The estimatio of the m parameter of the Nakagami distributio Li-Fei Huag Mig Chua Uiversity Dept. of Applied Stat. ad Ifo. Sciece 5 Teh-Mig Rd., Gwei-Sha Taoyua City Taiwa lhuag@mail.mcu.edu.tw Je-Je Li Mig Chua Uiversity Dept. of Applied Stat. ad Ifo. Sciece 5 Teh-Mig Rd., Gwei-Sha Taoyua City Taiwa jjli@mail.mcu.edu.tw Abstract: This paper itroduces the Nakagami-m distributio which is usually used to simulate the ultrasoud image. The gamma distributio is used to derive the momet estimator ad the maximum likelihood estimator because the Nakagami distributio has o momet geeratio fuctio ad too complicate likelihood fuctio. The momet estimator of m is ormally distributed with a smaller bias ad a larger stadard deviatio. The secod order maximum likelihood estimator of m is also ormally distributed with a larger bias ad a smaller stadard deviatio. The cofidece iterval for the ratio of medias from two idepedet distributios of Nakagami-m estimators is costructed. The momet estimator provides a quick uderstadig about the m parameter, while the secod order maximum likelihood estimator provides a full uderstadig about the m parameter. Key Words: The Nakagami distributio, The momet estimator, The maximum likelihood estimator, The distributio of the Nakagami-m, The ratio of medias. Itroductio The Nakagami distributio is usually used to simulate the ultrasoud image. Smolikova et al. say that aalysis of backscatter i the ultrasoud echo evelope, i cojuctio with ultrasoud B-scas, ca provide importat iformatio for tissue characterizatio ad pathology diagosis[6]. Pavlovic et al. preset the joit probability desity fuctio (PDF) ad PDF of maximum of ratios mu() = R-/r() ad mu(2) = R-2/r(2) for the cases where R-, R-2, r(), ad r(2) are Rayleigh, Ricia, Nakagami-m., ad Weibull distributed radom variables[4]. Agrawal ad Karmeshu propose a ew composite probability distributio, i. e. Nakagami-geeralized iverse Gaussia distributio (NGIGD) with four parameters[]. Che applies order statistics to aalyze the performace of ordered selectio combiig schemes with differet modulatio receptios operatig i Nakagami-m fadig eviromets ad all the results are validated by comparig the special case Rayleigh distributio with the fadig figure m = i the Nakagami-m distributio[2]. Peppas presets a closed-form expressio for the momets geeratig fuctio of the halfharmoic mea of two idepedet, ot ecessarily idetically distributed gamma radom variables with arbitrary parameters[5]. With the shape parameter m ad the spread parameter Ω, the probability desity fuctio of the Nakagami distributio is as follows. f() = 2 Γ(m) (m Ω )m 2m e m Ω 2 Huag ad Johso [3] provide theorems to costruct a cofidece iterval for ratio of percetiles from two idepedet distributios. 2 The estimators of Nakagami-m 2. The momet estimator If N Nakagami(m, Ω), let G = N 2. The = g /2 ad G = N 2 gamma(m, Ω m ). J = d dg = 2 g 2 f() = 2 Γ(m) (m Ω )m 2m e m Ω 2 f(g) = 2 Γ(m) (m Ω )m g 2m 2 e m Ω g 2 g 2 = Γ(m) (m Ω )m g m e m Ω g With the shape parameter α ad the scale parameter β, the probability desity fuctio of the gamma distributio is as follows. f(g) = Γ(α)β α gα e g β E-ISSN: 2224-2902 67 Volume 3, 206
