A GOODNESS-OF-FIT TEST FOR THE INVERSE GAUSSIAN DISTRIBUTION USING ITS INDEPENDENCE CHARACTERIZATION

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1 Sankhyā : The Indian Journal of Statistics 2001, Volume 63, Series B, pt. 3, pp A GOODNESS-OF-FIT TEST FOR THE INVERSE GAUSSIAN DISTRIBUTION USING ITS INDEPENDENCE CHARACTERIZATION By GOVIND S. MUDHOLKAR University of Rochester RAJESHWARI NATARAJAN Southern Methodist University and YOGENDRA P. CHAUBEY Concordia University SUMMARY. The class of inverse Gaussian (IG) distributions share substantial analytical elegance with the class of normal distributions, and they are widely used as models in many diverse areas of applied research. It is well-known that independence of the sample mean and sample variance characterizes a normal population. This was used by Lin and Mudholkar (1980), see also Mudholkar et al. (1996), for developing tests of normality. An analogous property, namely independence of the maximum likelihood estimates of the two parameters, characterizes the inverse Gaussian distribution. In this paper we use this characterization to develop the analogous goodness-of-fit test for the inverse Gaussian model. Monte Carlo methods are used in the construction and evaluation of the test. 1. Introduction The family of inverse Gaussian distributions is described by the probability density function { } λ 1/2 { f(x µ, λ) = 2πx 3 exp λ } 2µ 2 (x µ)2, x > 0, µ > 0, λ > 0, (1.1) x Paper received December 1999; revised November AMS (1991) subject classifications. Primary: 62F03, secondary 62E17, 62E20, 62E25. Key words and phrases. Goodness-of-fit, independence characterization, inverse Gaussian distribution.

2 goodness of fit test for the inverse gaussian distribution 363 where the parameter µ is the mean and the parameter λ is the scale. The corresponding cumulative distribution function (c.d.f.) can be written as ( ) λ x { } 2λ ( ) F (x µ, λ) = Φ x µ 1 + exp Φ µ λ x x µ + 1, (1.2) where Φ(.) denotes the c.d.f. of the standard normal variable. The coefficients of skewness and kurtosis of the family are β1 = 3 µ λ, β 2 = µ λ. (1.3) In the Pearson (β 1, β 2 ) plane, the inverse Gaussian points fall on the straight line β 2 = 3 + 5β 1 /3 which lies between the Type III (gamma) and the Type V lines. In terms of (β 1, β 2 ), IG(µ, λ) family is very close to the lognormal family. The inverse Gaussian distribution is now widely used in modeling positive and positively skewed data in such diverse areas of applied research as cardiology, hydrology, demography, linguistics, employment service, labor disputes and finance; see Chhikara and Folks (1989) and Seshadri (1999). In a recent paper, Huberman et al. (1998) use the data from America On Line, to provide an interesting application in the area of internet to the distribution of the number of links an internet user follows before the page value reaches a threshold. In spite of extensive use in applications, there is no adequate composite goodness-of-fit test which can be used to assess appropriateness of the inverse Gaussian model. The current status of the problem of testing adequacy of an inverse Gaussian model is approximately the same as that of the testing normality in the early sixties, when Kolmogorov is reported to have considered it as the problem of premier importance in statistics. Prior to the first omnibus test of composite hypothesis of normality by Shapiro and Wilk (1965), the e.d.f. tests such as Kolmogorov-Smirnov, Anderson-Darling and Cramér-von Mises, were adapted for the purpose using the plug-in method. Essentially, the same is true today for the composite inverse Gaussian hypothesis; see Edgeman (1990), Edgeman, et al. (1988), O Reilly and Rueda (1992), and Pavur et al. (1992). It may be noted that in addition to the plug-in modification of the e.d.f. tests, many approaches have been used to construct goodness-of-fit tests for the composite hypothesis of normality. These include the regression method of Shapiro and Wilk, the use of characterization properties as in entropy test of Vasicek (1976) and Z-tests of Lin and Mudholkar (1980), and Mudholkar et

