DISCRIMIATIG BETWEE THE OGISTIC AD THE ORMA DISTRIBUTIOS BASED O IKEIHOOD RATIO F. Aucoi 1 ad F. Ashkar ABSTRACT The problem of discrimiatig betwee the ormal ad the logistic distributios with ukow parameters ad based o the likelihood ratio statistic is cosidered. The mai iterest is i determiig the probability of correct selectio (PCS) betwee these two distributios. Asymptotic approximatios to these PCS s are derived ad the assessed usig Mote Carlo (MC) simulatios. The MC simulatios suggest the asymptotic approximatios to be ureliable for sample sizes below 100. Results also reveal a clear difficulty i discrimiatig betwee the two distributios for sample sizes smaller tha 50. KEYWORKDS: ikelihood ratio; Discrimiatio; ormal ad logistic distributios; Asymptotic approximatio; Mote Carlo simulatios. 1 Départemet de mathématiques et de statistique, Uiversité de Mocto, Mocto, B, Caada, E1A 3E9; PH (506) 858-498; FAX (506) 858-4396; EMAI frak.aucoi@gmail.com Départemet de mathématiques et de statistique, Uiversité de Mocto, Mocto, B, Caada, E1A 3E9; PH (506) 858-431; FAX (506) 858-4396; EMAI ashkarf@umocto.ca
1. ITRODUCTIO I may fields of applicatio requirig frequecy modelig, practitioers are faced with the problem of selectig a suitable statistical frequecy distributio to fit their data. This selectio is ofte doe o the basis of goodess-of-fit (GOF) statistics (e.g. likelihood ratio) ad diagostic plots (e.g. probability-probability plots). However, i most situatios, the available sample is small (0 to 50 observatios), which reders the selectio to be ot very reliable. The problem of choosig betwee frequecy distributios has bee addressed early i the statistical literature. Cox (1961, 196), Dumoceaux ad Atle (1973), ad Kappema (198, 1989) all proposed approaches based o the likelihood fuctio for discrimiatig betwee two distributios. Cox (1961, 196) tackled the problem of discrimiatig betwee the logormal ad the expoetial distributios based o the likelihood fuctio. He derived a asymptotic probability distributio for the likelihood ratio statistic, thus allowig calculatio of the "probability of correct selectio" (PCS) give a large sample. Ispired by Cox s work, Jackso (1968) derived asymptotic results for the case logormal vs gamma. White (198a, 198b) provided regularity coditios ad proofs for Cox s work. All of the previous results, beig based o asymptotic theory, require that the sample be fairly large for the PCS approximatios to be of ay use. Moreover, for may pairs of distributios, PCS approximatios caot be obtaied i closed form. The latter two facts, i cojuctio with the arrival of persoal computers, led some researchers to resort to Mote Carlo (MC) simulatios to determie the PCS. Usig simulatio, ad based o the likelihood ratio statistic, the case logormal vs Weibull (equivalet to ormal ad Gumbel) was addressed by Dumoceaux ad Atle (1973), while Bai ad Egelhardt (1980) covered the case Weibull vs gamma. Kappema (198) later studied the PCS for the pairs Weibull vs logormal, Weibull vs gamma, ad gamma vs logormal. Recetly, Qaffou ad Zoglat (008) derived asymptotic results for approximatig the PCS based o likelihood ratio i the case Gumbel vs ormal. Iterestigly, their MC simulatios showed the approximatio to be quite accurate eve for sample sizes as small as 0.
