ON THE ESTIMATION OF THE STRUCTURE PARAMETER OF A NORMAL DISTRIBUTION OF ORDER P

Transcription

1 STATISTICA, ao LX,. 1, 003 ON THE ESTIMATION OF THE STRUCTURE PARAMETER OF A NORMAL DISTRIBUTION OF ORDER P Agelo M. Mieo 1. INTRODUCTION I statistical iferece the usual hyotheses that we make o samle observatios are that they are draw from a oulatio distributed as a ormal, are homoskedastic ad ideedet. However, i may real situatios the hyothesis of ormality is ot met by data that we are rovided, so we have the roblem to fid alterative methods. I literature a used aroach is to aly robust methods. Aother aroach is to hyothesize a differet distributio for the observatios ad to look for derivig suitable methods by begiig from this hyothesis. This aroach ca be cosidered as alterative to the robust methods, sice rather tha referrig, imlicitly or exlicitly, to the theory of the so called outliers, seeks distributioal models more geeral tha the ormal oe, how is oited out by may researchers. Amog them, the most authoritative is certaily Sir Roald A. Fisher (19) that, agaist the ractice of excludig from aalysis the so called outliers, says this: as a statistical measure, however, the rejectio of observatios is too crude to be defeded ad uless there are other reasos for rejectio tha mere divergece from the majority, it would be more hilosohical to accet these extreme values, ot as gross errors, but as idicatios that the distributio of errors is ot ormal. I this sese, the family of ormal distributios of order (Viaelli, 1963; Luetta, 1963), kow as exoetial ower distributio i the aglo-saxo literature (Box ad Tiao, 199; Goi ad Moey, 1989), costitutes a valid alterative to the gaussia ormal distributio. A recet review o this distributio family ca be foud o Chiodi (000). I this aer it is faced the roblem of the arameter estimatio of a ormal distributio of order ad articularly that, harder, of the estimatio of the structure arameter, by comarig some of the most iterestig roosals existig i literature. I articular, after doig some cosideratios o the ormal distributio of order ad o the maximum likelihood estimators of its arameters, we comare three methods based resectively o the likelihood fuctio, o the rofile likelihood fuctio ad o the coditioal rofile likelihood fuctio ad a fourth suggestio based o the use of a articular idex of kurtosis.

2 110 A.M. Mieo. NORMAL DISTRIBUTION OF ORDER I 193 Subboti roosed a distributio family i which every comoet reresets a radom error distributio that geeralizes the ormal oe. Subboti, begiig from these two axioms: 1. the robability of a radom error deeds oly o the dimesio of the same error ad ca be exressed by a fuctio () havig the first derivative cotiuous i geeral;. the most robable value of a quatity, of which are kow direct measures, must ot deed o the used measure uit (i literature this axiom is kow as the Schiaarelli secod axiom); that are equal to those used by Gauss with the excetio of the secod art of the first axiom that i our case is more geeral (for this art Gauss settled dow the coditio that the best way to combie observatios is to use the arithmetic mea), has derived the distributio that has desity fuctio: mh m m f ( ) ex( h ) (1) (1/ m) I 1963 Luetta, followig the rocedure itroduced by Pearso (1895) to derive ew robability distributios, solves the differetial equatio: d log f log f - log a dx x c () that brigs to the desity fuctio: 1/ b f ( x ) ex( b x c ) (1 1/ ) (3) By usig a differet arameterizatio, the (3) assumes the form: 1 x µ f ( x ) ex 1/ (1 1/ ) (4) for -< x<, -< <, > 0 ad > 0, with E[ X ] xf ( x ) dx (5) the locatio arameter, 1/ 1/ {E[ X ]} { x f ( x ) dx } (6)

