Quadratic Statistics for the Goodness-of-Fit Test of the Inverse Gaussian Distribution

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1 118 IEEE TRANSACTIONS ON RELIABILITY, VOL. 41, NO. 1, 1992 MARCH Quadratic Statistics for the Goodness-of-Fit Test of the Inverse Gaussian Distribution Robert J. Pavur University of North Texas, Denton Rick L. Edgeman, Member IEEE Colorado State University, Ft. Collins Robert C. Scott Bradley University, Peoria statistics. Monte Carlo simulation is used to determine approximate critical values for these three tests. This modified test procedure uses regression relationships between sample size and critical values. An example illustrates the application of the procedure and a Monte Carlo power study explores the usefulness of the procedure. Key Words - Anderson-Darling, Cramer-von Mises, Watson, Goodness-of-fit, Monte Carlo simulation. Reader Aids - Purpose: Widen state of the art Special math needed for explanations: Statistics Special math needed to use results: Same Results useful to: Reliability and quality control theoreticians and analysts. Abstract - This paper discusses the problem of using a quadratic test to examine the goodness-of-fit of an inverse Gaussian distribution with unknown parameters. Tables of approximate critical values of Anderson-Darling, Cramer-von Mises, and Watson test statistics are presented in a format requiring only the sample size and the estimated value of the shape parameter. A relationship is found between the sample size and critical values of these test statistics, thus eliminating a need to interpolate among sample sues given in the table. A power study showed that the proposed modified goodness-of-fit procedures have reasonably good power. 1. INTRODUCTION Review of statistical literature indicates a surge of interest in the inverse Gaussian (IG) distribution. A recent monograph by Chhikara & Folks [3] provides a useful guide to many of the methods and applications of the IG distribution, including numerous reliability and life testing results. Methods that are based on the IG distribution include regression [ 181 and an alternative to the analysis of variance known as analysis of reciprocals [7]. Control charts have recently been developed for this distribution [5, 61. Sequential sampling plans based on IG distributed process output have been developed [9]. Goodness-of-fit (GoF) tests of the Kolmogorov-Smirnov type for the IG distribution with unknown parameters have been developed [8, 111. These tests tend to be more powerful against certain alternatives than their more well-known competitors, the chi-square and modified Kolmogorov-Smirnov tests [4]. This paper presents a modified procedure for use with the Anderson-Darling, Cramer-von Mises, and Watson tests to perform a GoF test for the IG distribution with unknown parameters. The test statistics used in these three procedures are members of the general Cramer-von Mises family of GoF Notation & Acronyms Edf empirical Cdf GoF goodness of fit A-D Anderson-Darling W Watson C-VM Cramer-von Mises IG inverse Gaussian n sample size CY statistical significance level c1 mean of the IG distribution x parameter of the IG distribution 4 h/p which dictates the shape of the IG distribution implies a maximum likelihood estimator XI,...,X,, random sample from the IG distribution Z,,, ordered value i of the sample Z1, &,..., Z, A A-D test statistic W2 C-VM test statistic U* w test statistic F,,(Y) Edf of Y F(Y) Cdf of an IG r.v. D, represents the A2, W2, or U2 test statistic from a sample of size n D,,,,, a critical value of D, for specified values of 4 and a Of(+) D,(6 + 6, (1/6) + p2( l/n)); a modified value of D,,; 6, are real-valued coefficients that depend on 6 Other, standard notation is given in Information for Readers & Authors at the rear of each issue. 2. THE INVERSE GAUSSIAN DISTRIBUTION The IG pdf for a random variable X is: This pdf is unimodal and its shape depends only on 4 = A/p [12]. The mean and variance of X are p and p3/h, respectively. The Cdf is evaluated by applying (2), due to Chhikara & Folks [l] /92$03.00 O 1992 IEEE

