Jackknife empirical likelihood test for testing one-sided Lévy distribution

ABSTRACT

Based on U-empirical process we construct a jackknife empirical likelihood-based test for testing one-sided Lévy distribution. Adjusted jackknife empirical likelihood test also developed. The simulation study shows that the proposed tests have very good power for various alternatives. Finally, we illustrate the test procedure using three real data sets.

1. Introduction

Stable distribution introduced by French Mathematician Paul Lévy [13] have been used in literature when one need to captures asymmetry, tail behavior and high kurtosis. For the applications of stable distribution in various disciplines we refer to Fama [5], Lau et al. [10], Jurlewicz and Weron [8], Samoradnitsky [17] and Shu et al. [18] and the references therein. The characteristic function of a stable random variable with skew parameter, $\mathrm{\text{β}}\in [-1,1]$, can be written as

$$\psi (t,\alpha ,\mathrm{\text{β}},\sigma )=\exp \left(-|\sigma t{|}^{\alpha }(1-i\mathrm{\text{β}}\mathrm{s}\mathrm{g}\mathrm{n}(t\left)\mathrm{\Psi }\right)\right),$$

where, $\mathrm{s}\mathrm{g}\mathrm{n}\left(t\right)=-1,0,1$ when t<0, t = 0 and t>0, respectively, and $\mathrm{\Psi }=\tan \left(\frac{\pi \alpha }{2}\right)$ for $\alpha \ne 1$ and $\mathrm{\Psi }=-\frac{2}{\pi }\log |t|$ for $\alpha =1$. The characteristic exponent or tail index $\alpha \in (0,2]$ governs the tail behavior of the distribution. For $\alpha =2,1,0.5$, $\psi (t,\alpha ,\mathrm{\text{β}},\sigma )$ have closed form probability density function and the corresponding distributions are known as Normal, Cauchy and Lévy, respectively. Numerous tests are available in literature for testing normal and Cauchy distribution. We refer readers to Thode [19], Zamanzade and Arghami [20,21] and Mahdizadeh and Zamanzade [14,15] and the references therein for more details on these tests. However, to the best of our knowledge, no specific test is available for Lévy distribution. Hence in this article, we propose a test based on jackknife empirical likelihood (JEL) ratio for testing Lévy distribution.

For the sake of completeness, we briefly outline the properties of Lévy distribution. A positive random variable (r.v.) is said to have the Lévy distribution with scale parameter $\sigma \in {\mathbb{R}}^{+}$ if its probability density function (pdf) is of the form

$$f(x;\sigma )=\sqrt{\frac{\sigma }{2\pi }}{x}^{-3/2}{\mathrm{e}}^{-\sigma /2x},x>0$$

and the corresponding distribution function is given by

$$F\left(x\right)=2\left(1-\mathrm{\Phi }\left((\sigma /x{)}^{1/2}\right)\right),x>0,$$ (1)

where $\mathrm{\Phi }(.)$ is the distribution function of a standard normal r.v. The Lévy distribution is a special case of the inverted gamma distribution with the shape parameter 1/2 and scale parameter $2/\sigma$. An inverse Gaussian density with parameters $(\lambda ,\mu )$ converges to the Lévy density as $\theta \to 0$ where $\theta =\lambda /\mu$ and $\sigma =\lambda$ (cf. O'Reilly and Rueda [16]).

Rest of the article is organized as follows. In Section 2, based on U-empirical process we propose a test for testing one sided Lévy distribution and studied its asymptotic properties. We also propose jackknife empirical likelihood ratio test and adjusted jackknife empirical likelihood ratio test (AJEL) for testing the same. Result of a Monte Carlo simulation study is presented in Section 3 to assess the performance of the proposed tests. A numerical illustration of the proposed method is given in Section 4. Some concluding remarks are given in Section 5.

2. Proposed test

Motivated by the fact that there is no specific test is available for one-sided Lévy distribution, we propose JEL- and AJEL-based tests for one-sided Lévy distribution. We use the following characterization result to develop our test.

