A CONSISTENT TEST FOR EXPONENTIALITY BASED ON THE EMPIRICAL MOMENT PROCESS

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1 STATISTICA, ao LXIX,., 009 A CONSISTENT TEST FOR EXPONENTIALITY BASED ON THE EMPIRICAL MOMENT PROCESS Simos G. Meitais. INTRODUCTION The expoetial is oe of the most frequetly used distributios i survival a- alysis, reliability theory ad other life time pheomea. Due to its importace, a large umber of goodess of fit tests have bee desiged for expoetiality. May of these tests are reviewed by Spurrier (984), Ascher (990) ad Heze ad Meitais (005). Whe testig for goodess of fit, the spectrum of possible deviatios from the ull hypothesis of expoetiality is ofte restricted to a oparametric class of life distributios. Such classes may be defied via mootoicity properties of the failure rate fuctio, with the class of icreasig failure rate average (IFRA) distributios beig oe of the most frequetly ecoutered alteratives to the expoetial distributio. It will be see below that although the momets of a expoetiated radom variable belogig to the IFRA class have a certai mootoicity property, the IFRA class is too arrow for this property to be characteristic. This will be the poit of departure for cosiderig a ew class of life distributios ad for costructig expoetiality tests which is cosistet withi this ewly defied family of alteratives. Let X deote a oegative radom variable with distributio fuctio F ad fiite mea µ = EX ( ) <. It ca be show that if F belogs to the class of IFRA distributios, the g() t is icreasig i t > 0 (.) t t X g(): t = [ E( Y )]/[ te( Y )], ad Y = e. Coditio (.) although ecessary it is ot sufficiet for the IFRA class of distributios. Motivated by this fact, Bartoszewicz ad Skolimowska (007) elarged the IFRA class by itroducig the LIFRA as exactly that class of distributios for

2 50 S.G. Meitais which (.) holds. The letter L i the trasitio from the termiology IFRA to t tx LIFRA stads for Laplace (trasform) sice Y = e ad hece (.) essetially refers to a mootoicuity property ivolvig the Laplace trasform of the origial radom variable X. Bartoszewicz ad Skolimowska (008) showed that NBU LIFRA L-class, ad derived may iterestig results for LIFRA distributios such as closure properties, coectios with the otio of the total time o trasform, ad ifiite divisibility. I the preset paper we develop tests for expoetiality which are appropriate agaist the class of LIFRA distributios. To do so otice that if F is LIFRA, (.) is equivalet to Dt (): = m() t mt () m () t 0, t> 0, (.) mt () = EY ( ), t ad that i additio, the expoetial distributio is a limitig member of the LIFRA class satisfyig Dt () = 0, t> 0. (.3) To see (.3) recall that uder uit expoetiality, Y = e X is uiformly distributed i (0, ). The E( Y ) is the momet of arbitrary order of the uiform t (0,) distributio ad therefore m(t)=(+t). Hece our test statistic may be t viewed as a test based o the momet process E( Y ) of arbitrary order t > 0 of the suitably trasformed variate Y = e X, i compariso to the correspodig momet process of the uiform (0,) distributio. Sice as it will be see i the sequel, momets of all orders t > 0 will be take ito accout, the ew test is based o a cotiuum of momet coditios, as oppossed to simple momet based tests which are based o the first few momets of iteger order. The otio of a cotiuum of momet coditios has bee recetly exploited i the ecoometrics literature; see for istace Carrasco ad Flores (000, 00). I view of (.) ad (.3), it is reasoable to test H0 : F is expoetial, agaist H : F is LIFRA ad ot expoetial, by devisig a empirical versio, say D () t, of Dt ( ), ad reect H 0 i favor of H for large values of some distace measure based o D( t ). Sice however the expoetial distributio is ivariat uder scale trasformatios of the type

3 A cosistet test for expoetiality based o the empirical momet process 5 X cx, c > 0, oe has to stadardize the origial observatios X,..., X by their sample mea X = X =, =,,...,. = With this i mid, the obvious cadidate for D () t results by replacig the momet process mt ( ) i (.) by the empirical momet process m t Y Y e. t X / X () = = = The Dt ( ) i (.) is aturally estimated by D () t = m () t m () t tm'() t, ad the ull hypothesis is reected for large values of, T = D () t e dt. (.4) a, 0 at I other words we suggest the Laplace trasform (LT) with its well kow u- iqueess properties as a appropriate distace measure based o D( t ). This choice leads to a iterestig limitig iterpretatio. Specifically we cosider the behavior of the LT of D() t as the argumet a > 0 of the LT goes to ifiity, at i.e. as e approaches a Dirac type fuctio. To this ed rewrite the test statistic i (.4) as T = g() t e dt, (.5) a, 0 at gt () = D() t. Usig the expasio x t t = + + = e x ( x /) o( x ), x 0, i Y e X/ X we have by straightforward algebra gt () = + ot ( ) X + Xk 3 X t =, k X = X, as t 0. The previous equatio alog with a Abelia theorem for the LT (see Zayed, sectio 5.), yields after some further algebra