I the preset paper, the PCS for the pair ormal vs logistic based o the likelihood ratio statistic is studied, assumig that the parameters of both distributios are ukow. Both the ormal ad the logistic distributios are of the locatio-scale type, ad are symmetric, but the logistic distributio has thicker tails ad higher kurtosis tha the ormal. The applicatios of the ormal distributio are umerous ad well documeted, whereas the logistic distributio, which origiated with Verhulst's work o demography i the early 1800s, has had other applicatios i fields such as epidemiology, psychology, techology ad eergy (Marchetti, 1977; Fisher ad Pry, 1971; Modis, 199). The desity fuctio of a logistic radom variable X ~ (, ) OG α is give by x a exp f( x; α, )=, x (1) x a 1+ exp ad the maximum likelihood estimators (ME's) ˆα (locatio) ad ˆ (scale) ca be determied iteratively by maximizig the log-likelihood fuctio, give by for α ad simultaeously. l ( α, ) = l f ( x; α, ), () i i= 1 The desity fuctio of a ormal radom variable ~ (, ) X μ σ is give by 1 ( x μ) f ( x; μσ, )= exp, x σ π σ (3) where μ is a locatio parameter ad σ is a scale parameter. The maximum likelihood estimators (ME's) of μ ad the variace σ, based o a radom sample 1 X, X,..., X, are give by 1 1 ˆ μ = ˆ ˆ σ = μ X ad ( ) i Xi (4) i= 1 i= 1 I the preset paper, i order to discrimiate betwee the ormal ad the logistic distributios, asymptotic approximatios of the PCS are derived, ad their reliability is the 3
assessed usig MC simulatios. Simulatio results suggest the asymptotic approximatios based o the likelihood ratio to be ureliable for samples cosistig of less tha 100 observatios. Moreover, due to shape similarities betwee the ormal ad the logistic distributios, the results show a clear difficulty i discrimiatig betwee these two distributios for sample sizes less tha 50. It is to be oted that i istaces whe a o-egative radom variable is eeded, it is discrimiatio betwee the logormal ad the log-logistic distributios that might be of iterest, but i such a case the results of the preset study remai applicable because the ormal ad the logormal distributios (also the logistic ad the log-logistic) are liked by a simple logarithmic trasformatio. The log-logistic distributio has bee proposed, for example, by Ashkar ad Mahdi (003, 006) for fittig extreme hydrological variables such as flood flows or low stream flows. The logormal distributio has bee used i the statistical literature for all sorts of applicatios.. PCS BASED O THE IKEIHOOD RATIO STATISTIC: ASYMPTOTIC RESUTS Assume that the radom sample X1, X,..., X is kow to come from either a ormal X μ σ, or a logistic distributio, X ~ (, ) distributio, ~ (, ) OG α. The log-likelihood ratio statistic, T, is defied as the logarithm of the ratio of two maximized likelihood fuctios: μσ ad (, ) where (, ) T = l ( ˆ μσ, ˆ ) ( ˆ α, ˆ ) α correspod to the likelihood fuctios uder a ormal distributio ad a logistic distributio, respectively. The decisio rule for discrimiatig betwee the ormal ad the logistic distributios is to choose the ormal if T > 0, ad to reject the ormal i favor of the logistic, otherwise. Because both of these two distributios are of the locatioscale type, oe importat property of the T statistic is that it is idepedet of the parameters of both distributios (Dumoceaux et al, 1973). (5) 4
.1 OGISTIC AS THE TRUE DISTRIBUTIO It is first supposed that the X i s come from the distributio X ~ (, ) OG α, with ukow parameters α ad. With o loss of geerality, these locatio ad scale parameters ca be fixed at α = 0 ad = 1, so that X ~ OG (0,1). Based o Qaffou ad Zoglat (008), the followig theorem ca be stated: Theorem 1 et the radom sample X 1, X,..., X come from a logistic distributio OG( α, ), the the T statistic asymptotically follows a ormal distributio with mea E [ T ] ad variace [ ] Var T, which is derived below. The proof of this theorem is straightforward from the followig lemma ad the cetral limit theorem. emma 1 Uder Theorem 1, whe (White, 198b): (1) ˆ ˆ μ μ ad σ σ where μ ad σ are fuctios of α ad ad satisfy the followig relatio () If T [ ] T E T = l. ( μσ ) = ( μσ ) E l f X;, max E l f X;, μσ, ( μσ, ) ( α, ), the [ ] T E T is asymptotically equivalet to From this lemma, proof of theorem 1 is established by provig that [ ] T E T is asymptotically ormal based o the cetral limit theorem. As for the eeded quatities μ ad i lemma 1, ad E [ T ] ad [ ] ad performig the followig calculatio: Var T i theorem 1, they are derived by first referrig to lemma 1 σ 5