Table : The momet estimator of m m Ω ˆm.003686 29848.003686 0.75 0.753207 0.75 29848 0.753207 0.5 0.5006322 0.5 29848 0.5006322 0.25 0.2493589 0.25 29848 0.2493589 The momet geeratig fuctio: = = 0 0 M G (t) = E(e tg ) Γ(α)β α gα e ( Γ(α)β α gα e ( = ( βt )α β t)g dg βt β )g dg E(G) = M G(0) = α( βt) α ( β) t=0 = αβ E(G 2 ) = M G(0) = αβ 2 (α + ) = αβ( α )( βt) α 2 ( β) t=0 V ar(g) = E(G 2 ) E(G) 2 = αβ 2 (α+ α) = αβ 2 It ca be show that α = E(G)2 V ar(g), β = V ar(g) E(G) m = E(N 2 ) 2 V ar(n 2 ). Sice the momet geeratio fuctio of the Nakagami distributio does ot exist, it s ecessary to apply the gamma distributio i the process of fidig the momet estimator of m. If a radom sample of, 2,..., 000 is selected spread parameter Ω, the momet estimators of shape parameters are listed i Table. The value of Ω wo t affect the estimatio of m. 2.2 The maximum likelihood estimator Let be the sample size. The likelihood fuctio is L(α, β) = f(g i α, β) = = Γ(α)β α Γ(α)β α gα i The log likelihood fuctio is e g i β g i α e g i β. l L(α, β) = lγ(α)β α +(α ) l g i g i β l L(α, β) = α β β + g i β 2 = 0 ˆΩ g i =, which is the same with its momet estimator. l L(α, β) α = Γ (α) Γ(α) l β + l g i = Γ (α) Γ(α) l g i α + l g i = 0 Γ (α) Γ(α) g i l g i + l α = l The digamma fuctio is defied to be ψ(α) = d dα l Γ(α) = Γ (α) Γ(α). Now we oly eed to deal with the digamma fuctio. The likelihood fuctio of the Nakagami distributio is too complicate to fid the maximum likelihood estimator of m. Let g i l g i ψ(α) + l α = = l. Abramowitz ad Stegu fid that ψ(α) + l α = 2α + 2α 2 20α 4 +. Oe term is used to obtai the first order maximum likelihood estimator. = 2ˆα ˆm = ˆα = 2 E-ISSN: 2224-2902 68 Volume 3, 206
Table 2: The st ad 2d order maximum likelihood estimator of m m Ω ˆm ˆm 2 0.8687865.00858 29848 0.8687865.00858 0.75 0.62875 0.7499995 0.75 29848 0.62875 0.7499995 0.5 0.3945828 0.499879 0.5 29848 0.3945828 0.499879 0.25 0.763376 0.249923 0.25 29848 0.763376 0.249923 Two terms are used to obtai the secod order maximum likelihood estimator. = 2ˆα 2 + 2ˆα 2 2 2 ˆα 2 2 = 6ˆα 2 + Table 3: The bias ad the stadard deviatio of the momet estimator of m m Ω bias of ˆm stdev. of ˆm 0 0.06 29848 0 0.06 0.75 0 0.05 0.75 29848 0 0.05 0.5 0.002 0.04 0.5 29848 0.002 0.04 0.25 0.003 0.025 0.25 29848 0.003 0.025 2 ˆα 2 2 6ˆα 2 = 0 ˆm 2 = ˆα 2 = 3 + 9 + 2 2 If a radom sample of, 2,..., 000 is selected from a Nakagami distributio with the shape parameter m ad the spread parameter Ω, the maximum likelihood estimators of shape parameters are listed i Table 2. Where ˆm 2 is adjusted by its bias. The value of Ω wo t affect the estimatio of m. 3 The distributio of the Nakagamim estimator I order to compare the momet estimator ad the maximum likelihood estimator, their distributios eed to be foud. The Kolmogorov-Smirov test is applied to fid the distributio of the Nakagami-m estimator. 3. The momet estimator The momet estimator is ormally distributed with a smaller bias ad a larger stadard deviatio. Table 4: The bias ad the stadard deviatio of the secod order maximum likelihood estimator of m m Ω bias of ˆm 2 stdev. of ˆm 2 0.0 0.04 29848 0.0 0.04 0.75 0.05 0.03 0.75 29848 0.05 0.03 0.5 0.02 0.02 0.5 29848 0.02 0.02 0.25 0.03 0.0075 0.25 29848 0.03 0.0075 3.2 The maximum likelihood estimator The first order maximum likelihood estimator is ot discussed because it s too far away from the theoretical value of m. The secod order maximum likelihood estimator is ormally distributed with a larger bias ad a smaller stadard deviatio. E-ISSN: 2224-2902 69 Volume 3, 206