4 5 S.G. Meitais lim at = [ X S ]: = T, a 3 a,, X = ( ) = S X X. Cosequetly the test statistic whe properly stadardized possesses a limit value, as a. Iterestigly the limit statistic T, ivolves the distace betwee the sample mea squared ad the sample variace, a distace that vaishes uder H 0 (expoetiality), as.. CONSISTENCY AND LIMIT DISTRIBUTION By straightforward algebra we have from (.4) that T a, = + k, = Y + Yk + a = Y + a = ( Y. (.) The cosistecy of the test statistic may be proved as follows. By applicatio to (.) of the Law of Large Numbers for statistics with estimated parameters, e.g. Radles (98), we have T a, P ε ( µ ), as, a ε a ( µ ) = µ E E + E X + X + aµ X + aµ ( X + aµ ) O the other had Fubii s Theorem yields that for each µ > 0, Y X. (.) ε a ( µ ) = ( te ) dt, (.3) 0 at X / µ t () t = µ () t µ () t tµ '() t with µ (): t = E[( e )] deotes the momet process of e X / µ.

5 A cosistet test for expoetiality based o the empirical momet process 53 I view of (.) ad (.3), ε a ( µ ) is zero uder expoetiality but positive uder H which implies the cosistecy of the test which reects H 0 for large values of T a,. Hece we have implicitly proved the followig characterizatio: Amog all o egative X with fiite mea µ, the expoetially distributed radom variable is the oly oe which satisfies ε a ( µ ) = 0 for each a > 0. For the asymptotic ull distributio we will write whe two statistics are asymptotically equivalet. Uder the ull hypothesis H 0, the radom variable X follows a expoetial distributio with mea µ. Without loss of geerality let µ =. We will cosider the symmetrized versio S a, of T a,,, S Y Y = + (.4) * k a, + + k, = Y + Yk + a Y + a Yk + a ( Y ( Yk A typical Taylor expasio yields Sa, S a,, S X X = + k a, + + k, = X + Xk + a X + a Xk + a ( X ( Xk X hx Xk hx hxk Xh X Xh k Xk k, = + ( ) { ( + ) [ ( ) + ( )] + [ ( ) + ( )]} a hx ( ) = hx ( ;a) = x + a ( x, ad a h ( x) = h ( x;a) = ( x ( x 3. Sice X ( ) isaop() sequece, a applicatio of the Law of Large Numbers to the secod term i the right had side of S a, yields S X X a, ( ) X δa k, = X + Xk + a X + a Xk + a X + a Xk + a (.5)

6 54 S.G. Meitais δa = Eh [( X + X) h( X) + Xh ( X)] = E E + E X + X + a X + a ( X X + α E E E 3 ( X ( X + X ( X+ a) X (.6) However by recallig equatio (.) it follows that the first term of δ a coicides a () with ε a (), while the secod term is equal to a δε. Cosiquetly uder uit δ a expoetiality we have δ a = 0, ad substitutio i the right had side of (.5) implies that Sa, W( XJ, XK; a), ( ) < k x x W( x, x; a) = x + x + a x + a x + a ( x ( x The (commo) asymptotic distributio of S a, ad T a, follows the by the theory of U statistics, e.g. Severii (005), 3.4. Specifically, defie x = 0 σ a = g( x ) : W( x, x ) e dx, ad let : Var[ g( X )]. The D T,a N(0,4 σ a ). (.7) By straightforward algebra we have a+ x x a gx ( ) = e Γ(0, a+ x) + + aeγ(0, a), ( a + x) ( a + x) c t x Γ (, c x) = t e dt deotes the icomplete gamma fuctio. Although a aalytic expressio for may be tedious to derive, umerical values of the limit variace may be easily cal- σ a