3 364 g.s. mudholkar, r. natarajan and y. p. chaubey al. (1996). The purpose of this paper is to use the following characterization due to Khatri (1962), see also Seshadri (1983), as in Lin and Mudholkar (1980), for developing a goodness-of-fit test for the inverse Gaussian model: A random sample (X 1, X 2,..., X n ) is from an inverse Gaussian population if and only if, X = X i /n and V = { n i=1 (1/X i 1/ X)}/n are independently distributed, assuming that the expected values of X, X 2, 1/X and 1/ X i exist and are different from zero. The proposed test statistic is shown to be asymptotically normal and hence it provides a portable tool for meta-analytic studies as well. We present the construction of the test statistic in Section 2 and Section 3 is devoted to the analysis of the power properties of the test. The final section, Section 4, contains conclusions and miscellaneous remarks. 2. Construction of the Test For an IG(µ, λ) sample, (X 1, X 2,..., X n ), it is known that X has an IG(µ, nλ) distribution and nλv is independently distributed as a χ 2 n 1 variate, see Chhikara and Folks (1989). We employ the characterization mentioned in Section 1, and other properties related to X and V to construct a goodness-of-fit test for the composite inverse Gaussian hypothesis. If the sample size n is large then the sample may be divided into, say k, subsamples in order to obtain replications ( X i, V i ), i = 1, 2,.., k, which may then be used nonparametrically to test the independence of ( X, V ), and consequently the inverse Gaussian character of the population. However, this introduces an extraneous random element and non-uniqueness in the solution. Also, the rank tests, e.g. the one by Hoeffding (1948) or its asymptotic equivalent due to Blum et al. (1961), which are consistent against all dependence alternatives have too low power to be practically useful. Therefore, we use an approach similar to the construction of the test of normality in Lin and Mudholkar (1980). 2.1 The test statistic. Following Lin and Mudholkar (1980), we obtain n replications X i = j i X j/(n 1) and V i = 1 n 1 ( j i X 1 1 j X i ), i = 1, 2,..., n of ( X, V ) by deleting one observation at a time. Even though these n pairs are not independent, in view of the following lemma, the covariance between X i and V i can be used as a starting point to construct a measure of dependence between ( X, V ). Lemma If X 1, X 2,..., X n is a random sample from an IG(µ, λ) population and G n = 1 n ni=1 ( X i X)V i, then E(G n ) = 0.

4 goodness of fit test for the inverse gaussian distribution 365 Proof. We have G n = 1 n ( n X i X)V i i=1 = 1 n n 2 ( X i X i )V i. i=1 But E(X i ) = E( X i ) = E( X) = µ, and E(V i ) = 1/λ. Furthermore, both X i and X i are mutually independent and independent of V i. Hence, and the lemma follows. (n 1) (n 2) E(G n) = 1 n 2 n (µ µ) 1 λ = 0, It is well-known that the sample product moment correlation coefficient provides an optimal test of independence in a bivariate normal population and, for large n, ( X i, V i ) follow an approximately bivariate normal distribution, as can be seen in the proof of Theorem Hence we consider ni=1 ( r = X i X)V i ni=1 ( X i X) 2 n i=1 (V i V. ) 2 = ni=1 (X i X)V i ni=1 (X i X) 2 n i=1 (V i V. ) 2 (2.1.1) as the statistic for testing independence between X and V. Clearly, asymptotically, E(r) = 0 under the null hypothesis. The following theorem gives the asymptotic null distribution of r. Theorem Let X 1, X 2,..., X n be a random sample from a population with the first four positive moments and the first two negative moments finite. Then, as n, r is asymptotically normal with mean µ, where { } µ = µ 2/(µ 1) 2 µ 1µ 1 / σ22 σ 33, (2.1.2) σ 22 = nv ar( X), σ 33 = nv ar(v ) and µ j denotes the jth raw moment of X. For the inverse Gaussian population, the asymptotic variance of r reduces to 3/n. That is, for the IG population, as n, n r d N(0, 3). (2.1.3)

