-IQR. Hence, the latter estimators should not be used. For all other test statistics, including Tn,0, performance is comparable, or even better when using median and half-IQR. In any case, the choice of the estimation method can have a pronounced influence on the performance of the tests. – The power of Tn,0is comparable under both estimation methods. – For alternatives with finite first and second moments, the KL distance based test has the highest power among all competitors. On the other hand, its power breaks down completely for some alternatives without existing first moment, and it is low for some alternatives with infinite second moment. Hence, the test can not really be seen as an omnibus test. 123
Cauchy or not Cauchy? New goodness-of-fit tests... 63 Table 4 Percentage of 10,000 MC samples declared significant by various tests for the Cauchy distribution using ML estimation (α=0.05,n=20) Tn,1Tn,2Tn,3Tn,4Tn,5Tn,6 Tn,0KL KS C(0,1) 5 5 5 5 5 5 5 5 5 N(0,1) 16 30 34 28 10 3 26 76 5 CN(0.5) 81085229144 CN(0.8) 13 19 18 12 5 2 18 37 5 Student(2) 8 9 8 6 3 2 10 22 4 Student(3) 10 14 14 10 4 2 15 37 4 Student(5) 12 19 20 14 6 2 19 51 5 Student(10) 15 25 27 20 8 3 22 64 5 Stable(0.4,0) 53 64 72 77 80 81 76 0 42 Stable(0.7,0) 10 14 19 22 25 26 19 1 12 Stable(1.2,0) 7 6 5 4 2 2 7 11 4 Stable(1.5,0) 10 13 12 8 3 2 14 29 4 Stable(1.8,0) 14 23 25 18 7 3 21 55 5 Stable(0.5,1) 55 70 78 81 83 84 26 54 99 Stable(1.5,1) 14 21 22 17 11 7 16 50 14 Stable(2,1) 17 29 34 28 10 4 26 76 5 Tukey(0.2) 10 14 14 10 4 2 14 38 4 Tukey(0.1) 13 20 21 16 6 2 19 55 5 Tukey(0.05) 15 25 28 21 7 3 23 65 5 Tukey-L(-3) 34496067727560 035 Tukey-L(-2) 13212935404229 117 Tukey-L(-0.5)6764328164 Tukey-L(0.5) 31 55 64 59 24 8 37 96 11 Uniform 487481784314459922 Logistic 14 22 24 18 7 3 21 60 5 Laplace 8 10 10 8 4 2 10 35 3 Gumbel 18 31 35 30 16 9 24 74 13 ML(0.25) 99 99 100 100 100 100 92 77 100 ML(0.5) 8086909192924372100 ML(0.75) 52 63 66 66 65 64 12 70 96 Exponential 38 53 57 54 45 36 20 92 64 CM AD W D n,1Dn,2Dn,3Dn,4Dn,5Dn,6 C(0,1) 555555555 N(0,1) 3 2 29 23 17 10 7 4 2 CN(0.5)439732221 CN(0.8)32181485321 123
64 B. Ebner et al. Table 4 continued CM AD W D n,1Dn,2Dn,3Dn,4Dn,5Dn,6 Student(2) 3210742211 Student(3) 2 1 14 10 6 3 2 2 1 Student(5) 3 2 19 14 9 5 4 3 2 Student(10) 3 2 24 19 13 8 5 3 2 Stable(0.4,0) 33 77 77 71 79 83 86 87 88 Stable(0.7,0) 11 18 15 18 24 26 28 29 30 Stable(1.2,0) 437532222 Stable(1.5,0) 3 2 14 9 5 3 2 2 1 Stable(1.8,0) 3 2 23 17 11 7 5 3 2 Stable(0.5,1) 99 99 82 79 82 85 87 88 89 Stable(1.5,1) 12 9 22 17 11 9 7 6 5 Stable(2,1) 3 2 28 23 17 10 7 4 3 Tukey(0.2) 3 2 14 10 6 4 3 2 1 Tukey(0.1) 3 2 20 15 10 5 4 2 2 Tukey(0.05) 3 2 24 19 13 8 5 3 2 Tukey-L(-3) 277260587176798182 Tukey-L(-2) 153426273842454849 Tukey-L(-0.5) 3 2 7 5 3 2 2 1 1 Tukey-L(0.5) 6 5 50 50 43 28 19 11 7 Uniform 131066726548352414 Logistic 3 2 22 17 11 6 4 3 2 Laplace 2 1 10 8 5 3 2 2 1 Gumbel 1073125191412 9 7 ML(0.25) 100 100 100 100 100 100 100 100 100 ML(0.5) 100 100 96 93 92 94 95 95 96 ML(0.75) 93 91 72 66 62 64 66 67 68 Exponential 50 43 55 50 41 39 38 37 35 – Among the new tests, values of the tuning parameter around a=3 result in a quite homogeneous power against all alternatives. For the tests based on Dn,λ,λ=5 seems to be a good choice. Both classes of tests perform similarly for the ML estimator; for the median and half-IQR estimator, the latter is preferable. – Among the group of edf tests, Woutperform the other tests in most cases. – For symmetric alternatives without existing first moment as Stable(0.4,0), Stable(0.7,0), Tukey-L(-3) and Tukey-L(-2), the tests based on Tn,0,AD and Dn,6 perform best. 123
