On Interval Estimation for Exponential Power Distribution Parameters

Transcription

1 Joural of Data Sciece 8(08), O Iterval Estimatio for Expoetial Power Distributio Parameters A. A. Olosude ad A. T. Soyika Abstract The probability that the estimator is equal to the value of the estimated parameter is zero. Hece i practical applicatios we provide together with the poit estimates their estimated stadard errors. Give a distributio of radom variable which has heavier tails or thier tails tha a ormal distributio, the the cofidece iterval commo i the literature will ot be applicable. I this study, we obtaied some results o the cofidece procedure for the parameters of geeralized ormal distributio which is robust i ay case of heavier or thier tha the ormal distributio usig pivotal quatities approach, ad o the basis of a radom sample of fixed size. Some simulatio studies ad applicatios are also examied. :Shape parameter, Short tails, expoetial power distributio, cofidece iterval. Itroductio The higher the degree of cofidece, t he l arger t he p ercetage o f p opulatio values that the iterval is to cotai. Thus, if we wat a high degree of cofidece a d a high cofidece l evel, w e a re g oig f ar o ut i to t he t ail o f s ome d istributio. For example, if the distributio has heavier tails tha a ormal distributio ad we believe we are costructig a 99 percet cofidece i terval t o c otai a t l east 9 9 p ercet of the populatio, the percetage of the time we do that, say, 000 radomly costructed itervals actually cotai at least 99 percet of the populatio, the result obtaied ofte times could be far less tha 99 percet that we really required. Coversely, if the populatio distributio has much lighter tails tha a ormal distributio, the iterval could be much wider tha ecessary. It is importat to remember that if the

2 94 O Iterval Estimatio for Expoetial Power Distributio Parameters populatio X is ot ormal, the sample mea x is ot ormal as well, ad so the radom variable (x µ)/(s/ ) does ot have the commo t distributio, implyig that the commoly used t table values are ot ecessarily correct. I this paper we developed a robust cofidece iterval for the uivariate family of elliptical desity called expoetial power distributio (EPD) which has a ormal ad laplace as its sub-family ad it is well kow for the flexibility of its tails. Usig the widely kow pivotal quatities method, the resultig robust cofidece iterval we aticipated will replace the commo oe i literature about the ormal distributio especially whe the observed sample data i a experimet has tail heavier or thier tha the usual ormal distributio. The ui-dimesioal expoetial power distributio is defied as f(x; µ,, ) = ( ) Γ + + { exp x µ } () where the parameters µ R ad (0, ) are respectively scale ad locatio parameters ad (0, ) is the shape parameter which regulates the tails of the distributio such that whe = the desity () is ormal, but for = / we have double expoetial distributio. The distributio () was first itroduced by Subboti (93), it has bee used i robust iferece (see Box(953)) where the parameters of the distributio were estimated via momets. If a radom variable X has the pdf () the its mth momets ca be obtaied from the relatio ( ( ) ( )) E(X m ) = [ m ((z) µ) m ] + ((z) + µ) m z exp z 0 Γ( ) dz () I additio, its cetral momet estimates Agro (99, 995) are: E(X) = µ ; E X E(X) = Γ( ) Γ( ) ; V ar(x) = Γ( 3 Γ( ) ) E(X E(X)) 3 = 0 ; E(X E(X)) 4 = 4 4 Γ( 5 ) ; ad Kurtosis= Γ( 5 )Γ( ). Γ( ) Γ ( 3 ) The results idicate that the sample mea X is the estimate of the true mea µ while the shape parameter ca be umerically obtaied from the estimate of the kurtosis. Substitutig shape parameter estimate ito Var(X) we estimate the scale parameter. Also the log-likelihood fuctio for radom samples x, x,.., x from () is: LogL(µ,, ) = l ( Γ( + )+ ) i= The derivatives ( of (8) with respect to µ,, ad are LogL µ = (x i µ) ) (x i µ) ; LogL = + x i µ x i<µ LogL = [Ψ( + i= ) + ] xi µ l xi µ i= i= ; x i µ x µ ; ad (3)

