Berlin, Germany Published online: 15 Feb 2007.

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Communications in Statistics Simulation and Computation, 35: 531 545, 2006 Copyright Taylor & Francis Group, LLC ISSN: 0361-0918 print/1532-4141 online DOI: 10.1080/03610910600716332 Inference Power of One-Sample Location Tests Under Distributions with Equal Lévy Distance HERBERT BÜNING AND SALMAI QARI Institut für Statistik und Ökonometrie, Freie Universität Berlin, Berlin, Germany 1. Introduction In this article, we study the power of one-sample location tests under classical distributions and two supermodels which include the normal distribution as a special case. The distributions of the supermodels are chosen in such a way that they have equal distance to the normal as the logistic, uniform, double exponential, and the Cauchy, respectively. As a measure of distance we use the Lévy metric. The tests considered are two parametric tests, the t-test and a trimmed t-test, and two nonparametric tests, the sign test and the ilcoxon signed-rank tests. It turns out that the power of the tests, first of all, does not depend on the Lévy distance but on the special chosen supermodel. Keywords Lévy metric; Location alternatives; Measure of tailweight; Monte Carlo simulation; Non normality; Nonparametric tests; Parametric tests; Power comparison; Supermodel. Mathematics Subject Classification G2F03; G2G35. Since the famous Princeton study of Andrews et al. (1972), Monte Carlo simulations have gained more and more attention in studying the efficiency and robustness of statistical tests and estimates. The efficiency of such procedures strongly depends on the underlying model of the data mostly assumed to be the normal distribution. Meanwhile, there is an enormous number of articles investigating the robustness of tests and estimates in the case of departures from normality by using Monte Carlo simulations. Examples of the models considered are classical distributions like the uniform, logistic, exponential, double exponential, Cauchy, and often so-called supermodels, including the normal distribution and other classical distributions as Received June 24, 2005; Accepted November 23, 2005 Address correspondence to Herbert Büning, Institut für Statistik and Ökonometrie, Freie Universität Berlin, Boltzmannstr. 20, Berlin D-14195, Germany; E-mail: hbuening@wiwiss.fu-berlin.de oz Salmai Qari, Institut für Statistik und Ökonometrie, Freie Universität Berlin, Boltzmaunstr. 20, Berlin D-14195, Germany; E-mail: salmai.qari@wiwiss.fu-berlin.de 531

532 Büning and Qari special cases. All the models describe different tailweight and extent of skewness. Examples of supermodels are the distributions of Box and Tiao (1962), Johnson and Kotz (1970), the system of Pearson curves, (see Kendall and Stuart, 1969), the RST-distributions (see Ramberg and Schmeiser, 1972, 1974), and the contaminated normal distribution (CN), (see Büning, 1991). The question arises: Does the property of robustness of a test or an estimate depend on the specially chosen supermodel? Or more precisely: If we consider two distributions from different two supermodels but with the same distance to the normal distribution would we then obtain (nearly) the same results concerning the power of the test or the mean square error of the estimate? In this article we try to give an answer to this question for some tests in the one-sample location problem selected for the purpose of illustration. As a measure of distance between two distribution functions F and G we choose the Lévy metric. First, we select four classical symmetric distributions with different tailweight the uniform, logistic, double exponential, and the Cauchy and then calculate the Lévy distances of this four distribution functions to the normal. Second, we determine from each of the supermodels, CN and RST, one member which has the same Lévy distance to the normal as the normal to the uniform, logistic, double exponential, and the Cauchy, respectively. In order to fix such members we have to choose the parameters in the two supermodels in an appropriate way. To our knowledge there is up to now no such a robustness study of tests based on the Lévy distance. e select the parametric t-test and a trimmed version of it as well as two nonparametric tests, the sign test and the ilcoxon signed-rank test described in Sec. 2. In Sec. 3, we introduce the Lévy metric and calculate the Lévy distances between the normal distribution and some other distributions considered in our simulation study. In Sec. 4 a power study of the four tests for distributions with equal Lévy distance is carried out via Monte Carlo simulation which will give an answer of the question above. 2. One-Sample Location Tests 2.1. Model and Hypotheses e consider the one-sample location model: X 1 X n are independent, identically distributed random variables with absolutely continuous distributions function F x = F x which is symmetric about the location parameter. e wish to test H 0 = 0 versus H 1 > 0 ithout loss of generality we can set 0 = 0. 2.2. t-test and Trimmed t-test At first, we consider the well-known parametric t-test which is the uniformly most powerful unbiased test under normality, i.e., X i N 2 i = 1 n.