5 366 g.s. mudholkar, r. natarajan and y. p. chaubey Proof. It is easy to see that we can write G n as, G n = X ( 1 n n i=1 where X = X i /n, Ȳ = (1/X i )/n. Also, we have X n n i=1 ) 1 1 X i (n 1) 2 Ȳ n(n 2) (n 1) 2, (2.1.4) 1 = n 1 X i n r=0 m r, (2.1.5) nr X r where, m r = n 1 n i=1 Xi r denotes the rth raw sample moment. Using the fact that m r = O p (1/ n), r = 1, 2,... and the representation in (2.1.5), we can write (2.1.4) as G n = 1 ( m ) 2 n 2 X 2 XȲ + O p (n 3 ). (2.1.6) Furthermore, from jackknife theory, see Thorburn (1977), we get ( 1 ) ( X i n X) 2 σ 22 p 0, (2.1.7a) n(n 1) and ( 1 n ) n (V i V ) 2 σ 33 (n 1) i=1 2 p 0. (2.1.7b) Therefore, an appeal to Slutsky s lemma establishes the result in (2.1.2). Furthermore, using the well-known asymptotic distributions of the sample moments, e.g., Cramér (1946), it is easy to show that for the IG(µ, λ) population, as n, the vector ( X, Ȳ, m 2 ) is asymptotically normally distributed with mean (µ, 1/µ + 1/λ,, µ 2 ), and covariance matrix (1/n) = (σ ij )/n, where σ 11 = V ar(x) = µ 3 /λ, σ 12 = Cov(X, Y ) = µ/λ, σ 13 = Cov(X, X 2 ) = 3µ 5 /λ 2 + 2µ 4 /λ, σ 22 = V ar(y ) = 1/(µλ) + 2/λ 2, σ 23 = Cov(Y, X 2 ) = 2µ 2 /λ µ 3 /λ 2, and σ 33 = V ar(x 2 ) = 4µ 5 /λ + 14µ 6 /λ µ 7 /λ 3. Hence, by use of the multivariate version of Mann-Wald theorem to obtain the asymptotic distribution of (m 2 / X 2, XȲ ) and the distribution of a linear combination of these components, we obtain, ( ) n 5/2 d G n N 0, 6 µ3 λ 3, (2.1.8)

6 goodness of fit test for the inverse gaussian distribution 367 as n. Also, as n, σ22 p µ 3 /λ, and σ33 p 2/λ 2. The proof can be completed by using Slutsky s lemma. Now, it is well-known that Fisher s Z transformation is a remarkably effective normalizing transformation of the correlation coefficient from a bivariate population, see Gayen (1951) and Chaubey and Mudholkar (1984). Hence, as in the case of Z-test of normality, we consider Z = 1 ( ) 1 + r 2 log, (2.1.9) 1 r where r is given by (2.1.1), as the statistic for testing the composite inverse Gaussian hypothesis. Then using Mann-Wald theorem it is easy to prove the following: Corollary If X 1, X 2,..., X n is a random sample from IG(µ, λ) then nz d N(0, 3). (2.1.10) Remark Interestingly, the asymptotic null distributions of the Z-statistic for testing normality (see Lin and Mudholkar (1980)) and the Z-statistic proposed in this paper for testing IG assumptions coincide, i.e., both are asymptotically normal with mean zero and variance 3/n. 2.2 Moderate sample size refinements. An empirical study indicated that the convergence of the distribution of the test statistic Z to asymptotic normality is slow. Hence fine tuning of the null distribution for testing the inverse Gaussian hypothesis with samples of moderate size is necessary. A Monte Carlo experiment was conducted towards this end. In this experiment 1000 samples, each of size n = 12, 15, 18, 20, 22, 25(5)50, from an inverse Gaussian population IG(1, θ), with parameter values θ = 0.1, 0.5, 1, 2, 3, 5, 8, 10, 15, 20, 25 and 35, were simulated using the algorithm proposed by Michael et al. (1976). The process was repeated five times for each of the configurations. For each of the sample, Z as defined by (2.1.9) was computed. The five sets of 1000 Z s so obtained for each (n, θ) configuration were then used to get five independent estimates of the mean and variance of the null distribution of Z. An examination of these estimates w.r.t. n and for various values of θ showed that the mean of the Z-statistic was very close to zero. However, the variance of Z-statistic was observed to depend on the shape parameter and the sample size. A regression analysis