3 A. A. Olosude ad A. T. Soyika 95 Fially the expected fisher iformatio matrix of EPD are E( LogL µ ) = ( ) Γ( Γ ) E( LogL ) = ( + Ψ( + ) l ); ad ( E( LogL ) = + Ψ( + 3 ) + Ψ (+ ; E( LogL ) = ; ) ) ( ) (l ) + 4 Ψ ( + 3 ) + ( + Ψ ) Mieo ad Ruggieri (003a,b) developed codes i R programmig eviromet to estimate these parameters from ay give sample from (), this also icludes the parameter which has o explicit solutio. Cofidece Iterval Defiitio.0.(Cofidece Iterval). Let X, X,..X be a radom sample from the desity f(x θ). Let a = t (X, X,..X ) ad b = t (X, X,..X ) be two statistics satisfyig the relatio a b for which P r {a < Z(X θ) < b} γ, where γ does ot deped o θ; the radom iterval (a,b) is called 00γ percet cofidece iterval for Z(X θ); γ is called the cofidece coefficiet; a ad b are called the lower ad upper cofidece limits, respectively for Z(X θ). A value (t, t ) of the radom iterval (a, b) is also called a 00γ percet cofidece iterval for Z(X θ). Ram et.al. (00) i his paper provided ew cofidece itervals (CIs) based o F-approximatios as well as ormal approximatios. Riggs (05), showed from the simulatio results which idicated that the score ad likelihood ratio itervals are geerally preferable over the Wald iterval.. Pivotal Quatity I this preset study we used the pivotal quatity approach to derive the cofidece iterval for the expoetial power distributio. Defiitio..(Pivotal Quatity). A pivotal quatity (Z) for a parameter θ is a radom variable Z(X θ) whose value depeds both o (the data) X ad o the value of the ukow parameter θ but whose distributio is kow to be idepedet of θ. For the case of the ormal distributio N(µ, ), the pivotal quatity Z = X µ ad Z = (X µ) has distributio N(0, ) ad χ that are idepedet of µ ad ad as such both are pivotal quatity for µ ad respectively. We establish some importat theorem i this sectio that served as a foudatio to the results obtaied Propositio.: Let X has a pdf (), the Γ(, ). Where µ,,ad, X µ ad δ are locatio, scale ad shape parameters respectively. Values for Gamma(.) for various ca be obtaied from Abramowitz ad Stegu (963), Paris (00) ad Wiitzki (003). Proof: By trasformatio techiques, we have that f Y (y) = d dy g (y) f X (g (y)) = Γ(, ), y > 0; the pdf () is a three parameter family, θ = (µ,, ). We ca deduce from above propositio. that

4 96 O Iterval Estimatio for Expoetial Power Distributio Parameters Corollary.: Let X has a pdf () the, Z = x µ EP D(0,, ) (4) ad x µ Γ(, ) are pivotal quatities ad idepedet. Propositio.: If the radom variable X has the pdf (). The the proposed pivotal quatity (4) has the pdf g(z) = ( ) Γ( Z ) exp( Z ); < Z <, > 0. (5) Proof. Substitutig (4) ito () ad usig chage of variable techiques; the pdf (5) obtaied is idepedet of µ ad ad thus (4) is a pivotal quatity for µ ad. Remark.: pdf (5) geeralizes Normal, Laplace ad Weibull distributios. By simply substitutig for i (5) ad simplify the the results obtaied reflect the targeted distributio... Geeralized t distributio Propositio.3: Let Z EP D(0,, ) from (5) ad let V deotes a radom variable which is Γ( r, ); a ew radom variable t defie as T = Z V r (6) has the pdf f r (t ) = { Γ( r + r ) Γ( r )Γ( ) } { } { r t ( ) } ( r + ) ( + r t (7) ad is a pivotal quatity. Proof. Sice Z has a symmetric distributio about zero, so does the t ad its pdf will satisfy f r (t ) = f r ( t ). Assumig both Z ad V are stochastically idepedet the for t > 0 we have, [ ] Z P > t = [ ] Z V/r P V/r > t = [ Z P > V takig the egative derivative wrt t. The margial distributio f(t ) is the pdf i 7 which we called a geeralized t distributio. Remarks.: t ] r