Table 5: The Cramer-Rao lower boud for the momet estimator of m m bias I(m) CRLB stdev. 0 33377.9 0.007 0.06 0.75 0 785200.9 0.00 0.05 0.5 0.002 227203.0 0.00089 0.04 0.25 0.003 35955.0 0.00085 0.025 Table 6: The Cramer-Rao lower boud for the secod order maximum likelihood estimator of m m bias I(m) CRLB stdev. 0.0 33377.9 0.007 0.04 0.75 0.05 785200.9 0.00 0.03 0.5 0.02 227203.0 0.00088 0.02 0.25 0.03 35955.0 0.00082 0.0075 3.3 The Cramer-Rao lower boud Ay estimator ˆα whose bias is give by a fuctio b(α) satisfies the Cramer-Rao lower boud V ar(ˆα) [ + b (α)] 2, I(α) where l L(α, β) I(α) = E[ ] 2 α = E[ Γ (α) Γ(α) l g i α + l g i ] 2 = E[ Γ (α) Γ(α) + l α ]2 The square root of Cramer-Rao lower bouds for the momet estimators are summarized i the Table 5. The square root of Cramer-Rao lower bouds for the secod order maximum likelihood estimators are summarized i the Table 6. 3.4 The media of the Nakagami-m estimators I order to compare populatio medias, the cofidece itervals for the ratio of two medias from idepedet distributios of Nakagami-m estimators are costructed by way of two rewritte theorems based o oparametric methods i Huag ad Johso [3]. m ad i the theorems are just sample sizes. Theorem Let X, X 2,..., X m be a radom sample from F ( ); let Y, Y 2,..., Y from F 2 ( ); ad let samples be idepedet. If the populatio desity fuctio ( ), is positive ad cotiuous i a eighborhood of the media m i, for,2, lim m, m(m + ) = λ, (0 < λ < ), ad m 2 0, the the 00(-α)% cofidece iterval of θ = m /m 2 is F i ˆm ± Z 0.25 ˆm α/2 2 m ˆm 2 2 [ ˆF ( ˆm )] + 0.25 ˆm 2 2 ˆm 4 2 [ ˆF 2 ( ˆm 2)] 2 Theorem 2 Let X, X 2,..., X m be a radom sample from F ( ) which has a positive cotiuous derivative F ( ) i a eighborhood of m, let Y, Y 2,..., Y be a radom sample from F 2 ( ) which has a positive cotiuous derivative F 2 ( ) i a eighborhood of m 2, ad let two samples be idepedet. Let the ratio of medias θ = m /m 2 be ukow but fiite. Uder the coditio that lim m, m(m + ) = λ, (0 < λ < ), θ values of the itersectios of ±Z α/2 ad Z M,NP (θ) = ˆm θ ˆm 2 0.25 m[ ˆF ( ˆm )] 2 + 0.25θ2 [ ˆF 2 ( ˆm 2)] 2 form the 00(-α)% cofidece iterval. Medias are estimated by order statistics, ad ˆF i ( ˆm i) is estimated by the kerel estimator for,2. Oe thousad replicatios of the radom sample X i,, X i,2,..., X i,000 are selected idepedetly from a Nakagami distributio with the shape parameter m ad the spread parameter. Oe thousad replicatios of the radom sample Y i,, Y i,2,..., Y i,000 are selected idepedetly from a Nakagami distributio with the shape parameter ad the spread parameter. The true ratio of medias from two populatios should be m. The simulated 95% cofidece itervals are listed i the Table 7. Where the momet estimator ad the secod order maximum likelihood estimator are adjusted by their bias. 3.5 The advatage ad disadvatage of two kids of estimators The momet estimator has two advatages. At first, it s much easier to fid the momet estimator tha the maximum likelihood estimator. Secodly, the bias of the momet estimator is smaller tha the bias of the maximum likelihood estimator. The maximum likelihood estimator also has two advatages. At first, the stadard deviatio of the maximum likelihood estimator is closer to the Cramer-Rao lower boud tha the stadard deviatio E-ISSN: 2224-2902 70 Volume 3, 206