3 O the estimatio of the structure arameter of a ormal distributio of order 111 the scale arameter ad the structure arameter. It is easy to see how (1) coicides with (4) if we cosider the followig substitutios: =x-, m= e h=( 1/ ) -1. The distributios described by (4) have bee called by Viaelli (1963) ormal distributios of order. This distributio family, as is kow, describes both letokurtic (0 < < ) ad latykurtic ( > ) distributios, rovidig as secial cases the Lalace distributio for =1, the gaussia ormal distributio for = ad the uiform distributio for. It is worth otig how this distributio family is ofte used i literature oly for 1 <, sice these distributios reset heavier tails tha those oes of the ormal distributio (see amog others Hogg, 1974; D'Agostio ad Lee, 1977). However, the case of latykurtic ormal distributios of order is also iterestig by a ractical oit of view, sice i the reality we ca have samles with observatios that we ca thik draw either from letokurtic or latykurtic distributios, how verified by Cox (1967) that by checkig differet data sets says that the surrisig coclusio was that while there are frequet deartures from ormality these were about equally ofte toward logtailed ad toward short-tailed distributios, or by Box (1967), accordig to whom latykurtic distributios do occur i ractice because of deliberate or ucoscious trucatio ad these ought ot to be ruled out a riori. 3. ESTIMATION OF THE PARAMETERS OF A NORMAL DISTRIBUTION OF ORDER As it is kow, i Statistics the most used estimatio method is the maximum likelihood, because it rovides estimators with suitable roerties, at least asymtotically. The derivatio of the maximum likelihood estimators does ot give big roblems i the case of the ormal distributio of order arameters, eve though usually we obtai estimators ot exressible i a closed form. Ideed, let's suose to have a samle of i.i.d. observatios draw from (4): the the likelihood fuctio is give by: x 1/ 1 i i L( x ;,, ) [ (1 1/ )] ex ad the log-likelihood fuctio is give by: (7) x 1/ 1 i i l( x ;,, ) logl( x ;,, ) log[ (1 1/ )] (8) If we wat to determie the maximum likelihood estimators, we ca derive the log-likelihood fuctio resect to the three arameters (,, ) ad equal to zero the obtaied exressios:

4 11 A.M. Mieo l( x ) 1 1 xi sigx i 0 (9) i 1 l( x ) 1 x 0 1 i (10) i 1 l( x ) log (1 1/ ) i log i i log i 0 x x x x i 1 i 1 i 1 (11) with (.) the digamma fuctio, i.e. the first derivative of the logarithm of the gamma fuctio (Abramowitz ad Stegu, 197): l ( x ) '( x ) ( x ) (1) x ( x ) Equatios (9) ad (11) do ot give estimators i a closed form, while (10) gives the maximum likelihood estimator for : ˆ i 1 x i 1/ (13) The quatity ˆ is also called ower deviatio of order ad it ca be see as a geeral variability idex (Viaelli, 1963). It is also ossible to comute the iverse of the Fisher iformatio matrix (Agrò, 1995) that defies the asymtotic variace matrix of the maximum likelihood estimators ( ˆ, ˆ, ˆ ): I 1 ( )/ (1/ ) 0 0 ( 1) (1 1/ ) [ log (1 1/ )] [ log (1 1/ )] 1 0 (1 1/ ) '(1 1/ ) 1 (1 1/ ) '(1 1/ ) 1 3 [ log (1 1/ )] 0 (1 1/ ) '(1 1/ ) 1 (1 1/ ) '(1 1/ ) 1 (14) with (.) the trigamma fuctio, i.e. the secod derivative of the logarithm of the gamma fuctio. It has bee oted (Caobiaco, 000) how i geeral the asymtotic variace

