A scale-free goodness-of-t statistic for the exponential distribution based on maximum correlations

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Journal of Statistical Planning and Inference 18 (22) 85 97 www.elsevier.com/locate/jspi A scale-free goodness-of-t statistic for the exponential distribution based on maximum correlations J.

Grane Departament d Estadstica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 87 Barcelona, Spain Abstract We propose a goodness-of-t statistic for testing exponentiality based on

1 Journal of Statistical Planning and Inference 18 (22) A scale-free goodness-of-t statistic for the exponential distribution based on maximum correlations J. Fortiana, A. Grane Departament d Estadstica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 87 Barcelona, Spain Abstract We propose a goodness-of-t statistic for testing exponentiality based on Hoeding s maximum correlation. We study its small and large sample properties, we obtain its exact distribution, tables of exact and asymptotic critical values, and some power curves. We compare this statistic with the Gini statistic, the Shapiro Wilk statistic for exponentiality and the Stephens modication of the latter. c 22 Elsevier Science B.V. All rights reserved. MSC: 62G1; 62G3; 62E15 Keywords: Goodness-of-t; Maximum correlation; L-statistics; Exponentiality 1. Introduction Let F 1 and F 2 be two cdf s having second-order moments. The Hoeding maximum correlation + (F 1 ;F 2 ) is dened as the correlation coecient corresponding to the bivariate cdf H + (x; y) =min{f 1 (x);f 2 (y)}, the upper Frechet bound of F 1 and F 2, i.e., the upper bound of the Frechet class F(F 1 ;F 2 ) of bivariate cdf s with marginals F 1 and F 2, ordered according to their correlation coecients. H + is a singular distribution, having support on the one-dimensional set {(x; y) R 2 : F 1 (x)=f 2 (y)}, and ( )/ 1 + (F 1 ;F 2 )= F 1 (p) F 2 (p)dp ; (1) Work supported in part by CGYCIT PB96-14-C2-1 and 1997SGR-183 (Generalitat de Catalunya). Corresponding author. addresses: fortiana@cerber.mat.ub.es (J. Fortiana), aurea@porthos.bio.ub.es (A. Grane) /2/$ - see front matter c 22 Elsevier Science B.V. All rights reserved. PII: S (2)272-

2 86 J. Fortiana, A. Grane / Journal of Statistical Planning and Inference 18 (22) where F i is the left-continuous pseudoinverse of F i, i =E(F i ) and 2 i =var(f i );i=1; 2 (see, e.g., Cambanis et al., 1976). Here the notation E(F) represents the expected value of any randomvariable whose cdf is F, and analogously for var(f). + (F 1 ;F 2 ) is a measure of agreement between F 1 and F 2, since it equals 1 i F 1 = F 2 (a.e). Cuadras and Fortiana (1993) proposed the statistic + (F n ;F ) as a qualitative measure of goodness-of-t of an iid sample, with empirical cdf F n, to a given distribution F.In this paper, we implement this idea in the form of a specic test and study its properties. Given n randomvariables iid F, we will test H : F = F Exp(;); R + ; where Exp(;) is the exponential distribution with location parameter = and unknown scale parameter, F (x)=1 exp( x=); x. To this end, we propose the statistic Q n = s n y n + (F n ;F ); (2) where y n, sn 2 and F n are the empirical mean, variance and cdf, respectively, of the iid randomvariables y 1 ;:::;y n. The motivation for this denition is that, as shown below (see Lemma 2.1), we have the equality Q n = l nj y (j) y ; (j) (3) where (y (1) ;:::;y (n) ) are the order statistics of the sample, which enables us to borrow results fromthe theory of L-statistics. Our rst aimis to obtain the critical values of Q n. For small sample sizes, we will use its exact distribution under H, which is derived in Section 2. For large sample sizes, we will use the asymptotic distribution under H of the auxiliary function L a n = s n + (F n ;F ) a y n (4) which is obtained in Section 3. Next, in Section 4, we compute the power of the test based on Q n and compare it to those of the Shapiro Wilk test of exponentiality, the test based on the Stephens modication of the Shapiro Wilk statistic, the test based on the Gini statistic and the test based on another statistic also derived from Hoeding s maximum correlation (See Fortiana and Grane, 1999), designed for testing goodness-of-t to the two-parameter exponential distribution, denoted here by Qn. 2. Exact distribution of Q n under H Let y 1 ;:::;y n be n randomvariables iid F =Exp(;), with empirical distribution function F n, and let y (1) ;:::;y (n) be the order statistics.