2 PAVUR ET AL.: QUADRATIC STATISTICS FOR THE GOODNESS-OF-FIT TEST OF THE INVERSE GAUSSIAN DISTRIBUTION 119 Let X1, X,,..., Xn be a random sample of n values from an IG distribution. The maximum likelihood estimators of p and X are: Let Yl, Y2,..., Y, be a random sample from a distribution and let Zi = F( yi). For computational purposes, W, A, and U2 can be written in the following forms. r n 1-1 x = (l/n) (l/xi - 1/X). (4) L i= I 1 These estimators are computationally simple, complete, stochastically independent, and jointly sufficient [ 13, 161; they are uniform minimum variance unbiased for p and A [ 161. Since the IG distribution can be indistinguishable, in practice, from the lognormal distribution, it can be tempting to apply a log transformation to the data and use standard normal theory. However, [2] provides various advantages in choosing the IG distribution over the lognormal. Let P(x) be an IG Cdf with p and X estimated by (3) and (4). Since ficx(cx> = Fx(x), for a positive constant c, P(x) is invariant to the multiplication of an IG r.v. by a positive constant. For simulation purposes, this implies that c can be judiciously chosen so that it is only necessary to vary one of the IG parameters. Let c = l/px; it follows that pcx = 1 and Xcx = Xx/px. Hence, if suffices to vary only the shape parameter 4 in the Monte Carlo study to find the empirical critical values of the quadratic tests. Since E(x) is used in place of F(x), the critical values generated in the simulation study are a modified A-D, C-vM, or W GoF test procedure. 3. THE CRAMER-VON MISES FAMILY OF GOODNESS-OF-FIT TESTS The proposed tests are for the null hypothesis H,: The sample originates from an IG distribution The three test statistics for this hypothesis are members of the general Cramer-von Mises family of quadratic GoF statistics: + (2n + 1-2i)ln(l-Z(i))] (8) n U = w2 - n(z -.5), Z = Zi/n (9) i= 1 Tables of critical values of these statistics for a variety of distributions are in [17]. In all cases, if the computed value of the test statistic exceeds the tabled value at the designated s- significance level, then the hypothesized distribution is rejected as a model for the phenomenon. Through the use of transformations, these and other GoF tests can be extended to additional distributions [ 1. Ref [4] details one method of developing GoF tests based on the Edf, when the parameters of a distribution are unknown. This method substitutes maximum likelihood estimates of the unknown parameters into the Cdf, and then proceeds to apply a standard test, such as the A-D, C-vM, or W test, to the sample data. 4. MONTE CARLO APPROXIMATION OF CRITICAL POINTS Let the test statistic for a given test be represented by On, where On is used variously to represent A2, W2, or U2, and the subscript n denotes the sample size. Monte Carlo simulation techniques were used to approximate selected upper-tail percentage points of the distribution of On for a variety of sample sizes and shape parameters, 4, for the A-D, C-vM, and W GoF tests. For each combination of n and 4, 00 Monte Carlo samples of pseudo-random IG variates were generated using the algorithm in [15]. The selected values of n were: F, ( Y) is the Edf, defined as the proportion of observations less than or equal to Y. As noted in [4], when \k (y) = 1, then (5) represents the C-VM statistic; and when 9 (y) = (F( y) [ 1 - F(y)]) -I, then (5) represents the A-D statistic. A modification of W is the Watson statistic: 4, 5, 6, 8,, 12, 16,, 25,, 40. The 13 values of 4 selected for study are listed in the left vertical column of tables 1-3. The quantiles of order 1 - CY were estimated from the Edf of On for CY = 0., 0., 0.05, Thus 12 tables (3 GoF tests by 4 s-significance levels) were generated for the critical values of A, W2, and U. These tables are available from the authors. In the original Monte Carlo study, 40 values of 4 were used. Our power studies indicated that the 13 values in table 1 would suffice with minimal loss of power. The limiting case,

3 1 IEEE TRANSACTIONS ON RELIABILITY, VOL. 41, NO. 1, 1992 MARCH 4-00, was investigated by using 4 = oo. The study was performed on a Hitachi Data Systems 8083 mainframe computer using programs written in Fortran APPROXIMATE CRITICAL VALUES Modifications of tests based on Edf statistics have been made to find critical values for tests corresponding to distributional families [4]. Such modification of an Edf test statistic, say T, can be expressed as T e g(n) where - g(n) = h (1/fi) + &(1/n) + &(1/n2) & are estimated coefficients. () Not all of the terms in () need be used. For example, a modification of the Kolmogorov-Smimov test statistic T for the GoF of a s-normal distribution with unknown mean and variance is T(\/;; /\/;;) [41. Modifications of the form T g(n) typically do not rely on a, particularly for a < 0.. When \/;; is included in the modified test statistic it usually has a coefficient