The corresponding statistic is given by Power of One-Sample Location Tests 533 t = X n S where X = 1 n n i=1 X i and S 2 = 1 n n 1 i=1 X i X 2. Under H 0 = 0 the statistic t has a t-distribution with n 1 degrees of freedom. That means H 0 has to be rejected at level if t t 1 n 1. Under non normal data the statistic t is asymptotically standard normally distributed. Next, we consider a trimmed version of the t-test, the so-called trimmed t-test t g which was introduced by Tukey and McLaughlin (1963). Let be X 1 X n the ordered sample of X 1 X n, the fraction of trimming, and g = n the number of trimmed observations at both ends of the ordered sample. Furthermore, let 1 X g = n 2g n g i=g+1 X w = 1 [ gx n g+1 + gx n g + the X-variables, and X i be the -trimmed mean n g i=g+1 SSD w = g X g+1 X w 2 + g X n g X w 2 + X i ]the -winsorized mean of n g i=g+1 the winsorized sum of squared deviations. Then the trimmed t-statistic t g is defined by t g = X g 0 SSDw /h h 1 with h = n 2g X i X w 2 Under normality the statistic t g has approximately a t-distribution with h 1 degrees of freedom. 2.3. Sign Test and ilcoxon Signed-Rank Test The sign test and the ilcoxon signed-rank test are special cases of linear rank tests. In order to define linear rank statistics let D i = X i 0 and D i be the corresponding ordered absolute differences, i = 1 n. Furthermore, we define indicator variables i by { 1 if D i belongs to a positive difference D j i = 0 if D i belongs to a negative difference D j. Then linear rank statistics are of the form n L + n = g i i with real valued scores g i, i i = 1 n.

534 Büning and Qari The statistic Z = L+ n E L+ n ar L + n has a limiting standard normal distributed with E L + n = 1 2 ar L + n = 1 4 n g i and i=1 n g i 2 see e.g., Büning and Trenkler (1994, p. 92) i=1 Thus, critical values c 1 of L + n are given approximately by c 1 = E L + n + ar L + n z 1 where z 1 is the 1 -quantile of the standard normal distribution. Now, the sign statistic n + That means is defined by the scores g i = 1 i = 1 n. + n n = i The statistic n + is a sum of Bernoulli variables and has a binomial distribution with parameters n and p = 0 5. Thus we have under H 0 : i=1 E + n = n/2 and ar + n = n/4 Critical values of n + can be found in tables of the binomial distribution. The ilcoxon signed-rank statistic n + is defined by the scores g i = i i = 1 n, i.e., + n n = i i with E n + = n n + 1 /4 and ar n + = n n + 1 2n + 1 /24 under the null hypothesis. For sample sizes n 20, critical values of n + can be found in Büning and Trenkler (1994). It should be mentioned that the statistics n + and n + are discrete ones. Thus, in order to have exact level- -tests for power comparisons with the t-test and the trimmed t-test t g a randomized n + and n + test is applied in the simulation study in Sec. 4 for sample sizes n 20; for sample sizes n>20 we use the normal approximation. i=1 3. Lévy Metric Deviations from the ideal model, e.g., from the normal distribution, may be described by the Prohorov-, Kolmogorov- or Lévy distance. Here, we consider the Lévy distance in context with robustness and power studies of the tests from Sec. 2. The Lévy distance d L between two distribution F G is defined by d L F G = inf { G x F x G x + + x IR }

Power of One-Sample Location Tests 535 Figure 1. Lévy distance. Obviously, d L is a metric. The term 2d L F G is the maximum distance between the graphs of F and G, measured along a 45 direction, see Fig. 1 with = d L where F is within the two dotted lines about G (see Büning, 1991; Huber, 1981). Now, let us calculate the distance d L between the normal distribution and four other distributions: the uniform, logistic, double exponential, and the Cauchy. Because the Lévy distance is not location- and scale-invariant, we have to scale all the densities of the distribution functions considered in order to get a meaningful comparison of the Lévy distances to the normal distribution function. e do it in the following way: The location parameter of all densities is fixed to = 0 and the scale parameter is determined in such a way that f 0 is equal to 1/ 2 like the value of the standard normal density for x = 0. In the following we write N 0 for the normal distribution if = 0 H 0 and N if 0 H 1, analogously for the other distributions. Table 1 presents the Lévy distances between the normal distribution (N 0 ) and the logistic distribution (L 0 ), the double exponential (D 0 ), the uniform (U 0 ), and the Cauchy (CA 0 ). As already mentioned, we consider two supermodels for our power study in Sec. 4. First, the symmetric contaminated normal distribution, CN k, the density Table 1 Lévy distances d L L 0 D 0 U 0 CA 0 N 0 0.0164 0.0767 0.0890 0.0932