5 O the estimatio of the structure arameter of a ormal distributio of order 113 of the maximum likelihood estimator of the ormal distributio of order scale arameter is larger tha the corresodig estimator of the Lalace distributio ad ormal distributio scale arameter; i fact, the suosed loss of efficiecy of the maximum likelihood estimator for is oly due to the eed of estimatig the structure arameter. Ideed, by suosig to kow the "true" value of the arameter, the iformatio matrix is give by: with I I I (15) I I I E ( )/ (1/ ) l f ( x ) ( 1) (1 1/ ) (16) I l f ( x ) I E 0 (17) l f ( x ) I E (18) ad, by ivertig the iformatio matrix, we obtai: ( )/ (1/ ) 0 1 ( 1) (1 1/ ) I (19) 0 that for =1 becomes: I (i this case to obtai the first elemet of the matrix, simle limit), while for = becomes: I 1 1,1, we have to comute a I 1 0 0

6 114 A.M. Mieo At this oit, it is easy to see how the asymtotic variace of the maximum likelihood estimator of the ormal distributio of order scale arameter, both for =1 ad for =, results equal to that of the corresodig maximum likelihood estimator of the scale arameter of the Lalace ad ormal distributio, resectively. Therefore, it is evidet that if we have relimiary iformatio such that we ca believe with a reasoable certaity that the samle at disosal has bee draw from a ormal or a Lalace distributio, the it is eedless to use ormal distributio of order. Maybe this relimiary iformatio could also cocer the locatio ad scale arameter ad so we do ot eed to make iferece at all. However, usually we do ot disose of this iformatio ad the it is ecessary to cosider also ukow. For the estimatio of, besides the use of the maximum likelihood estimator (Agrò, 1995), have bee roosed other two rocedures (Agrò, 1999) based the former o the rofile log-likelihood (Bardorff-Nielse, 1988): 1 l P( x ;,, ) log log (1 1/ ) log 1 log x i 1 i (0) the latter o the coditioal rofile log-likelihood (Cox ad Reid, 1987): 1 x 1 i i l CP( x ;,, ) log log ( 1) log[ ] x i 1 i 1 x 1 i i ( 1) log (1 1/ ) log 1 log (1) I the coditioal rofile log-likelihood the arameters of iterest have to be orthogoal to the uisace arameters ad the, i our case, has to be orthogoal to, i such a way makig ossible, accordig to Agrò, fiite estimates of eve for samles with smaller size (=30) tha those cosidered whe we use the (0) or whe we aly directly the (8). However, we thik that the requiremet of fiite estimates of is ot covicig, sice it is ossible to have values of, by defiig i this way a ormal distributio of order corresodig to a uiform distributio, as we have already see. Ayway, it is ecessary to ote that i order to estimate exist other roosals i literature, based o the comutatio of articular idices of kurtosis (see for examle Mieo A.M., 1994, 1995 ad 1996). These estimatio rocedures are to be see as rocedures based o the method of momets that first look for determiig the most roer value of by meas of the samle observatios, afterwards go through the use of the maximum likelihood estimators to estimate the locatio arameter ad the scale arameter. The idices of kurtosis more used for this aim are:

7 O the estimatio of the structure arameter of a ormal distributio of order 115 (1/ ) (5/ ) () 4 [ (3/ )] VI (1/ ) (3/ ) (3) (/ ) 1 (4) 1 whose momet estimators are give by: ˆ 4 ( x 1 i M ) i [ ( x ) ] i 1 i M (5) i ( x i M ) ˆ 1 (6) VI i 1 x i M ˆ ˆ x 1 i M i ˆ [ x ] i 1 i M ˆ 1 (7) From these relatioshis it is evidet the ature of the structure arameter, that essetially is itself a idex of kurtosis, ad the fact that is ositively correlated with, as it ca be see from (14), too. With these cosideratios it is evidet as a estimatio method for based o the use of a idex of kurtosis results very romisig. I articular, i the simulatio study described i the ext sectio we have used the estimatio rocedure of based o the idex of kurtosis Vˆ I, with value of M i the (6) give by the estimate of the locatio arameter of the corresodig ormal distributio of order (for more details o the method ad articularly o the reaso why we have chose the idex Vˆ I amog the others see Mieo A.M., 1996). Before edig this sectio, it is worth otig that about the estimatio roblem of the ormal distributio of order arameters has bee roosed recetly a aroach based o the use of a geetic algorithm (Vitrao ad Baragoa, 001) that, however, does ot reset substatial imrovemets i comariso to ay other umerical method used to solve the estimatio roblem by meas of the likelihood fuctio.