3 J. Fortiana, A. Grane / Journal of Statistical Planning and Inference 18 (22) Lemma 2.1. The statistic T n = s n + (F n ;F ) is an L-statistic; T n = 1 n l nj y (j) ; where l nj =(n j) log(n j) (n j + 1) log(n j +1)+log(n); 1 6 j 6 n; (5) and log. Proof. Using (1) with F n and F ; + can be written as ( ) + (F n ;F )= 1 1 Fn (p) F s (p)dp y n ; n since the mean and the standard deviation of the exponential distribution is. The integral in the numerator of + 1 Fn (p) F (p)dp = i= (i+1)=n i=n log(1 p) y (i+1) dp = y (i+1) [(n i) log(n i) n i= (n i 1) log(n i 1) log n 1]: Letting j = i + 1 we have 1 Fn (p) F (p)dp = [(n j) log(n j) (n j + 1) log(n j +1) n and the result follows. + log (n)]y (j) + Proposition 2.1. The Q n statistic dened in (2) does not depend on the scale parameter; and can be written as y (j) Q n = l=1 a nlz l l=1 z ; l (6)

4 88 J. Fortiana, A. Grane / Journal of Statistical Planning and Inference 18 (22) where a nl = log(n=(n l)); l=1;:::;n 1; and z 1 ;:::;z n are iid Exp(; 1) random variables. Proof. Suppose x j = y j =; j =1;:::;n; so that Q n = l njx (j) x ; (j) (7) fromdavid (1981; pp. 2 21); for j =1;:::;n; the transformation z j =(n j +1)(x (j) x (j 1) ); where; by convention; x () =; gives n iid Exp(; 1) randomvariables; and the jth order statistic fromthe standard distribution can be written as x (j) = j (x (k) x (k 1) )= k=1 j k=1 Substituting (8) in (7); j k=1 Q n = l nj z k =(n k +1) j k=1 z = k=(n k +1) z k n k +1 : (8) log(n=(n j + 1)) z j z : j Dene a nj =log(n=(n j +1)); ;:::;n; then a n1 =. We obtain (6) making l=j 1 and rearranging terms. Let N = a njz j and D = z j, so that Q n = N=D. Lemma 2.2. The characteristic function of the vector (N; D) is (N;D) (t 1 ;t 2 )= (1 i(a nj t 1 + t 2 )) 1 (1 it 2 ) 1 with (t 1 ;t 2 ) R 2 ; i= 1. Proof. In matrix notation; N=a z;d=1 z; where a=(a n1 ;:::;a n; ; ) ; z=(z 1 ;:::;z n ) and 1 =(1;:::;1). We also consider the matrix A =(a; 1). Then; ( ) N = Az: D From the transformation formula of characteristic functions by the action of an ane map Az (t)= z (A t); for t =(t 1 ;t 2 ) R 2. Since z 1 ;:::;z n are randomvariables iid Exp(; 1); z (A t)= zj ((a nj t 1 ;t 2 )) zn ((;t 2 ));