Theorem 2.1 Feller [6] —

Let $X,X_1$ and $X_2$ be i.i.d. r.v.'s having stable law with exponent α, then

$$s^{1/\alpha }{X}_1+t^{1/\alpha }{X}_2\stackrel{d}{=}(s+t{)}^{1/\alpha }X,\mathrm{\forall }s,t>0$$

Considering $s=t=1/2$ and $\alpha =1/2$ in Theorem 2.1, particular case for Lévy distribution was obtained in the following result.

Theorem 2.2

Let X, Y and Z be independent and identically distributed positive random variables. The distributions of X and $\frac{Y+Z}{4}$ are the same if and only if X, Y and Z have one-sided Lévy distribution with arbitrary scale factor.

2.1. Test based on U-empirical process

Using Theorem 2.2, first we develop a test based on U-empirical process. Based on a random sample $X_1,X_2,\dots ,X_n$ of size n from Lévy class of distribution $\mathcal{F}$ we are interested to test the null hypothesis

$${\mathcal{H}}_0:F\in \mathcal{F}$$

against the alternative hypothesis

$${\mathcal{H}}_1:F\ne \mathcal{F}$$

with the same support ${\mathbb{R}}^{+}$. To test the above hypothesis we define a departure measure $\mathrm{\Delta }\left(F\right)$ given by

$$\mathrm{\Delta }\left(F\right)={\int }_0^{\mathrm{\infty }}(P(X_1+X_2\le 4t)-P(X_3\le t))\mathrm{d}F\left(t\right),$$

where the random variables $X_1,X_2$ and $X_3$ are independently distributed as $\mathcal{F}$. Using Theorem 2.2, it can be easily verify that $\mathrm{\Delta }\left(F\right)$ is zero under ${\mathcal{H}}_0$ and not zero under ${\mathcal{H}}_1$.

We propose the following test statistic for testing the above hypothesis

$$T_n^{\ast }={\int }_0^{\mathrm{\infty }}\left({\mathcal{L}}_n\right(t)-F_n(t\left)\right)\mathrm{d}{F}_n\left(t\right),$$ (2)

where $F_n\left(t\right)=\frac{1}{n}\sum _{i=1}^n\mathbb{I}\{X_i\le t\}$ is the empirical distribution function and ${\mathcal{L}}_n\left(t\right)$ is the U-empirical distribution function given by

$${\mathcal{L}}_n\left(t\right)=\frac{1}{\left(\begin{array}{c}n\\ 2\end{array}\right)}\sum _{i=1}^n\sum _{j=1,j<i}^n\mathbb{I}\left\{\frac{X_i+X_j}{4}\le t\right\},t<0.$$ (3)

Here $\mathbb{I}(.)$ denote the indicator function. An asymptotic equivalence test of $T_n^{\ast }$ is given by

$$T_n=\frac{1}{\left(\begin{array}{c}n\\ 3\end{array}\right)}\sum _{i=1}^n\sum _{j=1,j<i}^n\sum _{k=1,k<j}^n\mathbb{I}\left\{\frac{X_i+X_j}{4}\le X_k\right\}-\frac{1}{2}.$$ (4)

For $X_i,X_j$ and $X_k$, $1\le k<j<i\le n$, we have

$$\begin{array}{rl}E\left(\mathbb{I}\left\{\frac{X_i+X_j}{4}\le X_k\right\}\right)& =P\left(\frac{X_i+X_j}{4}\le X_k\right)\\ & ={\int }_0^{\mathrm{\infty }}P(X_i+X_j\le 4t)\mathrm{d}F\left(t\right).\end{array}$$

Also for continuous F, ${\int }_0^{\mathrm{\infty }}F\left(t\right)\mathrm{d}F\left(t\right)=1/2$. Hence $T_n$ is an unbiased estimator of $\mathrm{\Delta }\left(F\right)$. Test procedure is to reject ${\mathcal{H}}_0$ against ${\mathcal{H}}_1$ for large values of $T_n$.

Next, we study the asymptotic properties of $T_n$. Because $T_n$ a U-statistic, it converges in probability to $\mathrm{\Delta }\left(F\right)$ under ${\mathcal{H}}_1$ [12]. In the following theorem we obtain the asymptotic distribution of $T_n$.