Cauchy or not Cauchy? New goodness-of-fit tests... 65 Table 5 Percentage of 10,000 MC samples declared significant by various tests for the Cauchy distribution using ML estimation (α=0.05,n=50) Tn,1Tn,2Tn,3Tn,4Tn,5Tn,6 Tn,0KL KS C(0,1) 5 5 5 5 5 5 5 5 5 N(0,1) 40 76 90 95 96 96 64 100 16 CN(0.5) 1420201510 62214 6 CN(0.8) 285361605239484611 Student(2) 1220242320152352 5 Student(3) 1936454846383882 7 Student(5) 2651667171664896 9 Student(10) 32 65 81 86 88 86 57 100 13 Stable(0.4,0) 95 98 99 99 99 99 99 0 82 Stable(0.7,0) 23 34 40 45 47 49 44 0 18 Stable(1.2,0) 8 10 10 8 6 4 12 18 5 Stable(1.5,0) 19 34 40 39 33 25 35 53 7 Stable(1.8,0) 34 62 76 80 78 71 56 88 12 Stable(0.5,1) 98 100 100 100 100 100 46 6 100 Stable(1.5,1) 32 61 74 78 76 68 44 80 57 Stable(2,1) 41 77 90 95 97 96 65 100 17 Tukey(0.2) 20 36 46 49 48 42 37 87 7 Tukey(0.1) 27 55 70 76 77 73 51 98 9 Tukey(0.05) 33 66 81 88 89 87 57 100 13 Tukey-L(-3) 79929697989896 068 Tukey-L(-2) 33516369737663 029 Tukey-L(-0.5) 8 11 13 12 11 8 13 37 4 Tukey-L(0.5) 75 98 100 100 100 100 83 100 54 Uniform 95 100 100 100 100 100 91 100 91 Logistic 28 58 74 81 82 80 52 100 10 Laplace 10 21 30 36 38 36 19 94 4 Gumbel 44 79 91 94 96 94 59 100 57 ML(0.25) 100 100 100 100 100 100 100 0 100 ML(0.5) 100 100 100 100 100 100 76 6 100 ML(0.75) 97 99 100 100 100 99 14 38 100 Exponential 87 98 99 100 100 99 48 100 100 CM AD W D n,1Dn,2Dn,3Dn,4Dn,5Dn,6 C(0,1) 555555555 N(0,1) 91577778790919191 CN(0.5) 4319161310 9 8 7 CN(0.8) 6650485048464339 123
66 B. Ebner et al. Table 5 continued CM AD W D n,1Dn,2Dn,3Dn,4Dn,5Dn,6 Student(2) 3321161514151413 Student(3) 4438303334363535 Student(5) 5654475558606160 Student(10) 6 10 67 63 74 78 80 80 80 Stable(0.4,0) 81 99 100 99 99 100 100 100 100 Stable(0.7,0) 15 31 38 36 44 49 52 53 55 Stable(1.2,0) 4 3 10 8 6 5 5 4 4 Stable(1.5,0) 4 4 34 28 28 27 27 25 24 Stable(1.8,0) 6 9 64 60 67 68 69 68 67 Stable(0.5,1) 100 100 100 100 100 100 100 100 100 Stable(1.5,1) 43 46 62 60 66 66 67 67 66 Stable(2,1) 9 16 78 77 88 90 92 92 91 Tukey(0.2) 3 4 39 31 35 36 38 38 38 Tukey(0.1) 5 7 57 52 61 65 67 67 67 Tukey(0.05) 6 10 68 65 75 79 81 82 82 Tukey-L(-3) 609896949798999999 Tukey-L(-2) 236260576975788081 Tukey-L(-0.5)3312988777 Tukey-L(0.5) 27 50 97 99 100 100 100 100 100 Uniform 53 80 100 100 100 100 100 100 100 Logistic 5 8 60 56 66 70 72 73 72 Laplace 2 2 23 20 25 25 26 26 26 Gumbel 36 43 80 80 88 90 91 91 91 ML(0.25) 100 100 100 100 100 100 100 100 100 ML(0.5) 100 100 100 100 100 100 100 100 100 ML(0.75) 100 100 100 100 100 100 100 100 100 Exponential 96 97 98 99 99 99 99 99 99 6 Real data example: log-returns of cryptocurrencies In this section, we apply the tests for the Cauchy distribution to log-returns of various cryptocurrencies, namely Bitcoin (BTC), Ethereum (ETH), Ripple (XRP), Litecoin (LTC), BitcoinCash (BCH), EOS (EOS), BinanceCoin (BNB) and Tron(TRX). The Cauchy distribution is found to be a comparably good model for such data sets in Szczygielski et al. (2020). There, 58 hypothetical distributions have been fitted to 15 major cryptocurrencies, and the Cauchy model turned out to be the best fitting distribution for 10 cryptocurrencies (including all currencies considered here). In Szczygielski et al. (2020), the number of observations of the various data sets varied widely from 638 to 2255. Further, with very large data sets, each model will be rejected in the end. Hence, we decided to consider shorter time series: a series with daily observations 123