5 A. A. Olosude ad A. T. Soyika 97. But (7) is a geeralized versio of the usual studet t distributio because f(t ) becomes usual studets t-distributio whe = ad = thus reducig (4) to Z = x µ. [ Γ( r+ ) ] πrγ(r/) ( + t /r) (r+)/ (8) because of the preset parameter which further regulate its tails.. Sice f(t ) is idepedet of µ ad the it is a pivotal quatity. { } { 3. f r (t ) r t ( ) } ( r ( + r t + ) is symmetric ad bell shaped, but falls off to zero as t ± more slowly tha the ormal ad expoetial power desities. They have f(.) e z / ad g(.) e z / respectively. 4. To get the cdf of (7), the trasformatio ta θ = r T shows that the cdf of ( ) (7) is a icomplete beta F r (t ) = Beta r T ;, r. Corollary. (Geeralized t ): Let Z EP D(0,, ) from (4) ad let V deotes a radom variable which is Γ(, ) see propositio.; a ew radom variable T give as has the pdf f(t ) = T = Z V Γ( ) (Γ( ) ) t ( + t ) Usig the trasformatio ta θ = T, we obtai the CDF has F (t ) = Beta Proof. Substitute r = ito (7) ad (0) follows. Havig established that (6) ad (9) are pivotal quatities ) for µ ad, with their co- ad F fidece limits distributed as F r (t )=Beta ( r T ;, r (9) (0) ( T ) ;, ( T ;, (t )=Beta respectively the we proceed to obtai its cofidece itervals while restrictig our applicatio to cofidece iterval via t. ).. Cofidece Iterval for µ for a kow Give the t distributio, (ad its cdf F (t ) which has o close form), for ay idepedet ad idetically distributed radom sample (x)={x,..., x } EPD(µ,, ); we obtai the sufficiet statistics x = xi, ˆ = s = ay 0 < γ <, we compute t such that F (t ) = ( + γ)/ from ( ) i= (x µ) ad for

6 98 O Iterval Estimatio for Expoetial Power Distributio Parameters [ γ = P t x µ ] t ( ) EP D(0,, ) owig that F (t )=Beta t ;, Propositio.4: The cofidece iterval for the µ of the EPD whe variace E( ) = s ca be obtaied as { x s ( ) (t ) < µ < x + s ( ) } (t ) () Proof. Simplify P { t < x µ < t } γ () for µ...3 Cofidece Iterval for with µ ukow Propositio.5: Give the iterval ( t, t ). The the cofidece iterval for the of the EPD is give as s (beta[(f (t )),, ]) ; s (beta[ (F (t )),, ]) (3) where s i= = Proof. Simplify (x i x) P r { ( t ) < { x µ for ad recall from propositio. that 3 Numerical Illustratio 3. Evaluatig cofidece limits } < (t ) } γ (4) X µ Γ( ). Give iterval ( t, t ), we defied the 00γ percet cofidece iterval for pivot (9) as P r { t < T < t } γ (5) We the evaluate ( the differet values of t ad t at various values of γ from T ) F t (t )=Beta ;, for kow values of. Usig the code Rbeta.iv(γ,, ) writte i R eviromet. We thus have

7 A. A. Olosude ad A. T. Soyika 99 Table : Limits ( t, t ) for differet γ values from F t (t ) at various values γ limits F t (t ), = 0. F t (t ), = F t (t ), = F t (t ), =.5 F t (t ), = 0. -t t t t t t t t Simulatio Results We simulated 000,000 sample from populatio havig a expoetial power distributio with parameters locatio parameter µ = 4.000, scale parameter =.500 ad shape parameter = 3.546, the simulatio is from R package called ormalp see Mieo (003a,b). With ew code writte i R eviromet, we have table, the estimated cofidece iterval for the mea. I the table we also have cofidece iterval for the ormal case, that is, suppose we assume the data might have come from ormal populatio, the the estimated mea µ ormal = ad the variace ormal =.63. We ote from the table that the cofidece legth for the ormal case is bigger compare with expoetial power distributio at 90%, 95% ad 99%. This shows a better result for the latter. Table : Cofidece Iterval for the estimate of mea Limits Expoetial Power Distributio Normal Distributio 90%.386e e-03 95%.389e e-03 99%.389e I Table 3 we have cofidece iterval for the variace of expoetial power compare with ormal distributio. Just like the case of mea, the cofidece legth for the expoetial power is closer compare with ormal 4 Data sets Table 3: Cofidece Iterval for the estimate of variace Limits Expoetial Power Distributio Normal Distributio 90% % % Example. Forty-eight pairs of poultry birds were fed with iorgaic ad orgaic poultry feed Olosude (03), The comprehesive aalysis ad summary of the estimated parameters have bee carried out i, Olosude (03). The data set are the average