Table 7: The 95% cofidece iterval of m m Thm. Estimator 95% CI MME (0.9967,.009) MLE2 (0.9993,.0078) 2 MME (0.9970,.00) 2 MLE2 (0.9990,.0080) 0.75 MME (0.7487,0.7599) 0.75 MLE2 (0.7479,0.7542) 0.75 2 MME (0.7490,0.7600) 0.75 2 MLE2 (0.7480,0.7540) 0.5 MME (0.4980,0.5060) 0.5 MLE2 (0.4984,0.5025) 0.5 2 MME (0.4980,0.5060) 0.5 2 MLE2 (0.4980,0.5030) 0.25 MME (0.2479,0.2523) 0.25 MLE2 (0.2492,0.252) 0.25 2 MME (0.2480,0.2520) 0.25 2 MLE2 (0.2490,0.250) of the momet estimator. Secodly, the cofidece iterval costructed by the maximum likelihood estimator is shorter tha the cofidece iterval costructed by the momet estimator uder all circumstaces. 4 Coclusio The gamma distributio ca be used to derive ad estimate the shape parameter of the Nakagami distributio. The cofidece iterval for the ratio of medias from two idepedet distributios of the Nakagamim estimators ca be costructed. Because the momet estimator of m is easier to be calculated ad has a smaller bias, a quick uderstadig about m ca be obtaied by fidig its momet estimator. Sice the maximum likelihood estimator of m has a smaller stadard deviatio, a full uderstadig about m ca be obtaied by fidig its maximum likelihood estimator. R programs of the Nakagami variable simulatio ad parameters estimatio ad the cofidece iterval costructio are i the appedix. Refereces: [] R. Agrawal ad Karmeshu, Ultrasoic backscatterig i tissue: characterizatio through Nakagami-geeralized iverse Gaussia distributio, Computers i Biology ad Medicie 37(2), 2007, pp. 66 72. [2] JLZ. Che, Performace aalysis for ordered selectio combiig schemes i Nakagami-m eviromets, Joural of the Frakli Istitute- Egieerig ad Applied Mathematics 342(6), 2005, pp. 638 656. [3] LF. Huag ad RA. Johso, Cofidece regios for the ratio of percetiles, Statistics ad Probability Letters 76(4), 2006, pp. 384 492. [4] DC. Pavlovic, NM. Sekulovic, GV. Milovaovic, AS. Paajotovic, MC. Stefaovic ad ZJ. Popovic, Statistics for Ratios of Rayleigh, Ricia, Nakagami-m, ad Weibull Distributed Radom Variables, Mathematical Problems i Egieerig 203, DOI: 0. 55/203/252804. [5] K. Peppas, Momets geeratig fuctio of the harmoic mea of two o-idetical gamma radom variables ad its applicatios i wireless commuicatios, Joural of the Frakli Istitute-Egieerig ad Applied Mathematics 349(3), 202, pp. 845 860. [6] R. Smolikova, MP. Wachowiak ad JM. Zurada, A iformatio-theoretic approach to estimatig ultrasoud backscatter characteristics, Computers i Biology ad Medicie 34(4), 2004, pp. 355 370. Appedix A. R program of the Nakagami variable simulatio ad parameters estimatio DistTest=fuctio(Simulated, NM,BigOmega,WriteTo, MMEbias,MMEstdev, MLE2bias,MLE2stdev) data=read.table(simulated) data.mat=as.matrix(data) raw=matrix(0,row(data), col(data)) Be=NM/BigOmega for (i i :row(data)) for (j i :col(data)) raw[i,j]=qgamma(data.mat[i, j], NM, Be) akagami=matrix(0,row(data), col(data)) for (i i :row(data)) for (j i :col(data)) E-ISSN: 2224-2902 7 Volume 3, 206
akagami[i,j]=sqrt(raw[i,j]) mesti=matrix(0,row(data),7) TmEsti=matrix(0,2,row(data)) for (i i :row(data)) mesti[i,]=mea(raw[i,])ˆ2/ var(raw[i,])-mmebias mesti[i,5]=/(2*(log(mea(raw[ i,]))-mea(log(raw[i,])))) mesti[i,6]=(3+sqrt(9+2*(log( mea(raw[i,]))-mea(log(raw[i, ])))) )/(2*(log(mea(raw[i,]))-mea( log(raw[i,]))))-mle2bias mesti[i,7]=-col(data)*digamma( NM)/gamma(NM)-col(data)*log( /Be)+sum(log(raw[i,])) for (i i :row(data)) TmEsti[,i]=mEsti[i,] TmEsti[2,i]=mEsti[i,6] write.table(tmesti,writeto, col.ames=false, row.ames=false, sep=" ") out4=mea(mesti[,]) out7=sqrt(var(mesti[,])) ksres=ks.test(mesti[,]," porm",nm,mmestdev) out8=ksres$p.value