7 368 g.s. mudholkar, r. natarajan and y. p. chaubey yielded the following expression for variance: s 2 n,ˆθ = 3 ( ) exp( ˆθ), (2.2.1) n log n For moderate size samples, Z may be considered normally distributed with mean zero and variance as given by (2.2.1). Thus, we have: The test. Given a random sample X 1, X 2,..., X n of size n, compute the Z-statistic as given in (2.1.9). Then, the lower tail p-value for the composite IG hypothesis using empirical Edgeworth correction for n 20 is given by: p = P (Z/s n,ˆθ < z/s n,ˆθ = z 1 ) = Φ(z 1 ), (2.2.2) where ˆθ = (n 1)/(n XV ). The p-value for the upper tail is similarly obtained. For conducting the two-tailed test, following George and Mudholkar (1990), we use 2-sided p-value = 2 min(p, 1 p). (2.2.3) The Type I error control of the test conducted as above is illustrated in Table 1 for 5% level of significance. 3. Power Properties Non-null expectations. Obviously power function of the Z-test depends upon the non-null distribution of the test statistic Z. It is easy to see that, as n, r converges in probability to its population mean, say ρ. Although higher order parameters such as non-null variance or skewness may play a role, the following expectations under various distributional assumptions can be used to roughly assess the large sample behavior of the Z-test. The notations for the density functions given in this section follow those in Johnson et al. (1994). LN(ψ, σ) : Lognormal. For the lognormal family with p.d.f. f X (x) = [x { 1 2πσ] 1 (log x ψ) 2 } exp 2 σ 2, x > 0, (3.1) ρ = 0. (3.2) Thus the parameter ρ under lognormal alternatives agrees with the null value, i.e., ρ = 0. G(α, β) : Gamma (Type III). For the Gamma family with p.d.f. f X (x α, β) = xα 1 exp[ x/β] β α, α > 0, β > 0, x 0, (3.3) Γ(α)

8 goodness of fit test for the inverse gaussian distribution 369 ρ = α 2 2α 2 + α 2 (3.4) provided α > 2. The parameter ρ is negative for the given range of shape parameter. W (α, c) : Weibull Family. For the Weibull family with p.d.f. f X (x α, c) = c α (x/α)c 1 exp{( x/α) c }, x > 0, (3.5) ρ = δ 1 δ2 1, (3.6) where δ 1 = Num(δ 1) Den(δ 1 ) and δ 2 = Num(δ 2) Den(δ 2 ) + 3, d = 1/c, Num(δ 1 ) = Γ(1 + 2d) Γ(1 + d) 3 Γ(1 d), [ Den(δ 1 ) = [Γ(1 + d)γ(1 d) 1] Γ(1 + d) Γ(1 + 2d) Γ(1 + d) 2] 1/2, Num(δ 2 ) = Γ(1 2d){Γ(1 + d)} 2 + Γ(1 + 2d)/{Γ(1 + d)} 2 3{Γ(1 + 2d)Γ(1 d)} 2 + 2Γ(1 + d)γ(1 d) 1, Den(δ 2 ) = {Γ(1 + d)γ(1 d) 1} 2. It is seen that, in general ρ < 0. P (a, k) : Pareto Family. For the Pareto family with p.d.f. f X (x, a, k) = a k a x (a+1), k > 0, a > 0, x k, (3.7) ρ = (2a 1) a + 2, provided a > 2. (3.8) 8a 3 4a 2 7a + 2 This is positive for the given range of a. B(p, q) : Beta Family. For the beta family with p.d.f. f X (x) = 1 B(p, q) xp 1 (1 x) q 1, 0 x 1, (3.9) ρ = δ 1 δ2 1, (3.10) where δ 1 and δ 2 are calculated using the raw moments formulae: µ r = B(p + r, q) B(p, q) = Γ(p + r) Γ(p + q) Γ(p) Γ(p + q + r), (3.11)

