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Estimates for parameters and characteristics of the confining SU(3)-gluonic field in pions

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a r X i v :h e p -p h /0609135v 1 14 S e p 2006Estimates for parameters and characteristics of the confining SU(3)-gluonic field in pionsand kaonsYu.P.GoncharovTheoretical Group,Experimental Physics Department,State PolytechnicalUniversity,Sankt-Petersburg 195251,Russia1Introduction and preliminary remarksIn Ref.[1]for the Dirac-Yang-Mills system derived from QCD-Lagrangian there was found a family of compatible nonperturbative solutions which could pretend to decsribing confinement of two quarks.Further study of this family as well as its applications to the quarkonia spectra (charmonium and bot-tomonium)in Refs.[2,3]showed that the above family is in essence unique one [4]and allows one to modify gluon propagator in a nonperturbative way so that the modified propagator could correspond to linear confinement at large distances [5].Two main physical reasons for linear confinement in the mechanism under discussion are the following ones.The first one is that gluon exchange between quarks is realized with the propagator different from the photon one and existence of such a propagator is direct consequence of the unique confining nonperturbative solutions of the Yang-Mills equations [4].The second reasonis that,owing to the structure of mentioned propagator,gluon condensate (a classical gluonfield)between quarks mainly consists of soft gluons(for more details see[4,5])but,because of that any gluon also emits gluons(still softer),the corresponding gluon concentrations rapidly become huge and form the linear confining magnetic colourfield of enormous strengths which leads to confinement of quarks.Under the circumstances physically nonlinearity of the Yang-Mills equations effectively vanishes so the latter possess the unique nonperturbative confining solutions of the abelian-like form(with the values in Cartan subalgebra of SU(3)-Lie algebra)[4]that describe the gluon condensate under consideration.Moreover,since the overwhelming majority of gluons are soft they cannot leave hadron(meson)until some gluon obtains additional energy(due to an external reason)to rush out.So we deal with confinement of gluons as well.The approach under discussion equips us with the explicit wave functions that is practically unreachable in other approaches,for example,within framework of lattice theories or potenial ly,for each two quarks(me-son)there exists its own set of real constants(for more details see below) a j,A j,b j,B j parametrizing the mentioned nonperturbative confining gluonfield (the gluon condensate)and the corresponding wave functions(nonperturba-tive modulo square integrable solutions of the Dirac equation in this confin-ing SU(3)-field)while the latter ones also depend onµ0,the reduced mass of the current masses of quarks forming meson.It is clear that constants a j,A j,b j,B j,µ0should be extracted from experimental data.This circum-stance gives possibilities for direct physical modelling of internal structure for any meson and for checking such relativistic models numerically.So far all applications of the confinement mechanism under consideration have been restricted to the quarkonia[2–5].The aim of the present paper is to estimate the above parameters for the case of pions and kaons.Of course, when conducting our considerations we should rely on the standard quark model(SQM)based on SU(3)-flavor symmetry(see,e.g.,Ref.[6]or the oldies [8]).Further we shall deal with the metric of theflat Minkowski spacetime M that we write down(using the ordinary set of local spherical coordinates r,ϑ,ϕfor the spatial part)in the formds2=gµνdxµ⊗dxν≡dt2−dr2−r2(dϑ2+sin2ϑdϕ2).