536 Büning and Qari Table 2 d L N 0 CN 0 k /k 3 5 6 0.105 0.0164 0.3375 0.0767 0.340 0.0890 0.3665 0.0932 of which is given by f x = 1 1 e 1 x 2 1 2 + e 1 x 2 k 2 2 2 k with >0 0 1, and k>1, see Büning (1991, p. 18 ff). If we set f 0 = 1/ 2 and = 0weget = 1 +. That means that there k are various combinations of and k to obtain a value of for which the condition f 0 = 1/ 2 is fulfilled. Now, we select such values of and k so that the Lévy distances between CN 0 k and N 0 are the same as those between N 0 and the logistic L 0 0 0164, double exponential D 0 (0.0767), uniform U 0 (0.0890), and Cauchy CA 0 (0.0932) (see Table 1). The values of and k are given in Table 2. The second supermodel is the RST-distribution, RST( 1 2 3 4 ), which is defined by its p-quantiles, see Ramberg and Schmeiser (1972, 1974): x p = F 1 p = 1 + p 3 1 p 4 0 p 1 2 where 1 is a location parameter, 2 is a scale parameter, and 3 4 are form parameters. Here, we only consider the symmetric case 3 = 4. In the same manner as for CN 0 we determine values of 2 3 = 4 so that RST 0 2 3 = RST 0 2 3 = 4 has the same Lévy distances to N 0 as N 0 to the four classical distributions. Table 3 presents such combinations of 2 and 3 = 4. Altogether, we have four configurations (Config), each of them with one classical distribution (classical) and one member of the CN- and RST-distribution for which the Lévy distances are equal, see Table 4. Table 3 d L N 0 RST 0 2 3 Parameter 2 3 d L 0.3764 0.2880 0.0164 0.7400 0.8170 0.0767 0.7979 1.0000 0.0890 0.8178 1.0940 0.0932

Power of One-Sample Location Tests 537 Table 4 Four configurations CN 0 RST 0 Config. Classical k 2 3 d L 1 L 0 0.1050 3 0.3764 0.2880 0.0164 2 D 0 0.3375 5 0.7400 0.8170 0.0767 3 U 0 0.3400 6 0.7979 1.0000 0.0890 4 CA 0 0.3665 6 0.8178 1.0940 0.0932 The simulation study in Sec. 4 is based on these four configurations. Figure 2 presents the densities of the distributions of the four configurations. From the figures we see that the tailweights of the distributions in the configurations are very different except for configuration 1 where the Lévy distance between the four distribution and the normal is very small in contrast to the other configurations. In the Configurations 2, 3, and 4, the RST-distributions have Figure 2. Densities of the four configurations.

538 Büning and Qari very short tails like the uniform, where the logistic, double exponential, Cauchy, and contaminated normal distributions have longer tails than the normal but with different extent. That means that distributions with the same Lévy distance to the normal may have very different tailweight. 4. Simulation Study e investigate via Monte Carlo simulation (60,000 runs) the power of the tests from Sec 2 under the distribution functions in the configurations 1, 2, 3, and 4. Critical values of the t- and t g -statistics under non normality are found by simulation (100,000 runs). The fraction of trimming is 10%. e consider sample sizes of n = 20, 50, and 100, but here we only present results for the case n = 20. For the sign- and ilcoxon signed-rank test we apply randomization for n = 20 in order to achieve the nominal level = 5%. The location parameter is chosen as the expectation E(X) for all the distributions in the four configurations except, of course, for the Cauchy where is the median. Thus, the power is calculated for the sample X 1 X 2 X n with X i = X i + i = 1 n, where X 1 X 2 X n is generated from one of the normalized distributions in Sec. 3. e choose = 0 3 0 5 0 7. Configuration 1 Table 5 displays the power values of the four tests for selected values of under the distributions from configuration 1 and the normal distribution for comparison. e consider the case of sample size n = 20 and = 5%. Figure 3 shows the power curves separately for all four tests under the distributions from Configuration 1. Table 5 Configuration 1: power values for n = 20, = 5% N L CN RST = 0 3 t 0.3646 0.3186 0.3146 0.4054 t g 0.3441 0.3273 0.3320 0.3647 n + 0.2746 0.2775 0.2759 0.2766 n + 0.3508 0.3290 0.3343 0.3816 = 0 5 t 0.6942 0.6154 0.5840 0.7642 t g 0.6656 0.6277 0.6356 0.7015 n + 0.5373 0.5318 0.5302 0.5415 n + 0.6769 0.6305 0.6340 0.7276 = 0 7 t 0.9135 0.8442 0.7973 0.9557 t g 0.8928 0.8580 0.8665 0.9223 n + 0.7782 0.7605 0.7685 0.7881 n + 0.9021 0.8592 0.8602 0.9369