9 370 g.s. mudholkar, r. natarajan and y. p. chaubey δ 2 = It is seen that ρ is negative. (2p + q) δ 1 =, (3.12) pq(p + q + 1) ( µ 2 /µ2 + µ 2 µ2 3µ 2 µ 2 ) 1 + 2µµ 1 (µµ (3.13) 1)2 3.1 Power function: A Monte Carlo Experiment. To study the power of the test with moderate size samples, 50,000 samples each of size n = 20, 30, 40 were simulated from the Uniform U(0, 1) distribution, the five distributions considered above as well the reciprocals of the five distributions. The empirical power function of the two-tailed Z-test, as well as the two one-tailed tests, is presented in Tables 1 and 2 for α = 5%. Obviously, in general, the two-tailed test is in order. However, if the nature of possible alternative is known a priori then the appropriate one-tailed test may be used, as discussed in Section 3.8 of Seshadri (1999). 3.2 Existing solutions. The modification of the Anderson-Darling A 2 test is the recommended solution among the e.d.f. tests and its merit is empirically confirmed in Pavur et al. (1992). We compared our two-tailed Z-test with the modified A 2 test for several IG populations as well as other distributions in another Monte-Carlo experiment. The results are tabulated for samples of size n = 20, 40 in Table 1 for α = The Type I error rate of the modified Anderson-Darling test statistic is very conservative for large values of the shape parameter, i.e., θ > 2. However, it has higher power in general for distributions considered in Table Examples. Folks and Chhikara (1978) discuss and illustrate the appropriateness of IG distribution in applications using four data sets. They present Q-Q plots for these data sets, and test goodness-of-fit of the IG model using Kolmogorov-Smirnov statistic. The data sets are D1: shelf life of a food product in days, D2: toughness of MIG welds; D3: precipitation from Jug Bridge, Maryland in inches; D4: runoff amounts at Jug Bridge, Maryland. We have tabulated some summary statistics such as the sample coefficients of skewness and kurtosis as well as the shape parameter for the four data sets in Table 3. We also computed the modified Anderson-Darling statistic as given in Pavur et al. (1992) which agreed with the decisions based on the Kolmogorov-Smirnov statistic given in Folks and Chhikara (1978). The Q-Q plots suggest less than satisfactory IG fits for the first three datasets; but IG fit for the fourth dataset appears to be almost ideal. The results of Kolmogorov-Smirnov tests confirm the Q-Q plot conclusions for

10 goodness of fit test for the inverse gaussian distribution 371 Table 1. The Empirical power (in percentages) comparison of the Z-statistic with the modified A 2 statistic for α = 5% Left-tailed Right-tailed Two-tailed test Distribution One-sided Z-test One-sided Z-test Z A 2 Z A 2 n=20 n=40 n=20 n=40 n=20 n=40 IG(1,1) IG(1,2) IG(1,8) Uniform Exp: G(1,1) G(2,1) G(4,1) W(1,2) W(1,4) LN(0,0.5) LN(0,1) P(1,1) P(2,1) B(2, 0.5) B(2, 2) B(2, 5) indicates power < 5%, SE < 0.22% Based on the Monte Carlo experiment with 50,000 replications for each entry Distributional labels as in Section 3. Table 2. The Empirical power (in percentages) comparison of the Z-statistic with the modified A 2 statistic for α = 5% for the distributions of the reciprocal variates corresponding to those considered in Table 1 Left-tailed Right-tailed Two-tailed test Distribution One-sided Z-test One-sided Z-test Z A 2 Z A 2 n=20 n=40 n=20 n=40 n=20 n=40 RIG(1,1) RIG(1,2) RIG(1,8) RG(1,1) RG(2,1) RG(4,1) RW(1,2) RW(1,4) RLN(0,0.5) RLN(0,1) RP(1,1) RP(2,1) RB(2, 0.5) RB(2, 2) RB(2, 5) indicates power < 5%, SE < 0.22% Based on the Monte Carlo experiment with 50,000 replications for each entry Distributional labels are reciprocals of distributions considered in Section 3.