Theorem 2.3

Let X, $X_1$ and $X_2$ be i.i.d. r.v's with distribution function F. As $n\to \mathrm{\infty },$ $\sqrt{n}(T_n-\mathrm{\Delta }(F\left)\right)$ approaches a Gaussian with mean zero and variance ${\sigma }^2,$ where ${\sigma }^2$ is given by

$${\sigma }^2=\mathrm{V}\mathrm{a}\mathrm{r}\left(2P\left(\frac{X+X_2}{4}\le X_3|X\right)+P\left(\frac{X_1+X_2}{4}\le X|X\right)\right).$$ (5)

Proof.

Write $T_n=T_{n1}-\frac{1}{2}$, where

$$T_{n1}=\frac{1}{\left(\begin{array}{c}n\\ 3\end{array}\right)}\sum _{i=1}^n\sum _{j=1,j<i}^n\sum _{k=1,k<j}^n\mathbb{I}\left\{\frac{X_i+X_j}{4}\le X_k\right\}.$$ (6)

$E\left(T_n\right)=\mathrm{\Delta }\left(F\right)$. Hence the asymptotic distributions of $\sqrt{n}(T_n-\mathrm{\Delta }(F\left)\right)$ and $\sqrt{n}(T_{n1}-E(T_{n1}\left)\right)$ are same. Now $T_{n1}$ is a U-statistic with kernel $h^{\ast }(X_1,X_2,X_3)=\mathbb{I}\{\frac{X_1+X_2}{4}\le X_3\}$ of degree three. Symmetric version of $h^{\ast }(.)$ is given by

$$h(X_1,X_2,X_3)=\frac{1}{3}\left(h^{\ast }\right(X_1,X_2,X_3)+h^{\ast }(X_1,X_3,X_2)+h^{\ast }(X_2,X_3,X_1\left)\right).$$

Using central limit theorem for U-statistics (see Theorem 1, Chapter 3 of Lee [11]), as $n\to \mathrm{\infty }$, $\sqrt{n}(T_{n1}-E(T_{n1}\left)\right)$ converges in distribution to Gaussian with mean zero and variance $9{\sigma }_1^2$, where ${\sigma }_1^2$ is the asymptotic variance of $T_n$ and is given by

$${\sigma }_1^2=\mathrm{V}\mathrm{a}\mathrm{r}(E\left(h\right(X_1,X_2,X_3\left)|X_1\right)).$$

Consider

$$E\left(h\right(X_1,X_2,X_3)|X_1=x)=\frac{2}{3}P\left(\frac{x+X_2}{4}\le X_3\right)+\frac{1}{3}P\left(\frac{X_1+X_2}{4}\le x\right).$$

Hence, we obtain the variance expression specified in the theorem.

Next, we obtain the asymptotic null distribution of the test statistic. Note that under ${\mathcal{H}}_0$, $\mathrm{\Delta }\left(F\right)=0$. Hence we have the following result.

Corollary 2.4

Let X be positive random variable with distribution function specified in Equation (1). As $n\to \mathrm{\infty },$ $\sqrt{n}{T}_n$ converges in distribution to Gaussian with mean zero and variance ${\sigma }_0^2$ where ${\sigma }_0^2$ is given by

$${\sigma }_0^2=\mathrm{V}\mathrm{a}\mathrm{r}\left({\int }_0^{\mathrm{\infty }}2\left(1-F\left(\frac{X+y}{4}\right)\right)\mathrm{d}F\left(y\right)+F\left(X\right)\right).$$ (7)

Proof.

Under ${\mathcal{H}}_0$, using the characterization of one-sided Lévy distribution given in Theorem 2.2, we obtain

$$P\left(\frac{X_1+X_2}{4}\le x\right)=P(X\le x).$$

Hence from Equation (5), we have

$${\sigma }_0^2=\mathrm{V}\mathrm{a}\mathrm{r}\left({\int }_0^{\mathrm{\infty }}2\left(1-F\left(\frac{X+y}{4}\right)\right)\mathrm{d}F\left(y\right)+F\left(X\right)\right),$$

which completes the proof.