(1) Besides,we have|δ|=|det(gµν)|=(r2sinϑ)2and0≤r<∞,0≤ϑ<π, 0≤ϕ<2π.Throughout the paper we employ the Heaviside-Lorentz system of units with ¯h=c=1,unless explicitly stated otherwise,so the gauge coupling constant g and the strong coupling constantαs are connected by relation g2/(4π)=αs.Further we shall denote L2(F)the set of the modulo square integrable complex functions on any manifold F furnished with an integration measure,then L n2(F)will be the n-fold direct product of L2(F)endowed with the obvious scalar product while†and∗stand,respectively,for Hermitian and complex conjugation.Our choice of Diracγ-matrices conforms to the so-called standard representation and isγ0= 100−1 ,γb= 0σb−σb0,b=1,2,3,α=γ0γ= 0σσ0 ,(2)whereσb denote the ordinary Pauli matrices andσ=σ1i+σ2j+σ3k.At last ⊗means tensorial product of matrices and I3is the unit3×3matrix. When calculating we apply the relations1GeV−1≈0.1973269679fm,1s−1≈0.658211915×10−24GeV,1V/m≈0.2309956375×10−23GeV2,1T≈0.6925075988×10−15GeV2.Finally,for the necessary estimates we shall employ the T00-component(volu-metric energy density)of the energy-momentum tensor for a SU(3)-Yang-Mills field which should be written in the chosen system of units in the formTµν=−F aµαF aνβgαβ+1√r+A1,−A3t+13A8t=−a2√r−(A1+A2),A3ϕ+13A8ϕ=b1r+B1,−A3ϕ+13A8ϕ=b2r+B2,−23A8ϕ=−(b1+b2)r−(B1+B2)(4)with the real constants a j,A j,b j,B j parametrizing the family.Another part of the family is given by the unique nonperturbative modulo square integrable solutions of the Dirac equation in the confining SU(3)-field of(4)Ψ=(Ψ1,Ψ2,Ψ3)with the four-dimensional Dirac spinorsΨj representing the j th colour component of the meson,which may describe the relativistic bound states of two quarks(mesons)and look as follows(with Pauli matrix σ1)Ψj=e iωj t r−1 F j1(r)Φj(ϑ,ϕ)F j2(r)σ1Φj(ϑ,ϕ) ,j=1,2,3(5) with the2D eigenspinorΦj= Φj1Φj2 of the euclidean Dirac operator D0on the unit sphere S2,while the coordinate r stands for the distance between quarks.The explicit form ofΦj is not needed here and can be found in Ref. [5,7].For the purpose of the present Letter we shall adduce the necessary spinors below.SpinorsΦj form an orthonormal basis in L22(S2).The energy spectrum of a meson is given byω=ω1+ω2+ω3withωj=ωj(n j,l j,λj)=gA j+−Λj g2a j b j±(n j+αj) n2j+2n jαj+Λ2j,j=1,2,3,(6) where g is the gauge coupling constant,a3=−(a1+a2),b3=−(b1+b2),A3=−(A1+A2),B3=−(B1+B2),Λj=λj−gB j,αj=βj ,P j=gb j+βj,F j2=iC j Q j rαj e−βj r 1+gb jµ20−ω2j+g2b2j while C j is determined from the normalization con-dition ∞0(|F j1|2+|F j2|2)dr=1¯h c,a j→a j/√¯h c,B j→B j/√2¯h2(n j+|λj|)2z2 − λg2a j b j¯h3(nj+|λj|)7z+O(z2),(8)where f(n j,λj)=4λj n j(n2j+λ2j)+|λj|µ20+g2b2j.We may seemingly use(6)with various combinations of signes(±)before sec-ond summand in numerators of(6)but,due to(8),it is reasonable to take all signs equal to+which is our choice within the Letter.Besides,as is not com-plicated to see,radial parts in nonrelativistic limit have the behaviour of form F j1,F j2∼r l j,which allows one to call quantum number l j angular momentum for j th colour component though angular momentum is not conserved in thefield(4)[1,5].So for mesons under consideration we should put all l j=0. Finally it should be noted that spectrum(6)is degenerated owing to degen-eracy of eigenvalues for the euclidean Dirac operator D0on the unit sphere ly,each eigenvlalue of D0λ=±(l+1),l=0,1,2...,has multiplicity 2(l+1)so we has2(l+1)eigenspinors orthogonal to each other.Ad referen-dum we need eigenspinors corresponding toλ=±1(l=0)so here is their explicit formλ=−1:Φ=C2e−iϑ2 e iϑ2e−iϕ/2,λ=1:Φ=C2e iϑ2 −e−iϑ2e−iϕ/2(9)with the coefficient C=1/√µ=3j=1ωj(0,0,λj)=3 j=1 −g2a j b j|Λj| (10)and,as a consequence,the corresponding meson wave functions of(5)are represented by(7),(9).3.1Choice of quark masses and gauge coupling constantIt is evident for employing the above relations we have to assign some valuesto quark masses and gauge coupling constant g.In accordance with Ref.[6], at present the current quark masses are restricted to intervals1.5MeV≤m u≤5MeV,3.0MeV≤m d≤9MeV,60MeV≤m s≤170MeV,so we take m u=(1.5+5)/2MeV=3.25MeV,m d=(3+9)/2MeV=6MeV,m s=(60+170)/2MeV=115MeV.Under the circumstances,the reducedmassµ0of Table1will respectively take values m u/2,m d/2,m u m d/(m u+m d),m u m s/(m u+m s),m d m s/(m d+m s).As to gauge coupling constant g=√Ψ(γµ⊗I3)Ψ=(Ψ†Ψ,Ψ†(α⊗I3)Ψ)=(ρ,J).Electric formfactor f(K)is the Fourier transform ofρf(K)= Ψ†Ψe−i Kr d3x=3 j=1 Ψ†jΨj e−i Kr d3x=3 j=1f j(K)= 3j=1 (|F j1|2+|F j2|2)Φ†jΦj e−i KrαΓ(α),πe −iKr cos ϑsin ϑdϑ=2sin (Kr )/(Kr ),we shall obtainf (K )=3j =1f j (K )=3j =1(2βj )2αj +1K (K 2+4β2j)αj=3j =116β2j·K 23j =12α2j+3αj +1r 2Ψ†Ψd 3x =r 2d 3x,J y =3j =1(F ∗j 1F j 2−F ∗j 2F j 1)Φ†jσ2σ1Φj r 2d 3x.