5 J. Fortiana, A. Grane / Journal of Statistical Planning and Inference 18 (22) where zj ((w 1 ;w 2 ))=(1 i(w 1 + w 2 )) 1 ; j =1;:::;n: Since Q n is a quotient of linear combinations of iid Exp(; 1) randomvariables, to obtain its exact probability density function we can adapt the technique used in Dwass (1961) and Matsunawa (1985). Proposition 2.2. The exact probability density function of Q n is given by f(t)=() a n 3 nj k=1;k j (a nj a nk ) 1 (1 t=a nj ) n 2 1 (;anj)(t); (9) where a nj = log(n=(n j)); ;:::;n 1; and 1 denotes an indicator function. Proof. Using Lemma 2.2 from Matsunawa (1985); we obtain the characteristic function of (N; D): where (N;D) (t 1 ;t 2 )= C l = k=1;k j a 1 nj C l a nl [(1 it 2 ) ia nl t 1 ] 1 (1 it 2 ) 1 n ; l=1 ( 1 1 ) 1 : a nk a nj Inverting this characteristic function; following (2.16) in Matsunawa (1985); f (N;D) (w; s) = l=1 C l a 1 nj 1 (;+ )(w) 1 (;sanl )(w)e s ( s w a nl ) n 2 / (n 1); for (w; s) R 2 +. Applying a change of variables and noticing that C l a 1 nj = a n 3 nj k=1;k j (a nj a nk ) 1 ; we get the probability density function of Q n ; f(t)= + s f (N;D) (ts; s)ds: We have developed a Mathematica program implementing these algorithms. Fig. 1 shows two examples of probability densities of Q n, and Table 1 in the appendix contains exact critical values of Q n for a bilateral test at the 5% signicance level.

6 9 J. Fortiana, A. Grane / Journal of Statistical Planning and Inference 18 (22) Fig. 1. Probability density functions of Q n for n = 5, on the left, and for n = 1, on the right. 3. Asymptotic distribution of L a n under H Proposition 3.1. (i) The function L a n dened in (4) is a linear combination of the order statistics x (1) ;:::;x (n) from the Exp(; 1) distribution L a n = n c a njx (j) with coecients c a nj = l nj a; 1 6 j 6 n and l nj were dened in (5). (ii) Letting n a = E(L a n) and n;a 2 = n var(l a n), we have [ ( ) n n a n 1=n = log a] ; n! 2 n;a = 2 [A (n)+a 1 (n)a + a 2 ]; (1) where A (n) and A 1 (n) depend only on n. The sequence {n;a} 2 converges to a 2 = 2 (a 2 2 a + 2), when n. (iii) We have the following convergences in law: n[l a n n] a L n N(;2 a); (11) [ ] L a n n n a L N(; 1): (12) n n;a

7 J. Fortiana, A. Grane / Journal of Statistical Planning and Inference 18 (22) Proof. (i) From Lemma 2.1 L a n = 1 n l nj y (j) a y n = 1 n cnjy a (j) = n cnj a x (j) with c a nj = l nj a; 1 6 j 6 n: (ii) Fromthe general asymptotic theory of L-statistics (Shorack and Wellner, 1986), L a n can be written as where L a n = 1 J a n (t) F n (t)dt; ( ) ( 1 t Jn a (t)=n(1 t) log + log 1 t +1=n 1 1 t +1=n ) a and Fn is the pseudoinverse of the empirical distribution function. The expectation of L a n under H can be computed as a n = 1 J a n (t) F (t)dt; where F (t) = log(1 t); 6 t 6 1. Alternatively, it can be computed directly fromthe expression in (i) a n = n (l nj a)m j where m j = j k=1 (n k +1) 1 is the expected value of the jth order statistic from the standard exponential distribution. See Fortiana and Grane (1999) for more details. The variance of L a n is var(l a n)= 2 n;a=n; where 1 1 n;a 2 = Jn a (s)jn a (t)[min(s; t) st]df (s)df (t): (13) Since the integrand is symmetrical, 2 n;a =2 2 1 ( J a n (s) s J a n (t) ) t 1 t dt ds