11 372 g.s. mudholkar, r. natarajan and y. p. chaubey D3 and D4; but do not show significance in case of D1 and D2. The significance level 96% of the Z-test applied to D4, is strongly in agreement with the Q-Q plot. The p-values of Z-test corresponding to D1, D2 and D3 are 3.8%, 28%, and 13% respectively. Thus, at 5% level of significance, the results of the Kolmogorov-Smirnov and the Z-tests disagree in case of D1 and D3. Table 3. Comparison of decisions based on α = 0.05 using goodness-of-fit tests for the four data sets Data Set b1 b 2 KS A 2 p-values of Z ˆθ D ; NR 0.91; NR 0.038; R D ; NR 0.38; NR 0.28; NR D ; R 0.792; NR 0.13 ; NR 3.75 D ; NR 0.19; NR 0.96; NR 1.79 Kolmogorov-Smirnov statistic for IG fit based on Folks and Chhikara (1978). Anderson-Darling statistic based on Pavur et al. (1992). Decisions of rejection for KS and A 2 were based on comparisons with Monte-Carlo Critical Percentiles given in the above papers. NR: Non-rejection, R: Reject Conclusion and Miscellaneous Remarks In this paper we have presented a goodness-of-fit test based on an inverse Gaussian characterization, for testing the composite IG hypothesis. As pointed out by a referee, it may be mentioned that, since the Z test is asymptotically valid as a test of independence, the proposed goodness-of-fit test may be expected to be fairly liberal, especially, for smaller sample sizes in some cases. The empirical results in Table 1 reflect this; compare the values in Table 1 for n = 20 and n = 40 for IG distributions. However, we may say that the Type I error control of the test is adequate, and its power is reasonable. The Type I error control may further be improved by correction for skewness and kurtosis. However, it involves expressions too intricate to be practically worthwhile. We carried out another comparison of Z-test similar to that in Table 1 for α = The conclusions from that study were very similar to those from Tables 1 and 2, hence we did not include it in this paper. Some other observations follow: 1. The Z-statistic is asymptotically normal, though in finite samples its null distribution depends on the shape parameter θ. The fact that the inverse Gaussian distributions do not constitute a location-scale family seems to manifest through this dependence.