(14)Magnetic moment of meson is M =V m d 3x ,where V is volume of meson.Then at l j =0we have J y =0for any spinor of (9),whileV m x,y,z d 3x =0because of turning to zero either integral over ϑor the one over ϕ,which is easily to check.As a result,magnetic moments of mesons under consideration with the wave functions of (5)(at l j =0)are equal to zero,as should be according to experimental data [6].3.4Numerical resultsWe employed relations(10)and(13)for obtaining estimates of the confiningSU(3)-gluonicfield parameters in mesons under discussion and,to imposemore restrictions,we considered in(10)eachωj=µ/3,µis meson mass,though it is not obligatory.Also the experimental estimates,if any,of<r>were used from Refs.[6,11].The results are adduced in Tables1,2.It should√be noted that in accordance with SQM[8]π0=(dd)/uu andg a1b1(GeV)B1π0— 6.2816-0.0124758-0.2754160.3385π0— 6.28160.01215240.2737830.3385π±—u ud 2.108110.0128925-0.273365-0.3385 s, 6.12560.03483550.5245850.5303K0,s, 6.114970.04948560.1043980.36052Theoret.(MeV)Theoret.<r>(fm)π0—µ=ω1(0,0,−1)+ω2(0,0,−1)+ω3(0,0,−1)=134.9760.602594π0—µ=ω1(0,0,−1)+ω2(0,0,−1)+ω3(0,0,−1)=134.9760.606236π±—u ud139.569950.6050 s,µ=ω1(0,0,−1)+ω2(0,0,−1)+ω3(0,0,−1)=493.6770.564046K0,s,µ=ω1(0,0,1)+ω2(0,0,1)+ω3(0,0,1)=497.6720.560964thefield(4)with the3-dimensional SU(3)-Lie algebra valued1-forms of electricE and magnetic H colourfields and also with T00-component(volumetric energy density)of the energy-momentum tensor(3)E= λ3(a1−a2)+λ8(a1+a2)√2r2,H=− λ3(b1−b2)+λ8(b1+b2)√2sinϑ,(15)T00≡T tt=1r4+b21+b1b2+b22r4+BG(E,E)= r2,H= b21+b1b2+b22Γ(18)so we can rewrite(16)in the form T00=T coul00+T lin00conforming to the con-tributions from the Coulomb and linear parts of the solution(4).The latter gives the corresponding split of n from(18)as n=n coul+n lin.The parameters of Tables1,2were employed when computing and for simplic-ity we put sinϑ=1in(16)–(17).Also there were used the following present-day whole decay widths of mesons under consideration[6]:Γ=1/τwith the life timesτ=8.4×10−17s,2.6033×10−8s,1.2386×10−8s,0.8953×10−10s (K0S-mode),5.18×10−8s(K0L-mode),respectively,whereas the Bohr radius a0=0.529177249·105fm[6].At last,as has been discussed in Refs.[3,5],we can estimate the quark velocities in the mesons under exploration from the conditionv q=11+ λBTable3Gluon concentrations,electric and magnetic colourfield strengths in pions.π0—r n coul n lin n E H(fm)(m−3)(m−3)(m−3)(V/m)(T)r00.172647×10530.142633×10550.144359×10550.768576×10210.118089×1015 10r00.172647×10490.142633×10530.142650×10530.768576×10190.118089×1014 dd:r0=<r>=0.606236fm,v d=0.9990800.1r00.159916×10570.139255×10570.299171×10570.739697×10230.116682×10161.00.216003×10520.511791×10540.513951×10540.271855×10210.707369×1014a00.275458×10330.182764×10450.182764×10450.970811×10110.133673×1010π±—u ud:r0=<r>=0.607418fm,v u=0.999500,v d=0.9990780.1r00.553136×10650.428537×10650.981673×10650.781449×10230.116271×10161.00.752978×10600.158112×10630.158865×10630.288321×10210.706251×1014a00.960236×10410.564627×10530.564627×10530.102961×10120.133462×1010 5Discussion and concluding remarks5.1DiscussionAs is seen from Tables3,4,at the characteristic scales of each meson the gluon concentrations are large and the correspondingfields(electric and mag-Table4Gluon concentrations,electric and magnetic colourfield strengths in kaons.K±—u us:r0=<r>=0.564046fm,v u=0.999536,v s=0.9839580.1r00.699798×10660.835931×10650.783392×10660.402965×10240.235429×10161.00.708323×10610.265950×10630.273033×10630.128203×10220.132793×1015a00.903288×10420.949724×10530.949724×10530.457820×10120.250942×1010 K0,s,r n coul n lin n E H(fm)(m−3)(m−3)(m−3)(V/m)(T)r00.377302×10600.148175×10610.185905×10610.348022×10220.116585×1015 10r00.377302×10560.148175×10590.148553×10590.348022×10200.116585×1014–mode):r0=<r>=0.560964fm,v d=0.999148,v s=0.984044K0—d ds(K0L0.1r00.218299×10670.857308×10650.226872×10670.348022×10240.116585×10161.00.216168×10620.269778×10630.291395×10630.109516×10220.654000×1014a00.275668×10430.963395×10530.963395×10530.391088×10120.123588×1010 