Using Corollary 2.4, we obtain a test based on normal approximation and we reject the null hypothesis ${\mathcal{H}}_0$ against ${\mathcal{H}}_1$ if

$$\frac{\sqrt{n}|T_n|}{{\hat{\sigma }}_0}\ge Z_{\alpha /2},$$

where ${\hat{\sigma }}_0$ is a consistent estimator of ${\sigma }_0$ and $Z_{\alpha }$ is the upper α-percentile point of standard normal distribution.

Implementation of the test based on normal approximation is not simple as it is very difficulty to find a consistent estimator of ${\sigma }_0^2$. This motivate us to develop an empirical likelihood-based test which is distribution free. Next, we discuss JEL- and AJEL-based test for testing one-sided Lévy distribution.

2.2. Jackknife empirical likelihood ratio test

To construct JEL-based test, first we obtain jackknife pseudo values using expression (4). The jackknife pseudo values are defined as

$$v_n^i=n{T}_n-(n-1)T_n^{(-i)},i=1,2,\dots ,n,$$ (8)

where $T_n^{(-i)}$ is the value of the test statistic computed from (4) using the same sample excluding the ith observation. The Jackknife empirical likelihood ratio for testing one-sided Lévy distribution is given by

$$l\left(p\right)=max\left\{\prod _{i=1}^n\left(n{p}_i\right):\sum _{i=1}^n{p}_i=1,p_i\ge 0,\sum _{i=1}^n{p}_i{v}_n^i=0\right\}.$$ (9)

Using the Lagrange multipliers method, we obtain $p_i$ as

$$p_i=\frac{1}{n}\frac{1}{1+{\lambda }_1{\hat{v}}_n^i},$$ (10)

where ${\lambda }_1$ satisfies

$$f\left({\lambda }_1\right)=\frac{1}{n}\sum _{i=1}^n\frac{{\hat{v}}_n^i}{1+{\lambda }_1{\hat{v}}_n^i}=0,$$ (11)

provided

$$\underset{1\le k\le n}{min}{v}_n^k<T_n<\underset{1\le k\le n}{max}{v}_n^k.$$ (12)

Hence we obtain the jackknife empirical log-likelihood ratio as

$$\log l\left(p\right)=-\sum _{i=1}^n\log \left(1+{\lambda }_1{\hat{v}}_n^i\right).$$

For large values of $l\left(p\right)$ we reject the null hypothesis ${\mathcal{H}}_0$ against the alternative hypothesis ${\mathcal{H}}_1$. To construct a critical region of the JEL-based test we find the asymptotic null distribution of the jackknife empirical log-likelihood ratio.

In likelihood theory, it is well-know that the asymptotic null distribution of likelihood ratio statistic is ${\chi }^2$ which is known as Wilks' theorem. Wilk's Theorem state the following: Suppose the dimensions of the parameter spaces are r and k under general and the null hypothesis, respectively. Let Λ be the likelihood ratio statistics. Under some regularity conditions and assuming ${\mathcal{H}}_0$, as $n\to \mathrm{\infty }$, $-2\log \mathrm{\Lambda }$ converges in distribution to ${\chi }^2$ with degrees of freedom equal to r−k. Next, we state analog of Wilk's theorem.

Theorem 2.5

Define $h(X_1,X_2,X_3)=\frac{1}{3}\left(\mathbb{I}\left(\frac{X_1+X_2}{4}\le X_3\right)+\mathbb{I}\left(\frac{X_1+X_3}{4}\le X_2\right)+\mathbb{I}\left(\frac{X_2+X_3}{4}\right.\right.\left.\left.\le X_1\phantom{\left(\frac{X_1+X_3}{4}\le X_2\right)}\right)\right)$ and ${\sigma }_1^2=\mathrm{V}\mathrm{a}\mathrm{r}\left(E\right(h(X_1,X_2,X_3)|X_1\left)\right)$. Assume $\mathbb{E}\left(h^2\right(X_1,X_2,X_3\left)\right)<\mathrm{\infty }$ and ${\sigma }_1^2>0$, then as $n\to \mathrm{\infty }$, $-2\log l\left(p\right)$ converges in distribution to ${\chi }^2$ with one degree of freedom.

Proof.