12 goodness of fit test for the inverse gaussian distribution The Z-test of normality, as mentioned earlier is geared towards detecting skew alternatives. Would the Z-test proposed in this manuscript detect some IG analog of skewness? This question is under investigation. 3. There is a certain kind of symmetry implicit in the positive and negative moments of the inverse Gaussian distribution. It would be interesting to study whether it might yield an analog of IG-symmetry, and consequently of IG-skewness. 4. Due to the symmetry mentioned above regarding positive and negative moments, we also considered distributions of the reciprocal variates corresponding to those considered in Section 3 for our empirical power comparisons. The results are reproduced in Table 2. Interestingly, the power of the Z test is higher than that of A 2 for most reciprocal distributions. Acknowledgements. The authors wish to thank the referees for their comments that greatly improved the presentation of the paper. References Blum, J.R., Kiefer, J. and Rosenblatt, M. (1961). Distribution free tests of independence based on the sample distribution function, Annals of Mathematical Statistics, 32, Chaubey, Y.P. and Mudholkar, G.S. (1984). On the almost symmetry of Fisher s Z, Metron, 42, Chhikara, R.S. and Folks, J.L. (1989). The Inverse Gaussian distribution. Marcel Dekker, New York. Cramér, H. (1946). Mathematical Methods of Statistics. Princeton University Press, Princeton, NJ. D Agostino, R.B. and Stephens, M.A. (1986). Goodness-of-Fit Techniques. Marcel Dekker, New York. Edgeman, R.L. (1990). Assessing the inverse Gaussian distribution assumption, IEEE Transactions on Reliability, 39, Edgeman, R.L., Scott, R.C. and Pavur, R.J. (1988). A modified Kolmogorov- Smirnov test for the inverse Gaussian density with unknown parameters. Communications in Statistics B Simulation and Computation, 17, Elderton, W.P. and Johnson, N.L. (1969). Systems of Frequency Curves. Cambridge University Press, Cambridge. Folks, J.L. and Chhikara, R.S. (1978). The Inverse Gaussian distribution and its Statistical Application - A Review, Journal of the Royal Statistical Society Series B, 40, Gayen, A.K. (1951). The frequency distribution of the product-moment correlation coefficient in random samples drawn from non-normal universes, Biometrika, 38, Geary, R.C. (1947). Testing for normality, Biometrika, 34, George, E.O. and Mudholkar, G.S. (1990). P -Values for two-sided tests, Biocmetrical Journal, 32, Hoeffding, W. (1948). A nonparametric test of independence, Annals of Mathematical Statistics, 19,

13 374 g.s. mudholkar, r. natarajan and y. p. chaubey Huberman, B.A., Pirolli, P.L.T., Pitkow, J.E. and Lukose R.M. (1998). Strong regularities in world wide web surfing, Science, 280, Johnson, N.L. and Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions. John Wiley, New York. Khatri, C.G. (1962). A characterization of the inverse Gaussian distribution, Annals of Mathematical Statistics, 33, Lin, C.C. and Mudholkar, G.S. (1980). A simple test for normality against asymmetric alternatives, Biometrika, 67, Michael, J.R., Schucany, W.R., and Haas, R.W. (1976). Generating random variables using transformation with multiple roots, American Statistician, 30, Mudholkar, G.S., Marchetti, C.E. and Lin, C.T. (1996). Independence characterizations and testing normality against restricted skewness-kurtosis alternatives. Technical Report, University of Rochester, NY; (To appear in JSPI (2002)). O Reilly, F.J. and Rueda, R. (1992). Goodness of fit for the inverse Gaussian distribution, Canadian Journal of Statistics, 20, Pavur, R.J., Edgeman, R.L. and Scott, R.C. (1992). Quadratic statistics for the goodness-of-fit test of the inverse Gaussian distribution, IEEE Transactions in Reliability, 41, Seshadri, V. (1983). The inverse Gaussian distribution: some properties and characterizations, Canadian Journal of Statistics, 11, Seshadri, V. (1998). The Inverse Gaussian Distribution: Statistical Theory and Applications. Springer Verlag, New York. Shapiro, S.S. and Wilk, M.B. (1965). An analysis of variance test for normality (complete samples), Biometrika, 52, Thorburn, D. (1977). On the asymptotic normality of the jackknife, Scandinavian Jounal of Statistics, 4, Vasicek, O. (1976). A test for normality based on the sample entropy, Journal of the Royal Statistical Society Series B, 38, Govind S. Mudholkar Department of Statistics University of Rochester Rochester, NY 14627, USA govind@metro.bst.rochester.edu Yogendra P. Chaubey Department of Mathematics and Statistics Concordia University Montreal, QC H4B 1R6, Canada chaubey@alcor.concordia.ca Rajeswari Natarajan Department of Statistics Southern Methodist University Dallas, TX , USA raji@mail.smu.edu