netic colour ones)can be considered to be the classical ones with enormousstrenghts.The part n coul of gluon concentration n connected with the Coulombelectric colourfield is decreasing faster than n lin,the part of n related to thelinear magnetic colourfield,and at large distances n lin becomes dominantwhile quarks in mesons under investigation should be considered the ultrarel-ativistic point-like particles.It should be emphasized that in fact the gluon concentrations are much greater than the estimates given in Tables3,4be-cause the latter are the estimates for maximal possible gluon frequencies,i.e.for maximal possible gluon impulses(under the concrete situation of pions and kaons).The latter also explains why gluon concentrations are much larger(about8orders of magnitude)for charged pions compared toπ0-meson:just the corresponding life times are different by the same orders so,accordingly,the conforming maximal gluon impulses are in inverse relation.The given picture is in concordance with the one obtained when considering charmonium in Refs.[2,3,5].As a result,the confinement mechanism developed in Refs.[1,4,5]is confirmed by the considerations of the present Letter.It should be noted,however,that our results are of a preliminary characterwhich is readily apparent,for example,from that the current quark masses(as well as the gauge coupling constant g)used in computation are known only within the certain limits and we can expect similar limits for the magnitudesdiscussed in the Letter so it is neccesary further specification of the parameters for the confining SU(3)-gluonicfield in pions and kaons which can be obtained,for instance,by calculating decay constants and weak formfactors for the given mesons with the help of wave functions discussed above.Also one can obtain the analogous estimates for vector mesons,for instance,by computing thewidths of radiative decays for them and so on.We hope to continue analysing other problems of meson spectroscopy elsewhere.5.2Connection with potential approach and string-like pictureThe results obtained above allow us to shed some light on the following two problems.As is known,during a long time up to now in meson spectroscopyone often uses the so-called potential approach(see,e.g.,Refs.[12,13]and references therein).The essence of it is in that the interaction between quarks is modelled on a nonrelativistic confining potential in the form a/R+kR+c0with some real constants a,k,c0and the distance between quarks R.On the other hand,also for a long time there exists the so-called string-like pictureof quark confinement but only at qualitative level(see,e.g.,book of Perkins in[8]).Up to now,however,it is unknown as such considerations might be warranted from the point of view of QCD.Let us in short outline as ourresults(based on and derived from QCD-Lagrangian directly)naturally lead to possible justification of the mentioned directions.Thereto we note that onecan calculate energy E of gluon condensate conforming to solution(4)in a volume V through relation E= V T00r2sinϑdrdϑdϕwith T00of(16)–(17) but one should take into account that classical T00has a singularity alongz-axis(ϑ=0,π)and we have to introduce some angleϑ0(whose physical meaning is to be clarified a little below)soϑ0≤ϑ≤π−ϑ0.Then let us choose V as shown in Fig.1,i.e.the one between two concentric spheres withradii R0<R restricted to interior of coneϑ=ϑ0.Fig.1.Integration domain for obtaining the gluonic energyWithout going into details(see also Ref.[4])we shall obtainE=E(R)=E0+aR0−2πB R0ln1+cosϑ01−cosϑ0 with constants A,B of(16).We recognize the mentioned confining poten-tial in(20)when identifying E0=c0and we can see that phenomenological parameters a,k,c0of potential are expressed through more fundamental pa-rameters connected with the unique exact solution(4)of Yang-Mills equations describing confinement.One can notice that the quantity k(string tension) is usually related to the so-called Regge slopeα′=1/(2πk)and in many if not all of the papers using potential approach it is accepted k≈0.18 GeV2,E0=c0≈−0.873GeV(see,e.g.,Refs.