8 116 A.M. Mieo 4. SIMULATION PLAN AND RESULTS I order to comare the four differet aroaches to estimate the arameter, described i the revious sectio, we have coducted a simulatio study by drawig 1000 samles of size 10(10)50, 100, 00 from a ormal distributio of order, with values of =(1.5(0.5)3.5), locatio arameter =50 ad scale arameter =. The samles have bee draw by usig the method of the iverse of the distributio fuctio, by exloitig the relatioshi likig ormal distributio of order ad gamma distributio (Mieo A. 1978). Besides the arameter, the locatio ad the scale arameters have bee estimated by usig the maximum likelihood estimators. To solve these otimizatio roblems we have chose the simlex method (Nelder ad Mead, 1965) imlemeted i the fuctio otim() of the R software (Ihaka ad Getleme, 1996). Sice the simlex method is a ucostraied otimizatio method, a suitable rearameterizatio o the fuctios to be otimized that reset costrais o e has bee ecessary (see for a similar examle Everitt, 1987,. 3-35). The startig oits for the simlex method have bee the least squares estimates, i.e. resectively the arithmetic mea, the stadard deviatio ad =. The obtaied results seem very iterestig. While it seems that there are ot substatial differeces amog the aroaches i the estimatio of the locatio arameter (results ot show), there are some differeces i the estimatio of the remaiig two arameters. I articular, for the scale arameter we ca see (table 1) how for large samle sizes (=100, 00) the whole log-likelihood ad the rofile log-likelihood seem to give ubiased estimates, while the method based o the idex Vˆ I results cometitive. For these samle sizes the coditioal rofile log-likelihood behave i a good way, but for =1.5. Cocerig the estimate variaces, all the methods seem to rovide very similar values of variace. For smaller samle sizes (=10(10)50) all the estimates seem biased, with values related to the whole log-likelihood ad to the rofile log-likelihood biased to the to, while the values related to the coditioal rofile log-likelihood ad to the fourth method based o the idex Vˆ I seem biased to the bottom. However, it is worth otig that for sake of comariso we have used the relatioshi (13) for all the four methods, by relacig with the corresodig maximum likelihood estimate; but for small samle sizes some correctios are ecessary, sice the maximum likelihood estimator of is biased, as it is kow; i articular have bee roosed correctios that recall that oe used i the case of the maximum likelihood estimator of the variace (see Mieo A.M., 1996), or asymtotic correctio (see Chiodi, 1988), that seem to adjust, at least artially, the bias. However, these correctios would ot adjust the bias of the estimates derived by usig the whole log-likelihood or the rofile log-likelihood, sice they would icrease the mea values reorted i table 1. Cocerig the variaces of these estimates, it is worth otig as the lowest are that derived from the fourth method (estima-

9 O the estimatio of the structure arameter of a ormal distributio of order 117 tio based o the idex Vˆ I ), but for =10 where the lowest variaces are that derived from the coditioal rofile log-likelihood, that ayway have the great drawback described ext. TABLE 1 Arithmetic meas (M) ad variaces (V) of the ower deviatio of order estimates for the four used methods (I=whole log-likelihood, =rofile log-likelihood, =coditioal rofile log-likelihood, =method based o the idex VI ˆ ) =1.5 =.0 =.5 =3.0 =3.5 M( ˆ ) V( ˆ ) M( ˆ ) V( ˆ ) M( ˆ ) V( ˆ ) M( ˆ ) V( ˆ ) M( ˆ ) V( ˆ ) 10 I 0 I 30 I 40 I 50 I 100 I 00 I For the structure arameter, we ca ote as all the methods show some samles that joi the theoretical bouds that we imosed ca assume the arameter, i.e. 1 ad + (we have imosed to as lower boud 1 because eve if could assume values u to 0, defiig i this way ormal distributios of order of some statistical iterest sice we have cusidate distributios with tails heavier tha the Lalace distributio, the estimatio for 0< < 1 ivolves remarkable comutatioal roblems): we are ot articularly worried about this occurrece, sice for =1 ad for, as we have see, we have secial robability distributios very used i Statistics. It is evidet that a good estimator has to behave i such a way that these values do ot hae very ofte whe we have draw samles with a fiite value of 1. The simulatio results cocerig the estimatio of are reorted i tables ad 3.