of unity due to interest in the asymptotic points of G. The regression model (11) provides a good fit for each fixed value of 4 and a for critical values in the 12 tables generated in the Monte Carlo study : In (ll), Dn,d,a is the empirical critical value for specified n, 4, a; E is a random error term; and the pi are regression coefficients. For this particular model, R2 > 0.97 for most values of 4. The fit of this model was compared with the fit of models that included an intercept term and other terms involving n, such as l/n2 and lln3. The marginal contributions of these terms did not warrant their inclusion in (11). The fitted regression equation can be written in the form: If the test statistic 0, 2 Dn,4,a, then the null hypothesis, Ho, is rejected. Thus, say for CY = 0.01, one can compute the ex- + $, ( l/fi) + &( l/n)] and compare pression 0: = on[\/;; this value to p,ol, rejecting Ho if the absolute value of D,* is greater than the absolute value of p,ol. The 12 tables from the Monte Carlo simulation study of the previous section were used to generate appendix tables 1, 2, 3 which correspond to the A-D, C-vM, W tests, respectively. While some accuracy is sacrificed as compared to the original 12 tables, these tables offer the advantage of allowing for any n, without interpolation. 6. POWER OF THE PROPOSED TESTS When using tables 1-3, the following generally conservative procedure can be used to determine the rejection for any of the test procedures and specified sample size. a. Denote by $L and the tabled values of 4 that bracket 6 = i/fi. b. Reject the null hypothesis, Ho, iff the test procedures using the regression coefficients in the row designated by 4L and both indicate rejection to be appropriate. A more tedious, but also more sensitive, approach is to use linear interpolation on both the value of On[& + 8, (l/\/;;) + &( l/n)] and the critical values given in the table. This procedure was used to assess the approximate power of the A-D, C-vM, and W tests against each of four alternative distributions as well as the IG distribution with 4 = 1. Results of this investigation are in tables 4-6 in the appendix. Alternative distributions used were the uniform (0,l); exponential with mean = 1; lognormal with mean = e and variance = e3 - e2; and 2-parameter standard Weibull with shape parameter = 2. lo4 random samples for each of the sample sizes,,, were examined for each of these distributions. The results of this power study indicate that performance of these tests is superior to those in [8, 111. In general, shape parameters can be selected to provide either good or poor discrimination with the IG distribution. The parameters for the lognormal and Weibull distributions were chosen so that these distributions and the IG distribution would be similar in shape. A more extensive power study would be necessary to provide power results for other parameter values. The power study revealed excellent discriminatory ability for all three of the tests against the exponential and uniform alternatives, poor discriminatory ability against the lognormal alternative, and moderate power against the Weibull altemative. All three test procedures achieved an empirical s-significance level indistinguishable from the stated s-significance level for the IG distribution with 4 = 1. Additional simulation results indicated that for very large or very small 4, these tests provide conservative results, ie, empirical s-significance levels can be somewhat smaller than stated s-significance levels. Generally the tests appear able to distinguish between the IG and distributions of very different shape, but are relatively unable to discriminate between the IG and distributions of similar shape. In stochastic modeling settings it is usually the former distinction that is important. Although the empirical power results for the A-D test were somewhat higher than those for the C-VM and W tests, power distinctions among the three procedures were unimportant. 7. ENDURANCE OF BALL BEARINGS The A-D, C-VM and W GoF tests of section 3 for the IG are applied to test data [14] on the endurance of deep-groove ball bearings. Twenty-three ball bearings were used in the life test study and yielded the results recorded below, in millions of revolutions to failure:

4 ~~ ~ ~ ~~~ ~ PAVUR ET AL.: QUADRATIC STATISTICS FOR THE GOODNESS-OF-FIT TEST OF THE INVERSE GAUSSIAN DISTRIBUTION Conformance of these data to an IG distribution assessed by a modified Kolmogorov-Smirnov test is affirmed [3]. Each of the GoF tests is used to test whether this phenomenon could be reasonably represented by the IG. Initially, we determine fl = x = for the 23 observations. Next, standardized failure times are obtained by dividing each of the 23 original values by E. Application of (4) to the original data yields = so that 6 = Application of (7)-(9) yields A =.97535, W2 =.07319, and U* = Critical values of these statistics, to which these calculated values must be compared, are now determined for ct = Use the interpolation method of the preceding section; d = 3.77 is bracketed in the 4 columns of tables 1-3 by 4L = 3 bu = 4. Let C(4) be the critical value of the test statistic. For the A-D test we have: 4 D24) C(4) o TABLE 1 Empirical Critical Values of the Anderson-Darling Statistic for the Inverse-Gaussian Distribution d 81 8* lo TABLE 2 Empirical Critical Values of the Cramer-von Mises Statistic for the Inverse-Gaussian Distribution d lo Since ID,* (3.77)l = 1.57 < lc(3.77) = , application of the A-D test leads to failure to reject Ho, that the data have originated from an IG process. Note that D,* (3.77) and C( 3.77) are interpolated values. Similar computations with the C-VM and W test procedures also lead to failure to reject the null hypothesis. While the conclusion of each of these three test procedures is that the data can be represented by an IG distribution, it must be recognized that a relatively small sample size was used and that the power of these procedures is somewhat low. For a sample of size, tables 4-6 indicate a range of powers from to 0.8. In view of these power results, the conclusion should be considered preliminary until further accumulation of evidence. APPENDIX: Empirical Critical Values & Power Study d lo TABLE 3 Empirical Critical Values of the Watson Statistic for the Inverse-Gaussian Distribution 8,

5 122 IEEE TRANSACTIONS ON RELIABILITY, VOL. 41, NO. 1, 1992 MARCH TABLE 4 Empirical Power Results for the Anderson-Darling Test s-significance Level Sample Size Distribution n TABLE 6 Empirical Power Results for the Watson Test s-significance Level Sample Size Distribution n Exponential,569,782, ,458, ,369,631, ,221,480,675,889 Exponential,511, ,954,391,615,765,927,4,533,704,895,178,391,572,817 Lognormal,272,5,344,433,157,193,235,312,094,122,157,223,029, ,5 Lognormal,248,259,295,378,134,1,183,248,073, , ,028,040,067 Uniform,741,952,993,646,917,984,557,876,971,999,382,770,931,996 Uniform,688,913,981,999,570, ,998,475,798,936,996,7.653, Weibull,364,577, ,263,474,625,824,196,394, ,1,2,388,634 Weibull.325 SO8,639,816, ,522, ,426,6,068,165,270,480 Inverse Gaussian.226,8,7,213,118,112,1, , ,063,016,015 Inverse Gaussian IO,216,4.8,218,9,2,7.117,053,054, ,0,011,015 Entries represent the proportion of rejections out of lo4 Monte Carlo samples of size n. Entries represent the proportion of rejections out of lo4 Monte Carlo samples of size n. TABLE 5 Empirical Power Results for the Cramer-von Mises Test REFERENCES s-significance Level Sample Size Distribution n Exponential Lognormal Uniform Weibull Inverse Gaussian,556, ,974,261, ,421,7,937,989,340,552, ,7,0,5,211,4,701,843,961,148,185,223,299,609, ,242,448, ,5,7,6,111,369,632,795,943, ,151,212,519,8, ,178,365,513,740,053, ,062,236,493,681,891,0.046,063,099,358,736,9,994,086,225,362,605,011,014,015 Entries represent the proportion of rejections out of lo4 Monte Carlo samples of size n. [ 11 R. S. Chhikara, J. L. Folks, Estimation of the inverse Gaussian distribution function, J. Amer. Statistical Assoc., vol 69, 1972 Jun, pp [2] R. S. Chhikara, J. L. Folks, The inverse Gaussian distribution as a lifetime model, Technometrics, vol 19, 1977 Nov, pp [3] R. S. Chhikara, J. L. Folks, The Inverse Gaussian Distribution: Theory, Methodology, and Applications, 1989; Marcel Dekker. [4] R. B. D Agostino, M. A. Stephens, Goodness-of-fit Techniques, 1986; Marcel Dekker. [5] R. L. Edgeman, Inverse Gaussian control charts, Australian J. Stati3tic3, vol 31, 1989 Apr, pp [6] R. L. Edgeman, Control of inverse Gaussian processes, Quality Engineering, vol 1, 1989 Jun, pp [7] R. L. Edgeman, An altemative to analysis of variance for reliability data, Quality & Reliability Engineering Int l 1, vol6, 1990 Aug, pp 3-7. [8] R. L. Edgeman, Assessing the inverse Gaussian distribution assumption, IEEE Trans. Reliability, vol 39, 1990 Aug, pp [9] R. L. Edgeman, P. Salzberg, A sequential sampling plan for the inverse Gaussian mean, Staristische Hefie, vol 33, 1991 Spring, pp [lo] R. L. Edgeman, R. C. Scott, Ldliefors s tests for transformed variables, BrazilianJ. Probability & Statistics, vol 1, 1987 Nov-Dec, pp [ll] R. L. Edgeman, R. C. Scott, R. J. Pavur, A modified Kolmogorov- Smimov test for the inverse Gaussian density with unknown parameters, Communications in Statistics - Simulation, vol 17, 1988 Dec, pp [12] N. L. Johnson, S. Kotz, Distributions in Statistics - Continuous Univariate Distributions-I, 1970; John Wiley & Sons. [13] R. F. Kappenman, On the use of a certain conditional distribution to derive unconditional results, Amer. Statistician, vol 33, 1979 Feb, pp