[12,13]).Also one often uses parametrization a=−4αs(R0)/3,whereαs(R0)is the strong coupling con-stant at R0so when R<R0potential description is not applicable.If using (20)and the results obtained in Table1we can in series computeϑ0,αs(R0),R0 for all mesons under discussion and also for the ground state of charmonium ηc(1S)for that we use the parametrization from Refs.[3–5]with replacing a i→a i/√4πsince the system of units in those references is dif-ferent from the one used here.Results of computation are presented in Table 5.Table5Parameters determining the confining potential for pions,kaons and charmonium ground state.Particleαs(R0)π0—64.56◦0.757dd0.532×10−3π±—u ud0.600×10−3K±—u us0.357×10−2K0,s,60.272◦0.960c0.163×10−2Fig.2.Formation of string-like picture between quarkssystem when inserting potential of form(20)into Dirac(Pauli,Schr¨o dinger) equation.So,we draw the conclusion(mentioned as far back as in Refs.[2]) that the potential approach seems to be inconsistent.Now if there are two quarks Q1,Q2and each of them emits gluons outside of its own coneϑ=ϑ1,2(see Fig.2)then we have soft gluons(as mentioned in Section1)in regions I,II and between quarks so a characteristic transverse size D of the gluon condensate is decreasing with increasing quark masses as we just now saw.For heavy quarks the gluon configuration between them practically transforms into a string.As a result,there arises the string-like picture of quark confinement but the latter seems to be warranted enough only for heavy quarks.It should be emphasized that string tension k is determined just by parameters b1,2of linear magnetic colourfield from solution(4)[see (20)]which indirectly confirms the dominant role of the mentionedfield for confinement.5.3Concluding remarksConsiderations of the present Letter as well as ones of Refs.[2,3]show that in meson spectroscopy the approach based on the unique family of compatible nonperturbative solutions for the Dirac-Yang-Mills system derived from QCD-Lagrangian may be employed for both light mesons and heavy quarkonia. Under the circumstances there are no apparent obstacles to apply the approach to any meson.We hope further studies along this direction to confirm the given point of view.AcknowledgementsAuthor is grateful to the referee whose valuable remarks promoted essential improvement of the initial version of the Letter.References[1]Yu.P.Goncharov,Mod.Phys.Lett.A16(2001)557.[2]Yu.P.Goncharov,Europhys.Lett.62(2003)684;Yu.P.Goncharov and E.A.Choban,Mod.Phys.Lett.A18(2003)1661.[3]Yu.P.Goncharov and A.A.Bytsenko,Phys.Lett.B602(2004)86.[4]Yu.P.Goncharov,Phys.Lett.B617(2005)67.[5]Yu.P.Goncharov,in:New Developments in Black Hole Research,ed.P.V.Kreitler(Nova Science Publishers,Inc.,New York,2006),Chapter3,pp.67–121(hep-th/0512099).[6]S.Eidelman,et al.,Particle Data Group,Phys.Lett.B592(2004)1.[7]Yu.P.Goncharov,Pis’ma v ZhETF69(1999)619;Phys.Lett.B458(1999)29.[8]Yu.B.Rumer,A.I.Fet,Theory of Unitary Symmetry,Nauka,Moscow,1970;F.E.Close,An Introduction to Quarks and Partons,Academic Press,London,1979;D.H.Perkins,Introduction to High Energy Physics,Addison–Wesley,London,1987.[9] A.Deur,Nucl.Phys.A755(2005)353;A.Deur,et.al.,hep-ph/0509113.[10]A.P.Prudnikov,Yu.A.Brychkov and O.I.Marichev,Integrals and Series.Elementary Functions(Nauka,Moscow,1981).[11]R.Alkofer,A.Bender,C.D.Roberts,Int.J.Mod.Phys.A10(1995)3319;B.V.Geshkenbein,Phys.Rev.D61(2000)033009.[12]W.Roberts,B.Silvestre-Brac,Phys.Rev.D57(1998)1694;R.Bonnaz,B.Silvestre-Brac and C.Gignoux,Eur.Phys.J.A13(2002)363.[13]N.Brambilla,et al.,CERN Yellow Report,CERN-2005-005Geneva:CERN,2005(hep-ph/0412158).。

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RETC各步骤简要说明

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RETC各步骤简要说明1、Type of ProblemUser specifies identifying header for the output file and selects whether direct or inverse problem is solved. The user also has to specify which type of input data are available for inverse procedure. Experimental hydraulic data need to be provided to solve the inverse problem (i.e., optimization of parameters in hydraulic model). For the direct problem, RETC calculates the hydraulic functions from user-specified model.用户指定确认输出文件的标题，同时选择需要解决的直接或逆向的问题。