10 118 A.M. Mieo TABLE Arithmetic meas (M) ad variaces (V) of the structure arameter estimates for the four used methods (I=whole log-likelihood, =rofile log-likelihood, =coditioal rofile log-likelihood, =method based o the idex VI ˆ ). =1.5 =.0 =.5 =3.0 =3.5 M( ˆ ) V( ˆ ) M( ˆ ) V( ˆ ) M( ˆ ) V( ˆ ) M( ˆ ) V( ˆ ) M( ˆ ) V( ˆ ) 10 I 0 I 30 I 40 I 50 I 100 I 00 I I table we have reorted the meas ad variaces of ˆ cosiderig all the values of ˆ, but ˆ > 10.0 that have bee cosidered as ˆ (for this roblem see Mieo A.M., 1996). From these values we ca see how for large samle sizes (=100, 00) the method that seems to give the best values of ˆ is that based o the use of the idex of kurtosis Vˆ I. For smaller samle sizes (=10(10)50) the fourth method seems agai better tha the others three, also cosiderig that the mea comuted o ˆ suffers of the great skewess of its samlig distributio: ideed, i a revious simulatio study (Mieo A.M., 1995) we have already oted as from the values of ˆ obtaied with the fourth method we get frequecy distributios with mode cetered o the true value of used to geerate the samles of seudo-radom observatios. Ayway, we cosider really iterestig the results show o table 3, that reorts the ercetage of values with ˆ > 10.0 (ad the ˆ ) ad values with ˆ < 1.01 (ad the ˆ =1). I fact, i this table we ca ote as i the case of coditioal rofile log-likelihood it is very high the ercetage of values with ˆ =1: for samle sizes =10 this ercetage varies from 80% to 90%, with decreasig

11 O the estimatio of the structure arameter of a ormal distributio of order 119 ercetages as icreases, but that are very high, ayway. Accordig to our oiio, this behaviour shows a fudametal iadequacy of the coditioal rofile log-likelihood to estimate, esecially whe we have samles with medium-small sizes. TABLE 3 Percetages of samles that reset estimates of the structure arameter greater tha 10.0 or lower tha 1.01 for the four used methods (I=whole log-likelihood, =rofile log-likelihood, =coditioal rofile log-likelihood, =method based o the idex VI ˆ ). =1.5 =.0 =.5 =3.0 =3.5 ˆ >10.0 <1.01 >10.0 <1.01 >10.0 <1.01 >10.0 <1.01 >10.0 < I I 30 I 40 I 50 I 100 I 00 I Cocerig the methods based o the whole log-likelihood ad o the rofile log-likelihood, these seem to have the oosite drawback, that is it seems very high the ercetage of samles with ˆ > 10.0, with results that seem better for the method based o the whole log-likelihood, that have yet the drawback of a greater comutatioal comlexity, at least i comariso to the aroach based o the rofile log-likelihood. The method to estimate based o the idex Vˆ I seems the best amog the four, havig either a great umber of samles with ˆ =1, either with ˆ. Therefore, by summig u all the cosideratios doe so far, the fourth method is surely to refer i comariso to the others three.