Power of One-Sample Location Tests 539 Figure 3. Power curves of Configuration 1: n = 20, = 5%. From Table 5 and Fig. 3 we get the following results: The power values for each of the four tests strongly depends on the underlying super model. The reaction of the four tests is very different under the three distributions with equal Lévy distances to the normal. The power of the t-test is mostly influenced by the distributions in contrast to the sign test where the power is nearly the same. Configuration 2 Table 6 displays the power values of the four tests under the distributions from Configuration 2 and again under the normal distribution for comparison. Figure 4 shows the power curves of the four tests under the distributions from Configuration 2. As in Configuration 1, we can state that the power values for each of the four tests strongly depends on the underlying super model except perhaps for the sign test where the power is again nearly the same for all distributions. The reaction of the four tests is more obvious in Configuration 2 than in Configuration 1, especially for the t-test. The tests t g and n + show nearly the same reaction. All the tests have

540 Büning and Qari Table 6 Configuration 2: power values for n = 20, = 5% N D CN RST = 0 3 t 0.3646 0.1964 0.1790 0.5167 t g 0.3441 0.2213 0.2761 0.4189 n + 0.2746 0.2437 0.2729 0.2796 n + 0.3508 0.2294 0.2689 0.4764 = 0 5 t 0.6942 0.3649 0.3059 0.8965 t g 0.6656 0.4162 0.4996 0.7916 n + 0.5373 0.4345 0.5155 0.5605 n + 0.6769 0.4273 0.4859 0.8388 = 0 7 t 0.9135 0.5541 0.4504 0.9948 t g 0.8928 0.6280 0.6827 0.9741 n + 0.7782 0.6225 0.7359 0.8307 n + 0.9021 0.6320 0.6770 0.9816 Figure 4. Power curves of Configuration 2: n = 20 = 5%.

Power of One-Sample Location Tests 541 smallest power for D, except for the t-test which has greatest loss in power under CN. But now, the sign test has highest power for D among the four tests. Configuration 3 Table 7 displays the power values of the four tests under the distributions from Configuration 3 and again under the normal distribution for comparison. Figure 5 shows the power curves of the four tests under the distributions from Configuration 3. From Table 7 and Figure 5 we get the following results: Under the RSTdistribution and the uniform the differences of power are very small. That is not surprising because we can see from Configuration 3 in Fig. 2 that the RST- and the uniform-distribution have nearly the same shape. Again, we see that the power of the tests is heavily influenced by the underlying supermodel, especially for the t-test. Configuration 4 Table 8 displays the power values of the four tests under the distributions from Configuration 4 and again under the normal distribution for comparison. Figure 6 shows the power curves of the tests under the distributions from Configuration 4. From Table 8 and the Fig. 6 we can state as in the Configurations 1, 2, and 3 that the power strongly depends on the assumed supermodel but now all the tests have smallest power under the Cauchy, a distribution with very long tails. Again, the t-test reveals highest and the n + -test smallest variations under the four distributions. Now, let us have a special look at the t-test. From Tables 5 8, we see that the t-test reacts differently on changes of the supermodels, most in Configuration 4 and Table 7 Configuration 3: power values for n = 20 = 5% N U CN RST = 0 3 t 0.3646 0.5395 0.1605 0.5424 t g 0.3441 0.4302 0.2655 0.4347 n + 0.2746 0.2802 0.2732 0.2821 n + 0.3508 0.4982 0.2580 0.5002 = 0 5 t 0.6942 0.9140 0.2644 0.9145 t g 0.6656 0.8097 0.4705 0.8115 n + 0.5373 0.5638 0.5122 0.5668 n + 0.6769 0.8552 0.4611 0.8557 = 0 7 t 0.9135 0.9971 0.3864 0.9974 t g 0.8928 0.9787 0.6420 0.9794 n + 0.7782 0.8379 0.7293 0.8401 n + 0.9021 0.9849 0.6440 0.9853