out3=mea(mesti[,5]) out4=mea(mesti[,6]) out5=sqrt(var(mesti[,5])) out6=sqrt(var(mesti[,6])) ksres2=ks.test(mesti[,6], "porm",nm,mle2stdev) out8=ksres2$p.value out9=mea((mesti[,7])ˆ2) list(meamme=out4, sdmme=out7,normalp=out8,meamle=out3,meamle2= out4,sdmle=out5,sdmle2=out6,normalp2=out8,fisheri=out9 ) A.2 R program of the cofidece iterval costructio RoMCI=fuctio(trueRatio,umer, deom, WriteTo, WriteTo2 ###, Thm, Thm2 ) mal=read.table(umer,header= FALSE, sep=" ") be=read.table(deom,header= FALSE, sep=" ") malsize=col(mal) besize=col(be) pairs=row(mal) malmat=as.matrix(mal) bemat=as.matrix(be) malker=matrix(0,pairs,+ malsize) beker=matrix(0,pairs,+ besize) Thm32=matrix(0,pairs,4) Thm42=matrix(0,pairs,4) ztheta=rep(0,) for (i i :pairs) malker[i,+malsize]=media( malmat[i,]) beker[i,+besize]=media( bemat[i,]) sortmal=sort(malmat[i,]) sortbe=sort(bemat[i,]) malup=sortmal[ceilig(malsize/ 2)+:malsize] mallow=sortmal[:floor(malsize/ 2)] beup=sortbe[ceilig(besize/ 2)+:besize] below=sortbe[:floor(besize/ 2)] a=c(sqrt(apply(mal[i,],,var)), (media(malup[:floor(malsize/ 2)])-media(mallow))/.34) A=mi(a) a2=c(sqrt(apply(be[i,],,var)), (media(beup[:floor(besize/ 2)])-media(below))/.34) A2=mi(a2) h=0.9*a*malsizeˆ(-0.2) h2=0.9*a2*besizeˆ(-0.2) for (j i :malsize) malker[i,j]=exp(-((malker[i, +malsize]-malmat[i,j])/h)ˆ2/ 2)/(sqrt(2*pi)*malsize*h) for (k i :besize) beker[i,k]=exp(-((beker[i, E-ISSN: 2224-2902 72 Volume 3, 206
+besize]-bemat[i,k])/h2)ˆ2/ 2)/(sqrt(2*pi)*besize*h2) Thm32[i,2]=malker[i,+malsize]/ beker[i,+besize] Thm32[i,]=Thm32[i,2]-.96* sqrt(0.25/(malsize*beker[i, +besize]ˆ2 *sum(malker[i,:malsize])ˆ2)+ 0.25*malker[i,+malsize]ˆ2/ (besize*beker[i,+besize]ˆ4 *sum(beker[i,:besize])ˆ2)) Thm32[i,3]=Thm32[i,2]+(Thm32[ i,2]-thm32[i,]) Thm32[i,4]=Thm32[i,]<trueRatio & Thm32[i,3]>trueRatio Thm42[i,2]=Thm32[i,2] for (l i :) theta=0.*(l-) ztheta[l]=(malker[i,+malsize]- theta*beker[i,+besize])/ i, :malsize])ˆ2)+0.25 *thetaˆ2/(besize*sum(beker[i, :besize])ˆ2)) ltheta=ztheta[ztheta>.96] utheta=ztheta[ztheta>-.96] lback=legth(ltheta)- uback=legth(utheta)- for (l i :) theta=0.*lback+0.0*(l-) ztheta[l]=(malker[i,+malsize]- theta*beker[i,+besize])/ i,:malsize])ˆ2)+0.25 *thetaˆ2/(besize*sum(beker[i, :besize])ˆ2)) ltheta=ztheta[ztheta>.96] lback2=legth(ltheta)- for (l i :) theta=0.*lback+0.0*lback2+ 0.00*(l-) ztheta[l]=(malker[i,+malsize]- theta*beker[i,+besize])/ i,:malsize])ˆ2)+0.25 *thetaˆ2/(besize*sum(beker[ i,:besize])ˆ2)) ltheta=ztheta[ztheta>.96] judge= ztheta[legth(ltheta)]-.96<.96-ztheta[legth(ltheta )+] lback3=legth(ltheta)-judge Thm42[i,]=0.*lback+0.0* lback2+0.00*lback3 for (l i :) theta=0.*uback+0.0*(l-) ztheta[l]=(malker[i,+malsize]- theta*beker[i,+besize])/ i,:malsize])ˆ2)+0.25 *thetaˆ2/(besize*sum(beker[ i,:besize])ˆ2)) utheta=ztheta[ztheta>-.96] uback2=legth(utheta)- for (l i :) theta=0.*uback+0.0*uback2+ 0.00*(l-) ztheta[l]=(malker[i,+malsize]- theta*beker[i,+besize])/ i,:malsize])ˆ2)+0.25 *thetaˆ2/(besize*sum(beker[ i,:besize])ˆ2)) utheta=ztheta[ztheta>-.96] judge= ztheta[legth(utheta)]+.96< -.96-ztheta[legth(utheta )+] uback3=legth(utheta)-judge Thm42[i,3]=0.*uback+0.0* uback2+0.00*uback3 Thm42[i,4]=Thm42[i,]<trueRatio & Thm42[i,3]>trueRatio write.table(thm32, WriteTo, col.ames=false, row.ames=false, sep=" ") write.table(thm42, WriteTo2, col.ames=false, row.ames=false, sep=" ") E-ISSN: 2224-2902 73 Volume 3, 206