Cauchy or not Cauchy? New goodness-of-fit tests... 67 Fig. 1 Histograms of cryptocurrency log-returns from January01, 2020 to June 10,2021, together with fitted Cauchy densities from January 01, 2020 to June 10, 2021, with sample size 527 (526 for ETH), and an even shorter series from January 01, 2021 to June 10, 2021, with sample size 161 (160 for ETH). All prices are closing values in U.S. dollars, freely accessible from CryptoDataDownload via the link www.cryptodatadownload.com/data. Returns are estimated by taking logarithmic differences. Days with zero trading volume are omitted. Figure 1shows histogramms of the datasets with larger time span, together with the densities of fitted Cauchy distributions. Visually, the Cauchy model seems to be a reasonable approximation. As in the above cited literature, we assume in the following the independence of daily log return data.This is justified in view of the very weak serial correlations of all datasets. Figure 2shows the autocorrelations of the first four datasets with larger time span, which are all quite small and not significant in most cases. The findings are similar for the other four cryptocurrencies and the shorter time span. 123
68 B. Ebner et al. Fig. 2 Autocorrelations of cryptocurrency log-returns from January01,2020 to June 10, 2021 Tables 6and 7report the results of the different tests for the Cauchy model for the two time series. The p-values are based on Monte Carlo simulation under the Cauchy model with 104replications. For the test based on the KL distance, we choose the window length m=100 for the longer time series, and m=50 for the shorter one. The results show that even if the Cauchy distribution fits better than many other distributional models, it is still not an acceptable fit in any of the cases, possibly with the exception of EOS data. The tests based on Tn,5,KLand Dn,5result in p-values of 0.000 for all currencies for the longer time series, and p-values smaller than 0.01 for all currencies for the shorter one. The edf based tests have generally larger p-values, with the Watson test having smallest, and Cramér–von Mises test having largest p-values. Overall, the EOS data seem to be most compatible with the hypothesis of a Cauchy distribution. For the shorter series, the p-values of all edf based tests are larger than 0.1, and the tests based on Tn,1,˜ Tn,0,KS and CM don’t reject H0on the 5%-level even for the larger data set. These tests also show relatively large values for Ripple (XRP) and BitcoinCash (BCH). On the whole, the results confirm the findings of the simulation study concerning the comparative power of the test. 7 Conclusion We have proposed a family of tests for the Cauchy family of distributions with desirable theoretical and computational features as consistency and competitiveness to existing procedures. The authors suggest in view of the results of the simulation study to use the tuning parameter a=4 in applications for good power against most alternatives. 123