8 92 J. Fortiana, A. Grane / Journal of Statistical Planning and Inference 18 (22) and, partitioning the region {(s; t) R 2 :6 s 6 1; 6 t 6 s} in three parts, A={(s; t) R 2 :6 s 6 1 1=n; 6 t 6 s}; B={(s; t) R 2 :1 1=n 6 s 6 1; 6 t 6 1 1=n}; C={(s; t) R 2 :1 1=n 6 s 6 1; 1 1=n 6 t 6 s}; we consider the following expansions: log(1 s +1=n) log(1 s)+ when s 1 1=n, and log(1 s +1=n) log(1=n)+ m ( 1) k+1 (n(1 s)) k ; (14) k k=1 m ( 1) k+1 (n(1 s)) k ; (15) k k=1 when s 1 1=n. We have used Mathematica to compute the integrals on the three regions, obtaining A (n) and A 1 (n) in (1). To attain the intended precision, (14) was expanded up to m = 2 and (15) up to m =5. Since Jn a is continuous and bounded a.e. (F ), we can permute the limit with the integral. Thus, to compute a 2 = lim n n;a, 2 we just need to substitute J a (t) = lim n J a n (t)= [1 + log(1 t)+a] for Jn a in integral (13). Analogously, we have 1 ( s ) a 2 =2 2 J a (s) J a t (t) 1 t dt ds 1 ( s ) =2 2 t [1 + log(1 s)+a] [1 + log(1 t)+a] 1 t dt ds =2 2 (1 a + a 2 =2): (iii) Convergence (11) is obtained fromtheorem1 of Shorack and Wellner (1986, pp ). Convergence (12) is immediate from (11) and from the fact that 2 a = lim n 2 n;a. To compute critical values we have used the normal approximation based on (12) which, having a higher convergence rate than (11), gives a more powerful test. This normal approximation is rather good (for example for n = 2 we obtain a relative error of :6% on the right tail and a relative error of 1:8% on the left tail). On the other hand, while it is actually possible to compute the exact critical values from (9), the computational cost required for solving equations involving piecewise polynomial functions of degree n increases steeply with the sample size.

9 J. Fortiana, A. Grane / Journal of Statistical Planning and Inference 18 (22) Fig. 2. Power curves of Q n for A2 family of alternatives, with = 1, (see Section 4.1 for notation) from three dierent critical regions (c.r.). Fig. 2 shows the power of the test based on Q n for the A2 family of alternatives, see Section 4.1 below for notation, computed from three dierent critical regions: the one computed from (9), the one computed from (12) and the one computed from (11). The critical value a is obtained by solving the equation: n a n = c ; n;a where c is the (1 )-quantile of the standard normal distribution. We have developed a set of programs in Mathematica to compute these critical values. We reproduce some of themin Table 2 in the appendix. 4. Power of Q n 4.1. Families of alternatives We consider alternatives of the form F(x; 1 ; 2 ). In each case, one of 1 ; 2 is the scale parameter and can be xed to a given arbitrary value, without loss of generality. (A1) Generalized Pareto ( F(x; a; k)=1 1 k ) 1=k a x ; a ;

10 94 J. Fortiana, A. Grane / Journal of Statistical Planning and Inference 18 (22) with 6 x, ifk 6, and 6 x 6 a=k, ifk. Observe that lim k F(x; a; k)= Exp(;a). See Choulakian and Stephens (1997) for notation and relations. (A2) Gamma with shape parameter and scale parameter, with density function 1 f(x; ; )= () x 1 e x= ; ; ;x : In this case, F(x;1;) = Exp(;). (A3) Weibull with shape parameter and scale parameter, with density function f(x; ; )= x 1 exp{ (x=) }; ; ;x : In this case, F(x;1;) = Exp(;). (A4) Generalized exponential with location parameter and scale parameter. F(x; ; )=1 e (x )= ; x ; with, +. Note that F(x;;) = Exp(;) Power comparisons Power against an alternative distribution F(x; 1 ; 2 ) has been estimated by the relative frequency of values of the statistic in the critical region for N = 5 simulated n-samples of F(x; 1 ; 2 ). For (x; 1 ; 2 ) we have taken distributions in each of the A1 A4 families with one xed parameter and let the other vary within its range. For each family we have taken 3 dierent values of the parameter. We have compared Q n with the Shapiro Wilk statistic for exponentiality n(x x (1) ) 2 W = (n 1) i=1 (x i x) ; (16) 2 the Stephens modication of the Shapiro Wilk statistic ( i=1 W S = x i) 2 n(n +1) i=1 x2 i n( i=1 x i) ; (17) 2 the Gini statistic and i=1 G = (2i n 1)x (i) n(n 1) x Q n = i=1 l ix (i) i=1 b ix (i) ; (18) (19) where, for i =1;:::;n, l i =(n i) log(n i) (n i + 1) log(n i +1)+log(n), with log =, and b i = i=n (n +1)=(2n). This statistic, also derived fromhoeding s maximum correlation, was designed to test for the two-parameter exponential family (see Fortiana and Grane, 1999). Its distribution is invariant under both translations and dilations. For (16) we have used the critical values computed in Shapiro and Wilk