8 00 O Iterval Estimatio for Expoetial Power Distributio Parameters weights (wi, wo) ad average cholesterol (ci, co) cotets of the eggs produced by the birds, the data are reproduce i the appedix. Table 4: Cofidece iterval for the meas of egg weights ad its cholesterol cotets assumig expoetial power distributio whe beta is estimated γ wi Ci wo Co 90% ±0.097 ± ±0.53 ± % ± ± ±0.535 ± % ± ± ±0.53 ± Table 3: Cofidece i terval f or t he m eas o f e gg w eights a d i ts c holesterol cotets assumig ormal distributio γ wi Ci wo Co 90% ±0.33 ±0.063 ± ± % ±0.469 ±0.075 ± ± % ±0.930 ± ±.879 ±.07 Table 4: Cofidece iterval for the variace of egg weights ad its cholesterol cotet assumig expoetial power distributio whe is estimated γ wi Ci wo Co Table 5: Cofidece iterval for the variace of egg weights ad its cholesterol cotet assumig ormal distributio Further examples attached. γ wi Ci wo Co Coclusio From the applicatio, obviously from practical experiece it commo to have data with desity fuctio that has heavier or thier tail tha the usual ormal distributio. Estimatig cofidece iterval with assumptio of ormal distributio will always give cofidece l egth w ider t ha ecessary a d o fte t he c ofidece be comes ureliable i hypothesis, as this is show i both simulatio ad real life data from expoetial

9 A. A. Olosude ad A. T. Soyika 0 power distributio. I this case assumig ormal distributio i the estimatio of cofidece i terval m ay l ead t o e rroeous c oclusio, e specially f or t hose w ho w ill be iterest i goig further to hypothesis testig. This article preseted a robust parametric method of evaluatig cofidece i terval f or t he m ea a d t he v ariace with kow shape parameters. The results further geeralized what is obtaiable i ormal ad the double-expoetial case. 6 Ackowledgmet We wish to appreciate the aoymous referee for the costructive review of the paper which has greatly improve the quality of the article. Thaks Refereces. Abramowitz, M ad Stegu, I.A. (970): Hadbook of Mathematical Fuctios (9th editio). Dover Publicatios, Ic. New York.. Achcar De Araujo Pereira JA (999). Use of Expoetial Power Distributios for Mixture Models i the Presece of Covariates. Joural of Applied Statistics, 6, Agro G (99). Maximum Likelihood ad Lp-orm Estimators. Statistica Applicata, 4(), Agro G (995). Maximum Likelihood Estimatio for the Expoetial Power Distributio. Commuicatios i Statistics (simulatio ad computatio), 4(), Box, G. E. P. (953), A ote o the Regio of Kurtosis. Biometrika, 40, Cochra, W.G.(977). Samplig Techiques 3rd Editio. New York Joh wiley ad sos Pages Daiele Coi (0).A method to estimate power parameter i expoetial power distributio via polyomial regressio. Joural of Statistical Computatio ad Simulatio 8. Johso, R.A. ad Wicher, D.W. (006). Applied Multivariate Statistical Aalysis. Eglewood Cliffs, NJ: Pretice-Hall, Ic. 9. Lidsey, J.K. (999). Multivariate Elliptically Cotoured Distributios for Repeated Measuremets. Biometrics 55, Mieo, A.M. (003a). The ormalp package. URL Mieo, A.M. ad Rugerri, M (003b). O the Estimatio of the Structure Parameter of a Normal Distributio of Order p. Statistica, 63, 09.. Nadarajah, S. (005). A Geeralized Normal Distributio. Joural of Applied Statistics Vol 3, No 7,

10 0 O Iterval Estimatio for Expoetial Power Distributio Parameters 3. Olosude, A. A.(03) O expoetial power distributio ad poultry feeds data: a case study. J. Ira. Stat. Soc. (JIRSS) Vol.(), MR36504 Zbl (Ira) 4. Paris, R.B. (00). Icomplete Gamma Fuctio. NIST Hadbook of mathematical fuctios. Cambridge Uiversity press. 5. Riggs, K. (05). Cofidece Itervals for a Proportio Usig Iverse Samplig whe the Data is Subject to False-positive Misclassificatio. Joural of Data Sciece. Vol.3(), Subboti, M. T. (93), O the Law of Frequecy of Error. Mathematicheskii Sborik, Tiwari, R.C., Yi Li, ad Zhaohui, Z. (00). Iterval Estimatio for Ratios of Correlated Age-Adjusted Rates.Joural of Data Sciece. Vol.8 (3), p Wiitzki, S.(003). Computig the icomplete gamma fuctio to arbitrary precisio. Lect. Not. Comp. Sci. 667:

11 A. A. Olosude ad A. T. Soyika 03 APPENDIX LIVESTOCK PRODUCTION, TEACHING AND RESEARCH FARM FEDERAL UNIVERSITY OF AGRICULTURE ABEOKUTA (Poultry Uit) KEY w C i o Egg weight Cholesterol /egg (mg/egg) Iorgaic subscript Orgaic subscript S/NO wi Ci wo Co Figure : data used

12 04 O Iterval Estimatio for Expoetial Power Distributio Parameters