Cauchy or not Cauchy? New goodness-of-fit tests... 75 By (27), the tightness of 1 √nn j=1ψl(Xj), l=1,2 and Slutzky’s Theorem, we have Z∗ n− ZnH P −→ 0, as n→∞. Hence Z∗ nand Znshare the same weak limit in H. Writing Zn(t)=n−1/2n j=1 Zn,j(t),t∈R, where Zn,j(t)=h(t,Xj)− 1 2t2+|t|+1e−|t|ψ1(Xj)+1 2t|t|+1e−|t|ψ2(Xj),t∈R,j=1,...,n,we have E[ Zn,1(t)]=0 and Zn,1, Zn,2,... are i.i.d. centred random elements in H. Straightforward calculations show that K(s,t)=E[ Zn,1(s) Zn,1(t)]has the stated formula and the weak limit of Znfollows by the central limit theorem in Hilbert spaces. To conclude the proof, we have by the Cauchy-Schwarz inequality Z∗ nHn−Z∗ nH≤∞ −∞ Z∗ n(t)4ω(t)dt1/2∞ −∞ ω( βnt) ω(t)−1 2 ω(t)dt1/2 . Since by assumption ω1/2is a weight function, we directly see using the CauchySchwarz inequality and the continuous mapping theorem that ∞ −∞ Z∗ n(t)4ω(t)dt1/2 is tight. Since by direct calculations ∞ −∞ ω( βnt) ω(t)−12 ω(t)dt =oP(1)holds, we see that Zn2 H= βnZ∗ n2 H+oP(1). Finally, by Slutzky’s theorem, Zn2 Hhas the same weak limit as Zn2 Hand the assertion follows. A.2 Proof of Theorem 4.1 Let Znand Z∗ nbe defined as in (11) and (26). Since by assumption we have (αn, βn)P −→ (0,1), we can show in complete analogy to the proof of Theorem 3.1, that n−1Zn2 H−Z∗ n2 HP −→ 0 holds. Let gj(·,·),j=1,2, be defined as in (22) and (23), and put Z0 n(t)=n−1/2n j=1h(t,Xj),t∈R.Then n−1/2Z∗ n(t)−Z0 n(t)=αn/nn j=1g1(t,Xj)+( βn−1)/nn j=1g2(t,Xj)follows and by the triangle inequality we have n−1/2(Z∗ n−Z0 n)2 H=∞ −∞ αn 1 n n j=1 g1(t,Xj)+( βn−1)1 n n j=1 g2(t,Xj)2 ω(t)dt ≤∞ −∞ |αn|1 n n j=1 g1(t,Xj)+| βn−1|1 n n j=1 g2(t,Xj)2 ω(t)dt ≤2|αn|21 n n j=1 g1(t,Xj)2 H+2| βn−1|21 n n j=1 g2(t,Xj)2 H. By the law of large numbers in Hilbert spaces and (αn, βn)P −→ (0,1), the right hand side is oP(1). Note that the existence of the mean values is guaranteed by the assumptions. Again, the law of large numbers in Hshows n−1/2Z0 n(t)a.s. −→ Eh(t,X), 123
76 B. Ebner et al. as n→∞, and by the symmetry of the weight function, we have Tn n=1 √nZn2 H P −→ ∞ −∞ |Eh(t,X)|2ω(t)dt =∞ −∞ Eit −2X 1+X2eitX 2 ω(t)dt, as n→∞. A.3 Covariance kernel under EISE estimators We have for the covariance kernel in Theorem 3.1 in case of EISE estimators KEISE(s,t;ν) =1 2s2+t2+|s−t|+1e−|s−t| −1 2(t2+|t|+1)(ν +1)2|s|(ν +2)ν3+(ν +1)(1−|s|)+|s|+3 ν3e−|s|−|t| +1 2(t2+|t|+1)(ν +1)(|s|(ν +1)2+3(ν +1)−|s|)−1 ν3 +(ν +1)(s2(ν +2)(ν +1)+2|s|(ν +2)+s2)e−(ν+1)|s|−|t| −1 2t(|t|+1)(ν +1)s(ν +2)3(ν +1) 2ν2+s(ν +1)2+s−4sgn(s)e−|s|−|t| +1 2t(|t|+1)s(ν +2)3(|s|(ν +1)3−3(ν +1)2+|s|(ν +1)+1) 4(ν +1)ν2 −s(ν +1)2+s−4sgn(s)(ν +1)e−|s|−|t| +1 2πt(|t|+1)(ν +2)3sJ 1(s)−2J2(s)e−|t| −1 2(s2+|s|+1)(ν +1)2|t|(ν +2)ν3+(ν +1)(1−|t|)+|t|+3 ν3e−|s|−|t| +1 2(s2+|s|+1)(ν +1)(|t|(ν +1)2+3(ν +1)−|t|)−1 ν3 +(ν +1)(t2(ν +2)(ν +1)+2|t|(ν +2)+t2)e−(ν+1)|s|−|t| −1 2s(|s|+1)(ν +1)t(ν +2)3(ν +1) 2ν2+t(ν +1)2+t−4sgn(t)e−|s|−|t| +1 2s(|s|+1)t(ν +2)3(|t|(ν +1)3−3(ν +1)2+|t|(ν +1)+1) 4(ν +1)ν2 −t(ν +1)2+t−4sgn(t)(ν +1)e−|s|−|t| +1 2πs(|s|+1)(ν +2)3tJ 1(t)−2J2(t)e−|s| +(s2+|s|+1)(t2+|t|+1)+s(|s|+1)t(|t|+1)(ν +2)2(5ν2+14ν+10) 64(ν +1)3·e−|s|−|t|, 123
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