Goodness-of-fit testing for the Cauchy distribution with application to financial modeling

1 Introduction

A Cauchy random variable with location parameter $\mu \in \mathbb{R}$ and scale parameter $\sigma >0$ , denoted by $X\sim C(\mu ,\sigma )$ , has probability density function

Siméon Denis Poisson discovered the Cauchy distribution in 1824, long before its first mention by Augustin-Louis Cauchy. Early interest in the distribution focused on its value as a counterexample which demonstrated the need for regularity conditions in order to prove important limit theorems (see Stigler, 1974). Thanks to this special nature, the Cauchy distribution is sometimes considered as a pathological case. However, it can be used as a model for describing a wealth of phenomena. This is exemplified in the sequel.

This probability law describes the energy spectrum of an excited state of an atom or molecule, as well as an elementary particle resonant state. It can be shown quantum mechanically that whenever one has a state which decays exponentially with time, the energy width of the state is described by the Cauchy distribution (Roe, 1992). Winterton et al. (1992) showed that the source of fluctuations in contact window dimensions is variation in contact resistivity, and the contact resistivity is distributed as a Cauchy random variable. Kagan (1992) pointed out that the Cauchy distribution describes the distribution of hypocenters on focal spheres of earthquakes. An application of this distribution to study the polar and non-polar liquids in porous glasses is given by Stapf et al. (1996). Min et al. (1996) found that Cauchy distribution describes the distribution of velocity differences induced by different vortex elements. An example in the context of quantitative finance is provided in Section 4.

Many statistical procedures, employed in the above mentioned applications, assume that the random mechanism generating the data follows the Cauchy distribution. A parametric procedure usually hinges on the assumption of a particular distribution. It is, therefore, of utmost importance to assess the validity of the assumed distribution. This is accomplished by performing a goodness-of-fit test. In this article, we suggest six tests of fit for the Cauchy distribution. They are modifications of a test based on Kullback-Leibler (KL) information criterion, previously studied by Mahdizadeh and Zamanzade (2017). Information theory deals with stochastic processes as sources of information, or as models of communication channels; see, for example, Stone (2015). It is known to be a powerful tool in the study of communication and control in the animal and the machine (Wiener, 1961). Being an essential part of probability theory, information theory is also closely related to statistical inference (Kullback, 1997). Vinga (2014) and Bensadon (2016) provide some applications in biological sequence analysis, and machine learning, respectively. A large body of literature has grown around developing goodness-of-fit tests using the information-theoretic measures such as the entropy and the KL distance. This approach has been successfully applied for many distributions, including normal, uniform, exponential, inverse Gaussian and Laplace, among others. See for example Vasicek (1976), Dudewicz and van der Meulen (1981), Grzegorzewski and Wieczorkowski (1999), Mudholkar and Tian (2002), Choi and Kim (2006), Al-Omari and Haq (2016), Al-Omari and Zamanzade (2017), Al-Omari and Zamanzade (2018), Mahdizadeh (2017a,b), Zamanzade and Mahdizadeh (2017a,b).

Section 2 is given to a review of the existing tests. The new goodness-of-fit tests are presented in Section 3. Power properties of these tests are assessed by means of Monte Carlo simulations. The results are reported in Section 4. To illustrate the suggested procedures, a real data set is analyzed in Section 5. We end in Section 6 with a summary.

2 Review of the existing goodness-of-fit tests

Given a random sample $X_1,\dots ,X_n$ from a population having a continuous density function $f\left(x\right)$ , consider the problem of testing $H_0:f\left(x\right)=f_0(x\text{;}\mu ,\sigma )$ for some $\mu \in \mathbb{R}$ and $\sigma >0$ , where $f_0(x\text{;}\mu ,\sigma )$ is given (1). The alternative hypothesis is $H_1:f\left(x\right)\ne f_0(x\text{;}\mu ,\sigma )$ for any $\mu \in \mathbb{R}$ and $\sigma >0$ .

The Cauchy distribution is a peculiar distribution due to its heavy tail and the difficulty of estimating its parameters (see Johnson et al., 1994). First, the method of moment estimation fails since the mean and variance of the Cauchy distribution do not exist. Second, the maximum likelihood estimates of the parameters are very complex. We therefore estimate $\mu$ and $\sigma$ by the median and the half-interquartile range which are attractive estimators because of their simplicity. Suppose $X_{\left(1\right)}⩽\cdots ⩽X_{\left(n\right)}$ are the sample order statistics, and ${\xi }_p(0<p<1)$ is the sample p th quantile. Then, the two estimators are given by

Recently, Mahdizadeh and Zamanzade (2017) proposed four new tests of fit for the Cauchy distribution. The first three of them are modifications of the tests introduced by Zhang (2002). The corresponding test statistics are