11 J. Fortiana, A. Grane / Journal of Statistical Planning and Inference 18 (22) Fig. 3. Power curves for the A1 family (a = 1), on the left, and for the A2 family ( = 1), on the right. Fig. 4. Power curves for the A3 family ( = 1), on the left, and for the A4 family ( = 1), on the right. (1972), for (17) the points computed in Stephens (1986), for (18) the points computed in Gail and Gastwirth (1978) and for (19) the points computed in Fortiana and Grane (1999) Power curves Figs. 3 and 4 show the power functions of the bilateral tests at the 5%-signicance level, for n = 2. Note that for the A4 alternative W and Q n are not suitable, because both statistics are location and scale-free. Q n is more powerful than W and Q n for all the families studied. When the parameter is greater than that of the null hypothesis Q n is less powerful than G and W S, while for smaller values of the parameter the power of Q n is between G and W S.

12 96 J. Fortiana, A. Grane / Journal of Statistical Planning and Inference 18 (22) Appendix A. Critical values of Q n are given in Tables 1 and 2 Table 1 Exact critical values of Q n, for =:5 n Lower Upper n Lower Upper Table 2 Critical values of Q n, for =:5, computed with the asymptotic approximation n Lower-tail Upper-tail References Cambanis, S., Simons, G., Stout, W., Inequalities for Ek(x; y) when the marginals are xed. Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete 36, Choulakian, V., Stephens, M., Goodness-of-t tests for the generalized Pareto distribution, preprint. Cuadras, C.M., Fortiana, J., Continuous metric scaling and prediction. In: Cuadras, C.M., Rao, C.R. (Eds.), Multivariate Analysis, Future Directions, Vol. 2. Elsevier Science Publishers B.V., North-Holland, Amsterdam, pp

13 J. Fortiana, A. Grane / Journal of Statistical Planning and Inference 18 (22) David, H.A., Order Statistics, 2nd Edition. Wiley, New York. Dwass, M., The distribution of linear combinations of random divisions of an interval. Trabajos de Estadstica e Investigacion Operativa 12, Fortiana, J., Grane, A., A location and scale-free goodness-of-t statistic for the exponential distribution based on maximum correlations, preprint. Gail, M., Gastwirth, J., A scale-free goodness-of-t test for the exponential distribution based on the Gini statistic. J. Roy. Statist. Soc. B 4, Matsunawa, T., The exact and approximate distributions of linear combinations of selected order statistics froma uniformdistribution. Ann. Inst. Statist. Math. 37, Shapiro, S., Wilk, M., An analysis of variance test for the exponential distribution complete samples. Technometrics 14, Shorack, G.R., Wellner, J.A., Empirical Processes with Applications to Statistics. Wiley, New York. Stephens, M.A., Tests based on regression and correlation. In: Stephens, M.A., D Agostino, R.B. (Eds.), Goodness-of-t Techniques. Marcel Dekker, New York, pp

Exact goodness-of-fit tests for censored data

Exact goodness-of-fit tests for censored data Aurea Grané Statistics Department. Universidad Carlos III de Madrid. Abstract The statistic introduced in Fortiana and Grané (23, Journal of the Royal Statistical