l( π) ( X μ) E l f ( X; μσ, ) = E l( σ) σ l(π ) E ( X μ) = l( σ ) (6) σ l( π ) α ψ '(1) μα μ = l( σ ) + σ σ σ σ where ψ is the trigamma fuctio give by ( ) ( ) ψ k = d log Γ k / dk (Abramowitz & Stegu, 197), for which it is kow that ψ (1) = π / 6. Maximizig with respect to μ ad σ, i.e. by equatig the derivatives with respect to μ ad σ to zero, the followig result is obtaied ( ) = ad = '(1) = 3 (7) μ α σ ψ π otatio: By the secod poit of lemma 1, E (T) ad Var (T) are calculated for. Usig the AM [ T] E = lim, ad AV AM [ T] [ ] E E f X f X Var T = lim, for sufficietly large: l (, μσ, ) l (,0,1) l( π) ( X μ) = E l( σ ) + X + l 1+ exp X σ l(π) l( ψ '(1) ) 1 = + E l ( 1 + exp [ X] ) = 0.014367 ( [ ]) (8) [ T ] Var AV Var f X f X l (, μσ, ) l (,0,1) ( X μ) = Var + X + l 1+ X σ = 0.04881615 ( exp[ ]) (9) 6
Expressio (9) had to be evaluated umerically, while the term E l ( 1+ exp[ X] ) (8) could be evaluated usig the properties of its momet geeratig fuctio; that is: i which implies that + x e Γ(1 t) Ml( 1 exp[ ]) () t = dx + X = (10) x t (1 + e ) Γ( t) E dm ( [ ]) ( t = 0) l 1+ exp X l ( 1 exp[ X] ) + = = 1 (11) dt Fially, the asymptotic probability of correct selectio, give that the true sampled distributio is logistic ad that the competig distributio is ormal, is give by ( ) ( T ) E T AM PCS = P[ T < 0] 1 Φ = 1 Φ =Φ 0.065 Var AV where Φ is the distributio fuctio of the stadard ormal distributio. ( ) (1). ORMA AS THE TRUE DISTRIBUTIO It is ow supposed that the X i s come from the distributio X ~ ( μ, σ ) with ukow μ ad σ. With o loss of geerality, the locatio ad variace parameters are fixed at μ = 0 ad σ = 1, so that ~ (0,1) X. As before, based o Qaffou ad Zoglat (008), the followig theorem ca be stated: Theorem et the radom sample X 1, X,..., X come from a ormal distributio ( μ, σ ), the the T statistic follows asymptotically a ormal distributio with mea E [ T ] ad variace Var [ ] T, which is derived below. Oce more, the proof of this theorem is straightforward from the cetral limit theorem ad the followig lemma: emma Uder Theorem, as (White, 198b): (1) ˆ α α ad ˆ where α ad are fuctios of μ ad σ ad satisfy the followig relatio 7
() If T = l [ ] T E T. ( α ) ( α ) E l f X,, = max E l f X,, α,. ( μσ, ) ( α, ), the [ ] T E T is asymptotically equivalet to It is ow possible to evaluate α ad by referrig to lemma ad performig the followig calculatio: X α X α E[ l f( X; α, )] = E + l( ) l 1+ exp μ α X α = + l( ) E l 1+ exp (13) However, a problem is ecoutered at this stage, sice the quatity E X α l 1+ exp (14) caot be computed aalytically. A approximatio of this quatity is obtaied by usig the X α delta method; i.e. by approximatig g( X) = l 1+ exp up to the secod order, which gives by a Taylor series expasio E μ α σ exp μ α μ α 1+ exp [ g( X) ] l 1+ exp + ad this leads to the followig approximatio: (15) [ l ( ; α, )] + l( ) l 1+ exp + E f X μ α σ exp μ α μ α (16) μ α 1+ exp 8
Due to the complexity of (16), the use of the Maple software to fid the roots of [ α ] de l f ( X;, ) dα= 0 was ecessary. Maple provided three roots, oe of which correspoded to α = μ. The other two roots were far too complex to be cosidered useful i this case. Moreover, the solutio α = μ is cosistet with the fact that both μ ad α correspod to locatio parameters of symmetric distributios; i.e. the ormal ad logistic distributios. Thus, maximizig with respect to α ad, i.e. by solvig [ l ( ; α, )] α= 0 ad [ α] de f X d results are obtaied: E AV [ T] de l f ( X;, ) d= 0 for α ad, the followig σ α = μ ad = (17) It is ow easy, usig umerical itegratio, to determie the asymptotic quatities ad AM Var [ T ] [ T] (as was doe aalytically i the previous sectio): E E l f( X;0,1) l f( X, α, ) l(π) X X α X α = E + + l( )+ l 1+ exp l(π ) 1 l() = + E l( 1+ exp( X ) ) = 0.03981169 [ T ] Var Var l f( X ;0,1) l f( X, α, ) l(π) X X α X α = E + + l( )+ l 1+ exp AM = 0.041959 (18) (19) 9