Constructing a test based on (12) entails estimating the unknown quantities. The non-parametric estimation of $H\left(f\right)$ has been studied by many authors. Vasicek (1976) introduced a simple estimator which has been widely used in developing tests of fit. His estimator is given by

It is difficult to derive the null distributions of (5)–(11) and (15) analytically. Monte Carlo simulations were then employed to determine critical values of a generic test statistic, say T. To this end, 50,000 samples were generated from $C(0,1)$ for each sample size $n=10,20,30,50,100,200$ . The estimators (3) and (4) were computed from any sample, and plugged into T. Finally, $1-\alpha$ quantile of the resulting values was determined which will be denoted by ${\mathcal{T}}_{1-\alpha }$ . The composite null hypothesis is rejected at level $\alpha$ if the observed value of T exceeds ${\mathcal{T}}_{1-\alpha }$ .

3 The proposed new tests

In this section, we introduce six new testing procedures for the Cauchy distribution. To clarify motivation of these tests, we first examine the entropy estimator component of statistic (15). It is worth noting that $H\left(f\right)$ can be expressed as $$H\left(f\right)={\int }_0^1\log \left(\frac{\mathrm{d}}{\mathrm{d}p}{F}^{-1}\left(p\right)\right)\mathrm{d}p\text{.}$$ Vasicek (1976) used the above representation to propose his nonparametric entropy estimator. In doing so, the involved derivative at each sample point $(X_{\left(i\right)},i/n)$ is estimated by $$d_i=\frac{X_{(i+m)}-X_{(i-m)}}{2m/n},$$ where the order statistics and window size are defined as in Section 2. Now, ${\mathit{HV}}_{m,n}$ is simply defined to be the mean of logarithm of $d_i$ ’s for $i=1,\dots ,n$ . Clearly, $d_i$ is not a correct formula when $i⩽m$ or $i⩾n-m+1$ . To fix this problem, the denominator and/or the numerator of $d_i$ should be adjusted. It is also possible to employ a fully different approach for entropy estimation. In the following, some improved entropy estimators are reviewed. These estimators are then incorporated in (15) to come up with new tests, which are expected to be more powerful.

Bowman (1992) studied the estimator

Van Es (1992) considered estimation of functionals of a probability density and entropy in particular. He proposed the following estimator

Ebrahimi et al. (1994) suggested two improved entropy estimators. The first one is equal to that of Vasicek plus a constant. This implies that the test based on this estimator is equivalent to ${\widehat{D}}_1$ . So it is not included in this study. The second estimator is given by

Correa (1995) proposed another entropy estimator defined as

The test statistics obtained by replacing ${\mathit{HV}}_{m,n}$ in (15) with ${\mathit{HB}}_n,{\mathit{HVE}}_{m,n},{\mathit{HE}}_{m,n},{\mathit{HC}}_{m,n},{\mathit{HY}}_{m,n}$ and ${\mathit{HA}}_{m,n}$ will be denoted by ${\widehat{D}}_2,{\widehat{D}}_3,{\widehat{D}}_4,{\widehat{D}}_5,{\widehat{D}}_6$ and ${\widehat{D}}_7$ , respectively. Again, Monte Carlo approach is adopted to compute critical values of the resulting tests. To calculate test statistics based on the KL distance (with the exception of ${\widehat{D}}_2$ ), the window size m corresponding to a given sample size must be selected in advance. In entropy estimation based on spacings, choosing optimal m for given n is still an open problem. For each n, the window size having smallest critical value tends to yield greater power. For sample sizes 10, 20, 30, 50, 100, and 200, window sizes producing the minimum critical values for different tests are given in Table 1. Table 2 contains 0.05 critical points of the tests considered in this study. For the KL distance based tests, the above mentioned optimal window sizes are used. These thresholds will be used in the next section to study the power properties.

The entropy estimators mentioned in this section are consistent. In proving this result for the estimators dependent on the window size, it is assumed that $m/n\to 0$ as $m\to \infty$ and $n\to \infty$ . See pertinent references for more details. The next proposition attends to optimal property of the tests based on the KL distance.

4 Power comparisons

In this section, performances of the proposed tests are evaluated via Monte Carlo experiments. Toward this end, we considered nine families of alternatives:

• t distribution with n degrees of freedom denoted by $t_n$ .

• Normal distribution with mean $\mu$ and variance ${\sigma }^2$ denoted by N $(\mu ,{\sigma }^2)$ .

• Logistic distribution with mean $\mu$ and variance ${\pi }^2{\sigma }^2/3$ denoted by Lo $(\mu ,{\sigma }^2)$ .

• Laplace distribution with mean $\mu$ and variance $2{\sigma }^2$ denoted by La $(\mu ,{\sigma }^2)$ .

