Testing the Validity of Laplace Model Against Symmetric Models, Using Transformed Data

In this paper, we ﬁrst present three characterizations of Laplace distribution and then we introduce a goodness of ﬁt test for Laplace distribution against symmetric distributions, based on one of the transformations. The power of the proposed test under various alternatives is compared with that of the existing tests, by simulation. To show the behavior of the proposed test in real cases, two real examples are presented.


Introduction
Laplace distribution L(µ, θ) has the following density function: where θ is a positive scale parameter and µ is the location parameter. This distribution was discovered by Laplace in 1774 for describing the errors of measurement. It is also known as the double exponential distribution. The Laplace distribution is widely used in many applications. It is therefore very important to test whether the underlying distribution has a Laplace form. Our goal in this article is constructing a new goodness of fit test for Laplace distribution using transformed data. [1] proposed the following test statistic for testing uniformity.
where u (1) ≤ u (2) ≤ ... ≤ u (n) are order statistics from a random sample of size n, F 0 is the uniform distribution function and | . | is the absolute value function. Alsô is the kernel estimate of f , K is the standard normal density and the bandwidth h is obtained from the normal optimal smoothing formula, h = 1.06sn − 1 5 , where s is the sample standard deviation.

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[1], [2] and [3] introduced some goodness of fit tests for exponentiality using transformed data. Also, they [4] introduced a goodness of fit test for normal distribution using transformed data. Then, [5] improved the exponentiality test using transformed data.
Main contribution of this paper is as follows. We first introduce a new goodness of fit test for Laplace distribution using transformed data. The properties of new test statistic are stated. Then, the critical values and power values of the proposed test are obtained and compared with the powers of the competing tests. We show that the proposed test has the higher power as compared with the other tests against the considered alternatives. Finally, two real examples to show the behavior of the proposed test in real cases are presented. Moreover, recently [6], [7] and [8] proposed new goodness of fit tests for some statistical distributions.
In Section 2, we introduce three characterizations of Laplace distribution and next, based on one of the transformations, we introduce a goodness of fit test for Laplace distribution against symmetric distributions. The power of the proposed test is obtained, by simulation, and the results are in Section 3 where the proposed test is compared with other tests. In Section 4, two real examples are presented and analyzed.

The test statistic
Given a random sample X 1 , ..., X n from a continuous symmetric probability distribution F with a density f (x), the hypothesis of interest is where µ and θ are unspecified and Θ = R × R + . The alternative to H 0 is In order to obtain a test statistic, we use the following theorem.
Theorem 1. Let X 1 and X 2 be two independent observations from a symmetric continuous distribution F . Then is Fisher's distribution with 2 and 2 degrees of freedom and | . | is the absolute value function.) Proof. [2] showed that the above transformations (i), (ii) and (iii) (with absolute value signs removed) characterize exponential distribution. It is also mentioned in [9]. We know that if X has a symmetric distribution, then X is distributed as Laplace(0, θ) if and only if |X| is distributed as exponential(θ). Thus the results follow.
To construct a test using the mentioned characterizations, we transform the sample data (X 1 , X 2 , ..., X n ) to By the above theorem, under the null hypothesis, each W ij has a uniform distribution, and it seems to be appropriate to use our proposed uniformity test (described in the Introduction) to test the uniformity of the distribution of W ij 's and thus to test that the distribution of X i 's is Laplace.
Briefly, the proposed procedure is as follows.
1. Find the maximum likelihood estimate of the location parameter (under the assumption that the observations come from Laplace distribution) as follows. The median of the sample is the estimate of µ. Thuŝ if n is even.
2. Make the transformation y i = x i −μ for i = 1, 2, ..., n. Next transform the y i 's to 3. Compute the test statistic as where n = n(n − 1) andf (.) and F 0 (.) are as introduced in preceding Section. 4. Find the critical point C * at a given significance level (α), for sample size n. If DA is greater than C * , the null hypothesis is rejected at level α.
The test statistic is invariant with respect to location and scale transformations. To see this, note that if x i 's are multiplied by a constant c > 0 and are added by a constant d thenμ is multiplied by c and is then added by d. Thus that is, w ij is not changed. Thus the test statistic remains invariant.

Remark 1.
Since the proposed test statistic is invariant to transformations of location and scale and the parameter space for Laplace distribution is transitive, the distribution of the proposed test statistic does not depend on the unknown parameters and the test is exact. Consequently, we can conclude that the actual size of the proposed test is equal to the nominal size. Therefore, we don't need to provide the size of the test for comparison. In our power comparisons, the actual size of the tests are equal to the nominal size i.e., 0.05.
We think that the transformed data approach is useful because 1. The estimation of the scale parameter is not necessary, in fact by transformation the scale parameter is eliminated and the test statistic does not depend on it. For example in testing exponentiality using transformed data the scale parameter (λ) is eliminated and the test statistic does not depend on λ, thus we don't need to estimate it. Here, for Laplace distribution the scale parameter (θ) is eliminated by transformation and the test statistic does not depend on θ, then it is not necessary to estimate scale parameter for computing the proposed test statistic. Note that the statistics based on empirical distribution function (EDF) depend on both parameters and they must be estimated. 2. The data that we obtain by transformation, although not independent, are much more numerous than the original data. For example for n = 10 the number of transformed data is n = n(n − 1) = 90 and we deal with a larger sample.

Simulation study
In this section, the performance of the proposed test is investigated using Monte Carlo simulations. For small to moderate sample sizes 5, 10, 15, 20, 25, 30 and 50, we used Monte Carlo methods with 10,000   Table 1.
Since the tests based on the empirical distribution function are important tests which are commonly used in practice, we compare the power of the proposed test with the power of these tests. These tests are the Cramer-von Mises W 2 , the Watson U 2 , the Anderson-Darling A 2 , the Kolmogorov-Smirnov D, and the Kuiper V . For further information about these tests see [10].
The estimated powers of the present test, based on 10,000 samples of size n equal to 10 and 20, are given in Table  2. The maximum power in each row is bold faced.
We see that the proposed test has the highest power against all considered alternatives distributions (with exception of Cauchy distribution), as compared with the other tests. Generally, we observe that the proposed test performs quite well, as compared with the other tests. Thus, in applications, the use of the proposed test is recommended for achieving a higher gain in power.

Real data examples
Two real examples to show the behavior of the proposed test in real cases is presented in this section.  The following data are 100 breaking strengths of yarn presented by [11]. [12] concluded that the Laplace assumption is not rejected for any of EDF tests, even at a significance level of 0.10. Our procedure can be used to investigate the above conclusion. The value of the proposed test statistic is T = 0.0416 and the critical value is 0.061 at 0.05 level. Since the value of the proposed test statistic is smaller than the critical value, the Laplace assumption is accepted and our procedure confirms the results obtained by EDF methods.

Example 2.
[13] considered the following dataset, consisting of 33 differences in flood levels between stations on a river. They suggested that the Laplace distribution might provide a good fit. Puig and Stephens (2000) concluded that D, W 2 , and U 2 just reject the Laplace distribution for the data at 0.05 level. For this example we findμ = 10.13, T = 0.1414 and the critical value is 0.1072 at 0.05 level. Then the Laplace assumption is rejected as it is by D, W 2 , and U 2 tests.