In view of Theorem 2.3, the conditions $\mathbb{E}\left(h^2\right(X_1,X_2,X_3\left)\right)<\mathrm{\infty }$ and ${\sigma }_1^2>0$ are satisfied. Hence by Lemma A1 of Jing et al. [7] we have condition (12). Accordingly the proof of theorem follows from the Theorem 1 of Jing et al. [7].

Based on Theorem 2.5, we reject the null hypothesis against the alternatives at a level of significance α, if

$$-2\log l\left(p\right)>{\chi }_{1,\alpha }^2,$$

where ${\chi }_{1,\alpha }^2$ is the upper α-percentile point of ${\chi }^2$ distribution with one degree of freedom.

2.3. Adjusted jackknife empirical likelihood ratio test

Chen and Ning [2] combine the idea of the jackknife and adjusted empirical likelihood and proposed adjusted jackknife empirical likelihood ratio test. The adjusted jackknife empirical likelihood is given by

$$l\left(p\right)=max\left\{\prod _{i=1}^{n+1}{p}_i:\sum _{i=1}^{n+1}{p}_i=1,p_i\ge 0,\sum _{i=1}^{n+1}{p}_i\cdot {}^{ad}{v}_n^i=0\right\},$$ (13)

where

$${}^{ad}{v}_n^i=\left\{\begin{array}{ll}{v}_n^i& i=1,\dots ,n.\\ \frac{-a_n}{n}\sum _{i=1}^n{v}_n^i& i=n+1.\end{array}\right.$$ (14)

and $a_n$ is given as $max(1,\log (n\left)/2\right)$ (Chen et al. [3]).

Thus, the adjusted jackknife empirical log-likelihood ratio for testing one-sided Lévy distribution is given by

$$\log l^{ad}\left(p\right)=-\sum _{i=1}^{n+1}\log \left(1+{\lambda }_2{}^{ad}{\hat{v}}_n^i\right).$$

where λ satisfies

$$f\left({\lambda }_2\right)\equiv \frac{1}{n+1}\sum _{i=1}^{n+1}\frac{{}^{ad}{\hat{v}}_n^i}{1+{\lambda }_2\cdot {}^{ad}{\hat{v}}_n^i}=0.$$ (15)

The Wilk's theorem holds in this case as well and we state it as next result.

Theorem 2.6

Under the conditions stated in Theorem 2.5 and if $a_n=max(1,\log (n\left)/2\right),$ as $n\to \mathrm{\infty },$ $-2\log l^{ad}\left(p\right)$ converges in distribution to ${\chi }^2$ with one degree of freedom.

Proof.

Using the definition of jackknife pseudo value given in Equation (14), we obtain $T_n=\frac{1}{n}\sum _{i=1}^n{\hat{v}}_n^i$. Let $S^2=\frac{1}{n}\sum _{i=1}^n({\hat{v}}_n^i{)}^2$. By strong law of large number, we have $S^2={\sigma }_0^2+o\left(1\right)$. Now, as long as $a_n=o_p\left(n\right)$, we have $|{\lambda }_2|=O_p\left(1/\sqrt{n}\right)$. Consider

$$\begin{array}{rl}-2\log l^{ad}\left(p\right)& =\sum _{i=1}^{n+1}2\log \left(1+{\lambda }_2{}^{ad}{\hat{v}}_n^i\right)\\ & =2\sum _{i=1}^{n+1}({\lambda }_2{}^{ad}{\hat{v}}_n^i-({\lambda }_2{}^{ad}{\hat{v}}_n^i{)}^2/2)+o_p(1)\\ & =2n{\lambda }_2{T}_n-nS{\lambda }_2^2+o_p\left(1\right)\\ & =\frac{n{T}_n^2}{S^2}+o_p\left(1\right),\end{array}$$

where the second last identity follows from the fact that the $(n+1)$th term of the summation is $a_n{O}_p\left(n^{-3/2}\right)=o_p\left(n\right)O_p\left(n^{-3/2}\right)=o_p\left(1\right)$. Now, in view of Corollary 2.4, as $n\to \mathrm{\infty }$, $\frac{n{T}_n^2}{{\sigma }_0^2}$ converges in distribution to ${\chi }^2$ with one degree of freedom. Hence by Slutsky's theorem we have the result.