7 A cosistet test for expoetiality based o the empirical momet process 55 culated i the computer as a fuctio of the Laplace parameter a aloe. Such values are show i Table. TABLE Asymptotic variace for the test statistic S,a with parameter a a Asympt. variace SIMULATIONS This sectio presets the results of a Mote Carlo study coducted to assess the fiite sample behavior of the ew test. The asymptotic test has ( α ) 00% reectio regio Za, > z, α Za, = S a, /σ a ad z α deotes the ( α ) 00% quatile of the stadard ormal diststributio. For compariso purposes correspodig results are also show for the classical goodess of fit tests based o the empirical distributio fuctio (EDF). The Kolmogorov Smirov (KS) statistic is KS = max{ D, D }, D = max U( ), D = max U ( ) the Crámer vo Mises (CM) statistic is U( ) =, CM = + ad the Aderso Darlig (AD) statistic is AD = U + + U ( )log ( ) (( ) ) log( ( )), = U= Y ad U( ) deote the correspodig order statistics, =,,...,. I fact we have used the modified statistics 0. 0,5 0.6 KS = KS 0.6, CM CM + + = + ad AD = AD( + (0.6/ )), which were proposed by D Agostio ad Stephes (986) i order to accommodate differecies i sample size. Percetage poits for the modified EDF statistics may be foud i D Agostio ad Stephes (986), Table 4..

8 56 S.G. Meitais Distributios cosidered are the Gamma with desity θ x θ Γ ( ) x e, x the Weibull with distributio fuctio ( e ), the Liear failure rate with desity θ x θ x (+ θ x)e, the half Normal distributio with desity / (/ π ) exp( x /), the Logormal with desity ( θx π ) exp[ log x/ θ ], ad the Iverge Gaussia distributio with desity / 3/ ( θ/ π) x exp[ θ( x ) / x]. These distributios are deoted by G, W, LF, HN, LN ad IG, respectively. Table shows results (percetage of reectio rouded to the earest iteger) obtaied from 0,000 samples of size = 50. For simplicity we write Z a for our asymptotic test. Compariso of the figures i Table idicate that the momet based test is quite efficiet i discrimiatig betwee the expoetial distributio ad some stadard alteratives. I additio, ad although the power of the ew test varies cosiderably with the Laplace parameter α, it may be see that a compromise value, say a =.0 or a =.5, produces a test that outperforms the classical tests based o the EDF uder most samplig situatios, ad ofte by a wide margi. TABLE Percetage of reectio observed at omial level % (left etry), 5% (middle etry) ad 0% (right etry), with sample size = 50 Model test Z0.5 Z.0 Z.5 Z.0 Z4.0 Z5.0 KS* CM* AD* G(.0) G(.6) G(.8) W(.3) W(.5) LF(.0) LF(.0) LN(.0) LN(0.75) HN IG(.0) IG(.5) Departmet of Ecoomics Natioal ad Kapodistria Uiversity of Athes SIMOS G. MEINTANIS

9 A cosistet test for expoetiality based o the empirical momet process 57 REFERENCES S. ASCHER, (990). A survey of tests for expoetiality, Commuicatios i Statistics - Theory ad Methods, 9, pp J. BARTOSZEWICZ, M. SKOLIMOWSKA, (008). New classes of life distributios related to the Laplace trasform, Commuicatios i Statistics - Theory ad Methods, to appear. M. CARRASCO, J.P FLORENS, (000). Geeralizatio of GMM to a cotiuum of momet coditios, Ecoometric Theory, 6, pp M. CARRASCO, J.P. FLORENS, (00). Simulatio based method of momets ad efficiecy, Joural of Busiess & Ecoomic Statistic, 0, pp R. D AGOSTINO, M.A. STEPHENS, (986). Goodess of Fit Techiques, Marcel Dekker, New York. N. HENZE, S.G. MEINTANIS, (005). Recet ad classical tests for expioetiality: a partial review with comparisos, Metrika, 6, pp R.H. RANDLES, (98). O the asymptotic ormality of statistics with estimated parameters, Aals of Statistics, 0, pp T.A. SEVERINI, (005). Elemets of Distributio Theory, Cambridge Uiversity Press, New York. J.D. SPURRIER, (984). A overview of tests for expoetiality, Commuicatios i Statistics - Theory ad Methods, 3, pp A.I. ZAYED, (996). Hadbook of Fuctio ad Geeralized Fuctio Trasformatios. CRC Press, New York. SUMMARY A cosistet test for expoetiality based o the empirical momet process A test for expoetiality is proposed which is cosistet withi the ewly defied class of LIFRA life distributios. The test may be viewed as a test for uiformity based o a cotiuum of momet coditios. The limit ull distributio of the test statistic is derived, ad the fiite-sample properties of the proposed procedures are ivestigated via simulatio.

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