E ote that umerical itegratio is ecessary here because the term ( + ( X )) l 1 exp has o closed form. Fially, the asymptotic probability of correct selectio, give that the true sampled distributio is ormal ad that the competig distributio is logistic, is give by ( ) ( T) E T AM PCS = P[ T > 0] Φ =Φ =Φ 0.1938 V AV ( ) (0) 3. PCS CACUATED BY MC SIMUATIOS Whe the sample size is ot sufficietly large, the PCS s ivolved i the discrimiatio betwee the ormal ad the logistic distributios based o likelihood ratio ca be determied with more accuracy through MC simulatios. Whe the true distributio is logistic, for example, computatio of the PCS is performed as follows: 1. For a sample size, a radom sample { } x, x,..., x is geerated from a OG(0,1) distributio.. By maximum likelihood, both the logistic ad the ormal distributios are fitted to the sample { } 1 1 x, x,..., x, ad a realizatio, t, of the statistic T, is calculated ad stored. 3. Steps 1 ad are repeated may times (i this study, the repetitio was doe 10,000 times). 4. The approximate PCS uder the assumptio that the true distributio is logistic, is: PCS = Pr[ T < 0] ( umber of t values i step < 0)/10,000. 4. RESUTS The PCS results obtaied from the asymptotic approximatios ad by MC simulatios (carried out usig the statistical freeware R, 009) are preseted i Table 1 for various sample sizes. 10
Table 1: PCS s obtaied from both the asymptotic ad the MC simulatio results uder the assumptio that the true sampled distributio is logistic (ormal). Sample size () MC Asymptotic 10 0.31 (0.80) 0.58 (0.73) 0 0.4 (0.77) 0.61 (0.81) 30 0.49 (0.77) 0.64 (0.86) 40 0.53 (0.77) 0.66 (0.89) 50 0.57 (0.78) 0.68 (0.91) 60 0.61 (0.79) 0.69 (0.93) 70 0.63 (0.80) 0.71 (0.95) 80 0.65 (0.80) 0.7 (0.96) 90 0.67 (0.8) 0.73 (0.97) 100 0.69 (0.83) 0.74 (0.97) 150 0.76 (0.86) 0.79 (0.99) 00 0.81 (0.88) 0.8 (1.00) 300 0.88 (0.9) 0.87 (1.00) 400 0.9 (0.95) 0.90 (1.00) 500 0.95 (0.96) 0.93 (1.00) 1000 0.99 (0.99) 0.98 (1.00) Compariso betwee the MC simulatios ad the asymptotic results shows the asymptotic approximatios to be ureliable for sample sizes below 100 whe the true sampled distributio is logistic, ad for eve larger samples whe the true sampled distributio is ormal. It is also to be remembered that whe the true sampled distributio is ormal, the accuracy of the calculated asymptotic PCS's is affected by the Taylor series approximatio that was used i Eq. (15). Examiatio of the PCS's calculated by MC simulatio also reveals a clear difficulty i selectig betwee the ormal ad the logistic distributios for sample sizes smaller tha 50. For example, compariso of the results of Table 1 with those of Kappema (198) or Qaffou ad Zoglat (008) shows that selectio betwee the ormal ad the logistic distributios (logormal ad log-logistic) is much more difficult tha betwee the ormal ad the Gumbel (logormal ad Weibull). If both the Type 1 ad Type errors are to be less tha 10 %, the miimum sample size eeded to choose betwee the ormal ad the logistic models is betwee 300 ad 400 11
(Table 1), whereas the eeded sample size to choose betwee the ormal ad the Gumbel models is less tha 50 (Qaffou ad Zoglat, 008; Kappema, 198). Moreover, Table 1 shows that selectio based o likelihood ratio favors the ormal model over the logistic model, because the probability of correctly selectig the ormal model whe it is the true sampled distributio is systematically greater tha the probability of correctly selectig the logistic model whe it is the true sampled distributio. By compariso, the results of Kappema (198) ad of Qaffou ad Zoglat (008) did ot show a favorig of the ormal model over the Gumbel model i this respect. 5. COCUSIOS The ormal ad the logistic distributios are ofte cosidered as competig models whe the variable of iterest takes values from to +. I istaces whe a o-egative radom variable is eeded, it is choosig betwee the logormal ad the log-logistic distributios that might be of iterest. The results of the preset study have show that with small sample sizes such as those commoly ecoutered i practice, it is difficult to choose betwee the ormal ad the logistic models (logormal ad log-logistic) based o the likelihood ratio statistic. It would be iterestig to seek other methods that may provide higher discrimiatio power betwee the two models, although due to their shape similarity it is doubtful that sigificatly higher discrimiatio power could be achieved with relatively small samples. ACKOWEDGMETS The fiacial support of the atural Scieces ad Egieerig Research Coucil of Caada is gratefully ackowledged. 1
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