In adjusted jackknife empirical likelihood ratio test we reject the null hypothesis against the alternative hypothesis at significance level α if

$$-2\log l^{ad}\left(p\right)>{\chi }_{1,\alpha }^2.$$

3. Monte Carlo simulation study

We conduct a Monte Carlo simulation study to assess the performance of both jackknife and adjusted jackknife empirical likelihood ratio tests. The simulation study is done using R statistical software. The empirical type I error and powers of both these tests are estimated by taking 10,000 replications with samples sizes n = 25, 50, 75, 100 and 200. To find the empirical type I error, we generate sample of size n from one sided Lévy distribution with different values of σ. For these 10,000 replicas, proportion of test statistics falls in the critical region is computed and it gives the empirical type I error of the test. The empirical type I error calculated for both JEL and AJEL ratio tests is reported in Table 1. From Table 1 we observed that the empirical type I error for both these tests is close to 0.05 for all values of σ we considered.

Table 1. Empirical type I error of both JEL and AJEL tests.

	Lévy(0, 0.5)		Lévy(0, 1)		Lévy(0, 2)
n	JEL	AJEL	JEL	AJEL	JEL	AJEL
25	0.0582	0.0412	0.0545	0.0419	0.0513	0.0433
50	0.0544	0.0445	0.0532	0.0432	0.0502	0.0451
75	0.0522	0.0473	0.0518	0.0458	0.0499	0.0467
100	0.0509	0.0482	0.0511	0.0483	0.0501	0.0478
200	0.0501	0.0497	0.0504	0.0497	0.0496	0.0489

The empirical powers of both the tests are estimated under six families of alternatives with positive support, viz. (i) Burr (1.5,0.5,0.5), (ii) lognormal(0,1) (iii) chi-sq(3), (iv) half-normal(0,1), (v) gamma(3,2) and (vi) Weibull(2,1) distributions, all of which are considered as an alternative models to Lévy distribution for modeling skewed data. The empirical powers obtained for these alternatives are given in Table 2. From Table 2 we observed that the empirical power of both the tests approach to one as the sample size n increases.

Table 2. Empirical power of both JEL and AJEL tests.

Models	Burr(1.5, 0.5, 0.5)		lognorm(0, 1)		Chisq(3)
n	JEL	AJEL	JEL	AJEL	JEL	AJEL
25	0.3259	0.2122	0.2912	0.2421	0.5235	0.3928
50	0.4124	0.3512	0.6224	0.5693	0.6673	0.7237
75	0.4689	0.4962	0.7825	0.7214	0.9323	0.9254
100	0.6283	0.5723	0.8562	0.8383	0.992	0.9872
200	0.8652	0.8412	0.9862	0.9432	1	1
Models	Half normal(0, 1)		Gamma(3, 2)		Weibull(2, 1)
n	JEL	AJEL	JEL	AJEL	JEL	AJEL
25	0.5275	0.4225	0.8325	0.7633	0.7923	0.7713
50	0.7632	0.7312	0.9922	0.9792	0.9884	0.9742
75	0.9263	0.8938	0.9991	0.9981	1	1
100	0.9836	0.9773	1	1	1	1
200	1	1	1	1	1	1

We aslo generate r.v.'s from stable distribution with $\mathrm{\text{β}}=1$ and α values varies from 0.2 to 0.9 by using R-package ‘stabledist’. In Figure 1 we present the empirical power curve for both the JEL- and AJEL-based tests. The empirical power is obtained using samples of sizes n = 25 and 50 and 10000 repetitions. From Figure 1, one can observe that the empirical powers of the tests decreases as α approaches to 0.5 and further increases to one as values of α goes away from 0.5. As expected, for $\alpha =0.5$ the empirical power is closed to 0.05, which is the case for Lévy distribution.

Figure 1.

Empirical power curve of the Jackknife empirical likelihood ratio test (in black) and adjusted jackknife empirical likelihood ratio test (in red) for sample size (a) n = 25 and (b) n = 50.

4. Illustration

We illustrate the use of our test using three real data sets. First we consider the lifetime of pressure vessels data taken from Keating et al. [9]. The lifetime of pressure vessels constructed of fiber/epoxy composite material wrapped around metal liners may be modeled by using gamma distribution with shape parameter depends on factors like applied pressure and composite wall thickness [9]. We consider failure times of 20 similarly constructed vessels subjected to a certain constant pressure and the data is given in Table 3.