6 PAVUR ET AL.: QUADRATIC STATISTICS FOR THE GOODNESS-OF-FIT TEST OF THE INVERSE GAUSSIAN DISTRIBUTION 123 [ 141 J. Lieblein, M. Zelen, Statistical investigation of the fatigue life of deepgroove ball bearings, J. Research National Bureau of Standards, vol 57, 1956, pp [15] J. Michael, W. Schucany, R. Haas, Generating random variates using transformations with multiple roots, Amer. Statistician, vol, 1976 May, pp [16] J. K. Patel, C. H. Kapadia, D. B. Owen, Handbook ofstatisticaldistributions, 1976; Marcel Dekker. [17] M. A. Stephens, Tests based on Edf statistics, in Goodness-of-jt Techniques (R. B. D Agostino, M. A. Stephens, eds.), 1986, pp ; Marcel Dekker. [18] G. A. Whitmore, A regression method for censored inverse-gaussian data, Canadian J. Statistics, vol 11, 1983 Oct, pp AUTHORS Dr. Robert J. Pavur; Dept. of BCIS; University of North Texas; Denton, Texas 763 USA. Robert J. Pavur is an Associate Professor of Management Science at the University of North Texas. He received his PhD in Statistics from Texas Tech University in His work has appeared in the Canadian J. Statistics, Sankya, 7he American Statistician, 7he American J. Mathematical and Management Science, Communications in Statistics, and numerous other journals. Dr. Rick L. Edgeman; Center for Quality & Productivity Improvement; B219 Clark Bldg.; College of Business; Colorado State University; Fort Collins, Colorado USA. Rick L. Edgeman (M 90) is director of the Center for Quality & Productivity Improvement at Colorado State University. He received the PhD in Statistics in 1983 from the University of Wyoming and is the Book-Review Editor for Quality Progress. Rick is the author of more than 40 papers appearing in journals such as IEEE Trans. Reliability, Quality Engineering, Quality & Reliability Engineering Int 1, Quality Progress, Statistical Hefre, The American Statistician, Australian J Statistics, Inr *l J. Modelling & Simulation, and Communications in Statistics. Dr. Robert C. Scott; Economics Department; Bradley University; Peoria, Illinois USA. Robert C. Scott is chair n of the Economics department at Bradley University. He received the MS in Statistics in 1973 and the PhD in Econometrics in 1974 from the University of Iowa. His work has appeared in such journals as the J. American Statistical Assoc, J. Regional Economics, The American Statistician, Communications in Statistics, and Brazilian J. Probability & Statistics. Manuscript TR received 1989 Februruy 9; revised 1990 January 14; revised 1991 July 1. IEEE Log Number TR b Annual Reliability and Maintainability Symposium The P. K. McElroy Award for Best Paper was bestowed on Charles H. Stapper, John A. Fifield, and Howard L. Kalter for their paper High-Reliability Fault-Tolerant 16-Mbit Memory Chip that was given at the 1991 Symposium in Orlando. For more information, see the gold section of your copy of the 1992 Proceedings. Each year the Symposium presents The P. K. McElroy Award for the best paper at the previous Symposium. The Award consists of a plaque and a $00 honorarium. There are two criteria for best paper: The content of the written paper is lucid, excellent, and important to the theory and/or practice of R&M engineering. The verbal presentation of the paper at the Symposium is likewise lucid and excellent. P. K. McElroy was an intensely practical person. Papers that receive the Award must be able to make a difference to R&M engineers and/or managers. It is not enough that the content be competent and important; that competence and importance must be readily obvious in both the written and verbal presentations. Before the Symposium, the content of each written paper is examined by the Program Committee for technical excellence and clarity of exposition. The best of the papers are chosen and referred to a select group of past General Chair n of the Symposium. Each person in that group listens to each presentation, and that group choses the best paper to receive the P. K. McElroy Award.