12 10 A.M. Mieo 5. CONCLUSION I this aer we have see how the family of ormal distributios of order costitutes a valid geeralizatio of the hyothesis of ormality that usually is made ad that ofte it is ot sustaiable o data we have at disosal. The use of the ormal distributios of order costitutes also a arametric alterative to the robust methods, that referrig, imlicitly or exlicitly, to the theory of the so called outliers do ot seem suitable to a rofitable use i the scietific research. How is dagerous the ractice to elimiate automatically the outliers is testified by the followig real evet (Faraway, 000, ag. 70): NASA lauched the Nimbus 7 satellite to record iformatio o the terrestrial atmoshere. After several years of oeratio, i 1985 the British Atarctic Survey observed a large decrease of the level of the atmosheric ozoe over the Atarctica. NASA astoished o the fact that its satellite did ot record such aomaly ever: by examiig more careful the satellite data it was foud that the data rocessig rogram automatically discarded extremely low observatios, assumig that they were wrog recordigs. With good reaso we ca believe that this drawback retarded the discovery of the so called ozoe hole over the Atartica, delayig, as a result, the adotio of the correct olicies to try to reduce it (for examle, by baig CFC ). Cocerig the ormal distributios of order, however exist some ractical roblems still oe that have recluded a wide use, so far: oe of these roblems is the estimatio of the structure arameter. I this aer we have comared some iterestig roosals amog those existig i literature ad i articular the estimatio based o the whole log-likelihood, o the rofile log-likelihood, o the coditioal rofile log-likelihood ad o the idex of kurtosis Vˆ I. The results of a simulatio study show as the best method is that oe based o the idex of kurtosis Vˆ I, while, if for ay articular reaso, we wat to use a aroach based o the likelihood fuctio, the whole log-likelihhod seems behave better tha the rofile log-likelihood ad the coditioal rofile log-likelihood. Diartimeto di Scieze Statistiche e Matematiche Silvio Viaelli Uiversità di Palermo ANGELO M. MINEO REFERENCES M. ABRAMOWITZ, I.A. STEGUN (197), Hadbook of mathematical fuctios, Dover Pubblicatios, New York. G. AGRÒ (1995), Maximum likelihood estimatio for the exoetial ower distributio, Commuicatios i Statistics (simulatio ad comutatio), 4, art, G. AGRÒ (1999), Parameter ortogoality ad coditioal rofile likelihood: the exoetial ower fuctio case, Commuicatios i Statistics (theory ad methods), 8, art 8, O.E. BARNDORFF-NIELSEN (1988), Parametric statistical models ad likelihood, Lecture otes i statistics, Sriger, Heidelberg. G.E.P. BOX (1967), Discussio o Toics i the ivestigatio of liear relatios fitted by the method of least squares, Joural of the Royal Statistical Society, Serie B, 9, art 1,