542 Büning and Qari Figure 5. Power curves of Configuration 3: n = 20 = 5%. Table 8 Configuration 4: power values for n = 20 = 5% N CA CN RST = 0 3 t 0.3646 0.1352 0.1519 0.5498 t g 0.3441 0.2229 0.2526 0.4379 n + 0.2746 0.2632 0.2744 0.2792 n + 0.3508 0.2269 0.2519 0.5064 = 0 5 t 0.6942 0.2117 0.2550 0.9177 t g 0.6656 0.4056 0.4471 0.8151 n + 0.5373 0.4829 0.5129 0.5675 n + 0.6769 0.4071 0.4486 0.8594 = 0 7 t 0.9135 0.2954 0.3746 0.9973 t g 0.8928 0.5800 0.6158 0.9811 n + 0.7782 0.6779 0.7242 0.8434 n + 0.9021 0.5794 0.6246 0.9865

Power of One-Sample Location Tests 543 Figure 6. Power curves of Configuration 4: n = 20 = 5%. least of all in Configuration 1, which is illustrated by Fig. 7. The reason for that may be the increasing Lévy distances between the normal and the distributions from Configuration 1 up to Configuration 4. At the end of this section it should be noted that similar results are true for n = 50 and n = 100, although the influence of the supermodel on the power of the tests becomes smaller with increasing sample sizes, a fact which is not surprising. 5. Concluding Remarks As an answer to our question in the Introduction we may say that the power of tests strongly depends on the assumed supermodel. e have seen that distributions with the same distance to the normal distribution can produce very different power results. A better criterion for robustness as the Lévy distance may be the tailweight or skewness of a distribution in comparison to the normal. The distributions in our simulation study have different tailweight as already seen from Fig. 2 in Sec. 3. But how can we measure tailweight or skewness of a distribution? A comprehensive study of such so called selector statistics is carried out by Hüsler (1988). Because in this article only symmetric distributions are considered, we restrict our attention to

544 Büning and Qari Figure 7. Power of the t-test under the four configurations. a measure of tailweight, e.g., the measure M T which is defined as follows: M T = x 0 975 x 0 025 x 0 875 x 0 125 where x p is the p-quantile of the distribution, see Büning (1991, p. 262). Obviously, the measure M T is location and scale invariant. For the normal distribution we have M T = 1 704. alues of M T for all the distributions in our four configurations are given in Table 9. Table 9 M T for some distributions Config1 L CN RST M T 1.883 2.012 1.567 Config2 D CN RST M T 2.161 3.424 1.304 Config3 U CN RST M T 1.267 1.841 1.267 Config4 CA CN RST M T 5.263 3.446 1.254

Power of One-Sample Location Tests 545 From Table 9 we can see that the values of M T in each of the four configurations are very different although the distributions have the same Lévy distance to the normal. Otherwise, distributions may have the same tailweight but very different distances to the normal, see e.g., the RST-distributions in Configurations 2, 3, and 4. For these three RST-distributions the power of the four tests is nearly the same, see Tables 6 8. On the other hand, the distribution L in Configuration 1 and CN in Configuration 3 have nearly the same tailweight and different distances to the normal, but the power under these two distributions is more or less different for each of the tests. The question arises: Is the tailweight of distributions a better concept for power comparisons than the Lévy distance? A convincing answer might be of great interest. References Andrews, D. F., Bickel, P. J., Hampel, F. R., Huber, P. J., Rogers,. H., Tukey J.. (1972). Robust Estimation of Location: Survey and Advances. Princeton, N.J: University Press. Box, G. E. P., Tiao, G. C. (1962). A further look at robustness via Bayes s theorem. Biometrika 49:419 432. Büning, H. (1991). Robuste und Adaptive Tests. Berlin: alter De Gruyter. Büning, H., Trenkler, G. (1994). Nichtparametrische Statistische Methoden. Berlin: alter De Gruyter. Huber, P. J. (1981). Robust Statistics. New York: iley. Hüsler, J. (1988). On the asymptotic behaviour of selector statistics. Commun in Statist.-Theor. Meth. 17:3569 3590. Johnson, N. L., Kotz, S. (1970). Distributions in Statistics-Continuous Univariate Distributions-I. Boston: Houghton Mifflin. Kendall, G. M., Stuart, A. (1969). The Advanced Theory of Statistics. ol. 1. London: Charles Griffin. Ramberg, J. S., Schmeiser, B.. (1972). An approximate method for generating symmetric random variables. Commun. ACM 15:987 990. Ramberg, J. S., Schmeiser, B.. (1974). An approximate method for generating asymmetric random variables. Commun. ACM 17:78 82. Tukey, J. M., McLaughlin, D. H. (1963). Less vulnerable confidence and significance procedures for location based on a single sample: trimming/winsorization I. Sankhya A 25:331 352.