• Gumbel distribution with mean $\mu +\sigma \gamma$ (where $\gamma$ is Euler’s constant) and variance ${\pi }^2{\sigma }^2/6$ denoted by Gu $(\mu ,{\sigma }^2)$ .

• Beta distribution with mean $\alpha /(\alpha +\beta )$ denoted by Be $(\alpha ,\beta )$ .

• Gamma distribution with mean $\alpha \beta$ and variance $\alpha {\beta }^2$ denoted by Ga $(\alpha ,\beta )$ .

• Mixture of the normal and Cauchy distributions with mixing probability p denoted by NC $(p,1-p)$ . The distribution mixes N(0,1) and C(0,1) with weights p and $1-p$ , respectively.

• Tukey distribution with parameter h denoted by Tu $\left(h\right)$ . It is distribution of the random variable $Z\exp \{{\mathit{Zh}}^2/2\}$ with $Z\sim N(0,1)$ .

The members selected from the above families are $t_3,t_5$ , N(0,1), Lo(0,1), La(0,1), Gu(0,1), Be(2,1), Ga(2,1), NC(0.3,0.7) and Tu(1). For each alternative, 50,000 samples of sizes $n=10,20,30,50$ were generated, and the power of each test was estimated by the percentages of samples entering the rejection region. Tables 3–6 present the estimated powers of the fourteen tests of size 0.05, given in Sections 2 and 3, for different sample sizes (the results for $n=100,200$ are provided as Supplementary material). To provide enough space for the outputs, the reference to parameters of the distributions is only made in the case of t distribution. For each alternative, power entry associated with the best test among ${\widehat{D}}_i$ ’s is in bold. In addition, the highest power value from the other tests is in italic.

It is observed that no single test is uniformly most powerful. We note, however, that the tests based on the KL distance are generally more powerful than the other tests. Compare the bold and italic entries for each alternative. Given a distribution and sample size, difference of the italic entry from the bold one is reported in Table 7. The values are sizable for symmetric distributions like $t_3,t_5$ , N(0,1), Lo(0,1), La(0,1) and Gu(0,1). All of the tests perform poorly when the parent distribution is either NC(0.3,0.7) or Tu(1), and increasing the sample size does not give rise to marked improvement in power.

With the exception of sample size 10, $Z_A$ is generally the best among KS, $A^2,W^2,D_{n,\lambda },Z_K,Z_A$ and $Z_C$ tests. Moreover, it can be seen that either ${\widehat{D}}_2$ or ${\widehat{D}}_4$ has mostly the best performance among ${\widehat{D}}_i$ ’s.

5 Example

Heavy-tailed distributions, like Cauchy, are better models for financial returns because the normal model does not capture the large fluctuations seen in real assets. Nolan (2014) provides an accessible introduction to financial modeling using such distributions.

The stock market return is the return that we obtain from stock market by buying and selling stocks or get dividends by the company whose stock you hold. The stock market price is usually modeled by lognormal distribution, that is to say stock market returns follow the Gaussian law. The feature of stock market return distribution is a sharp peak and heavy tails. The Gaussian distribution clearly does not enjoy these attributes. So the Cauchy distribution may be a potential model. The German Stock Index (DAX) is the major stock market index in Germany which contains the stocks of 30 largest German companies trading on the Frankfurt Stock Exchange. The DAX evaluates the Prime Standard of those 30 major German companies trading on the Frankfurt Stock Exchange. We now apply the fourteen goodness-of-fit tests to a real dataset containing 30 returns of closing prices of the DAX. The data are observed daily from January 1, 1991, excluding weekends and public holidays. The data (rounded up to seven decimal places) are given in Table 8, which are obtained from datasets package in R statistical software. The Cauchy Q-Q plot appears in Fig. 1. The corresponding histogram, superimposed by a Cauchy density function, is also included. The location and scale parameters estimated from the data are $\widehat{\mu }=0.0009629174$ and $\widehat{\sigma }=0.003635871$ .

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Fig. 1

The Cauchy Q-Q plot of the 30 returns, and the corresponding histogram along with fitted Cauchy density.

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The values of all statistics are computed (see Table 9), and compared with the corresponding critical values in Table 2. By using any test, the null hypothesis that the data follow the Cauchy distribution is not rejected at 0.05 significance level.

6 Conclusion

This article concerns goodness-of-fit test for the Cauchy distribution. Six tests based on the KL information criterion are developed, and shown to be consistent. A simulation study is carried out to compare the performances of the new tests with their contenders. In doing so, five sample sizes and nine families of alternatives are considered. It emerges that the new tests are powerful against many symmetric distributions. The proposed procedures are finally applied on real data example.