13 O the estimatio of the structure arameter of a ormal distributio of order 11 G.E.P. BOX, G.C. TIAO (199), Bayesa iferece i statistical aalysis, J. Wiley, New York. R. CAPOBIANCO (000), Robustess asects of the geeralized ormal distributio, Quaderi di Statistica, Cetro er la formazioe i Ecoomia e Politica dello sviluo rurale, Diartimeto di Scieze Statistiche dell'uiversità di Naoli "Federico ", Diartimeto di Scieze Ecoomiche dell'uiversità di Salero,., M. CHIODI (1988), Sulle distribuzioi di camioameto delle stime di massima verosimigliaza dei arametri delle curve ormali di ordie, Istituto di Statistica di Palermo, Palermo. M. CHIODI (000), Le curve ormali di ordie come distribuzioi di errori accidetali: ua rassega dei risultati e roblemi aerti er il caso uivariato e er quello multivariato, i Atti della XL Riuioe Scietifica della SIS, Fireze, D.R. COX (1967), Discussio o Toics i the ivestigatio of liear relatios fitted by the method of least squares, Joural of the Royal Statistical Society, Serie B, 9, art 1,. 39. D.R. COX, N. REID (1987), Parameter orthogoality ad aroximate coditioal iferece, Joural of the Royal Statistical Society, Serie B, 49, R.B. D AGOSTINO, A.F.S. LEE (1977), Robustess of locatio estimators uder chages of oulatio kurtosis, Joural of the America Statistical Associatio, 7, B.S. EVERITT (1987), Itroductio to otimizatio methods ad their alicatio i statistics, Chama ad Hall, Lodo. J.J. FARAWAY (000), Practical regressio ad aova usig R, Published o the URL: htt://www. stat.lsa.umich.edu/~ faraway/book/. R.A. FISHER (19), O the mathematical foudatios of theoretical statistics, Philosohical Trasactios of the Royal Society of Lodo, Serie A,, R. GONIN, A.H. MONEY (1989), Noliear L -orm estimatio, M. Dekker, New York. R.V. HOGG (1974), Adative robust rocedures: a artial review ad some suggestios for future alicatios ad theory, Joural of the America Statistical Associatio, 69, R. IHAKA, R. GENTLEMAN (1996), R: A laguage for data aalysis ad grahics, Joural of Comutatioal ad Grahical Statistics, 5, G. LUNETTA (1963), Di ua geeralizzazioe dello schema della curva ormale, Aali della Facoltà di Ecoomia e Commercio di Palermo, 17, A. MINEO (1978), Protuari delle robabilità itegrali delle curve ormali di ordie r comrese fra k r r e criteri er la loro valutazioe e il loro imiego, Edigrahica Sud Euroa, Palermo. A.M. MINEO (1994), U uovo metodo di stima di er ua corretta valutazioe dei arametri di itesità e di scala di ua curva ormale di ordie, i Atti della XXXV Riuioe Scietifica della SIS,, Sa Remo, A.M. MINEO (1995), Stima dei arametri di itesità e di scala di ua curva ormale di ordie ( icogito), Aali della Facoltà di Ecoomia e Commercio di Palermo (Area Statistico- Matematica), 49, A.M. MINEO (1996), La migliore combiazioe delle osservazioi: curve ormali di ordie e stimatori di orma L, Tesi di Dottorato di Ricerca i Statistica Comutazioale e Alicazioi, Uiversità degli Studi di Naoli "Federico ". J.A. NELDER, R. MEAD (1965), A simlex method for fuctio miimizatio, The Comuter Joural, 7, K. PEARSON (1895), Cotributios to the mathematical theory of evolutio.. Skew variatio i homogeeous material, Philosohical Trasactios of the Royal Society of Lodo, Serie A, 186, M.T. SUBBOTIN (193), O the law of frequecy of errors, Matematicheskii Sborik, 31, S. VIANELLI (1963), La misura della variabilità codizioata i uo schema geerale delle curve ormali di frequeza, Statistica, 3, S. VITRANO, R. BARAGONA (001), The geetic algorithm estimates for the arameters of order ormal distributios, i Book of Short Paers CLADAG001, Palermo,

14 1 A.M. Mieo RIASSUNTO Sulla stima del arametro di struttura di ua distribuzioe ormale di ordie I questo lavoro si cofrotao quattro differeti arocci er la stima del arametro di struttura di ua distribuzioe ormale di ordie (sesso chiamata ella letteratura aglosassoe exoetial ower distributio). I articolare, abbiamo cosiderato la massimizzazioe della log-verosimigliaza, della log-verosimigliaza rofilo, della log-verosimigliaza rofilo codizioale e u metodo basato su u idice di curtosi. I risultati di uo studio di simulazioe sembrao idicare la sueriorità dell'ultimo aroccio. SUMMARY O the estimatio of the structure arameter of a ormal distributio of order I this aer we comare four differet aroaches to estimate the structure arameter of a ormal distributio of order (ofte called exoetial ower distributio). I articular, we have cosidered the maximizatio of the log-likelihood, of the rofile loglikelihood, of the coditioal rofile log-likelihood ad a method based o a idex of kurtosis. The results of a simulatio study seem to idicate the latter aroach as the best.