Table 3. The lifetime of pressure vessels.

274	28.5	1.7	20.8	871	363	1311	1661	236	828
458	290	54.9	175	1787	970	0.75	1278	776	126

Keating et al. [9] modelled the above data using gamma distribution where the re-estimated shape parameter comes out to close to 0.5. As Lévy's distribution is a special cases of inverse of Gamma distribution, we consider the inverse of the above observations and the calculated JEL- and AJEL-based test statistics value are 0.3161 and 0.2688 respectively. Both the test statistic value are less than 5% LoS value. Hence we can not reject the null hypothesis and conclude that the Lévy distribution is a reasonable choice for modeling inverse of the data, which is consistent with that of Keating et al. [9].

Next, we use the weighted average of rainfall (in mm) data in the month of January for the whole country starting from 1981 to 2011 released by Meteorological Department, Ministry of Earth Sciences, Government of India. This data is based on more than 2000 rain gauge readings spread over the entire country and are presented in Table 4 and it is available at www.data.gov.in. The calculated JEL- and AJEL-based test statistics values are 3.083 and 2.769, respectively, which are less than the critical value at 5% LoS. Hence, one sided Lévy distribution can be used to model the rainfall data.

Table 4. Weighted average of rainfall (in mm) data for the whole country for the month of January.

Year	Rainfall (in mm)	Year	Rainfall (in mm)	Year	Rainfall (in mm)
1981	29.3	1992	16.0	2003	7.6
1982	23.8	1993	18.2	2004	25.7
1983	18.5	1994	25.0	2005	28.1
1984	19.0	1995	31.3	2006	17.7
1985	23.2	1996	22.9	2007	1.7
1986	15.5	1997	14.3	2008	18.4
1987	13.2	1998	16.4	2009	12
1988	10.4	1999	13.7	2010	7.5
1989	15.4	2000	18.4	2011	6.8
1990	16.0	2001	7.3
1991	14.3	2002	15.7

Finally, we consider the ground water data considered by Duigon and Cooper [4] and Chang et al. [1]. Duigon and Cooper [4] collected data from various types of wells to study on availability of ground water to neighboring domestic wells near Bel Air, Harford county, Maryland. The data we consider is extracted from the original one, provides the well yields (or ‘specific capacity’ in gal/min/ft) based on Hillside location. The data is given Table 9 of Chang et al. [1] and we reproduced the same in Table 5. Chang et al. [1] considered gamma distribution for modeling this data and the estimated shape parameter is close to 0.5. The values of JEL- and AJEL-based test statistics for the inverse values of well yield are 0.036 and 0.033 respectively. Hence, we cannot reject the null hypothesis and one sided Lévy distribution can be used to model the inverse of water level in the wells near Bel Air.

Table 5. Well yields (in gal/min/ft) based on Hillside location.

0.22	1.33	0.75	0.18	0.01	0.16	0.28	0.87	0.02	0.10	0.03
0.05	0.86	5.00	0.04	4.00	0.37	0.38	0.11	0.10	0.02	0.01
0.05	0.17	0.46	0.16	1.33	0.14	2.86	0.13	7.50	4.50	0.03
0.003	0.05	0.02	0.04	0.75	0.52	5.00	0.35

5. Concluding remarks

A specific test for testing one-sided Lévy distribution is not available in literature. Motivated by this fact, using a simple characterization of one-sided Lévy distribution we proposed jackknife and adjusted jackknife likelihood ratio test for testing the same. We used U-empirical process to obtain the jackknife pseudo values. Monte Carlo simulation study show that the tests developed have well controlled error rate and very good power for various alternatives. Finally we illustrated our tests procedure using lifetime of pressure vessels, rainfall data of India and ground water level at Bel Air, Harford county, Maryland. On similar line of this article, one can develop JEL and AJEL ratio test for positive stable law for $0<\alpha <1$. Entropy-based tests for Lévy distribution and its comparison with the proposed tests also can be considered for future work.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Sudheesh K. Kattumannil  http://orcid.org/0000-0002-9605-3400