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One also specifies the maximum number of iterations (MIT) for the inverse problem and maximum number of iterations (KITER) to be printed into the output file. Use a large number (e.g., MIT=30) for the simultaneous fit. Improved estimates for the unknown coefficients are printed during the first KITER iterations of the least-squares analysis. This input parameter eliminates excessively long output files if many iterations are needed before convergence is reached. Results for the last iterations are always printed.用户同样需要指定逆向问题中迭代的最大值（MIT），以及输出文件中迭代的最大值（KITER）。

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confidence intervals总结置信区间SPSS计算解释 及APA 报告规范-zhaomf


Descriptive statistics (for CIs)
HOW TO REPORT (APA)?
APA, 2009, p117: When reporting confidence intervals, use the format 95% Cl [LL UL]
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2. For a comparison of two independent means, 1) p < .05 when proportion overlap of the 95% CIs is about .50 or less. (Proportion
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1. Identify what the means and error bars represent. Do bars show confidence intervals (CIs) or standard errors (SEs)? What is the experimental design?
Outstanding Issues Complex Experimental Designs
4*2(between) mixed design
References and Tools for this Presentation
• • American Psychological Association (2009). Publication Manual of the American Psychological Association. (6th ed.) American Psychological Association, Washington, DC: Author. Cumming, G., & Finch, S. (2005). Inference by eye: confidence intervals and how to read pictures of data. American Psychologist, 60(2), 170. Thompson, B. (2002). What future quantitative social science research could look like: Confidence intervals for effect sizes. Educational Researcher, 31(3), 25-32. SPSS TOOLS(excel): DANCE WITH p, C(confidence) .au/psy/research/cognitive-anddevelopmental-psychology/esci
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投资学第10版习题答案08

投资学第10版习题答案08CHAPTER 8: INDEX MODELSPROBLEM SETS1. The advantage of the index model, compared to the Markowitzprocedure, is the vastly reduced number of estimates required. In addition, the large number of estimates required for the Markowitz procedure can result in large aggregate estimation errors whenimplementing the procedure. The disadvantage of the index modelarises from the model’s assumption that return residuals areuncorrelated. This assumption will be incorrect if the index used omits a significant risk factor.2. The trade-off entailed in departing from pure indexing in favor ofan actively managed portfolio is between the probability (or thepossibility) of superior performance against the certainty of additional management fees.3. The answer to this question can be seen from the formulas for w0(equation 8.20) and w* (equation 8.21). Other things held equal, w0 is smaller the greater the residual variance of a candidate asset for inclusion in the portfolio. Further, we see that regardless of beta, when w0 decreases, so does w*. Therefore, other things equal, the greater the residual variance of an asset, the smaller itsposition in the optimal risky portfolio. That is, increased firm- specific risk reduces the extent to which an active investor will be willing to depart from an indexed portfolio.4. The total risk premium equals: + (× Market risk premium). Wecall alpha a nonmarket return premium because it is the portion of the return premium that is independent of market performance.The Sharpe ratio indicates that a higher alpha makes a securitymore desirable. Alpha, the numerator of the Sharpe ratio, is afixed number that is not affected by the standard deviation ofreturns, the denominator of the Sharpe ratio. Hence, an increase in alpha increases the Sharpe ratio. Since the portfolio alpha is the portfolio-weighted average of the securities’ alphas, then,holding all other parameters fixed, an increase in a security’s alpha results in an increase in the portfolio Sharpe ratio.5. a. To optimize this portfolio one would need:n = 60 estimates of means n = 60 estimates of variances 770,122=-n n estimates of covariances Therefore, in total: 890,1232=+n n estimatesb.In a single index model: r i r f = α i + β i (r M – r f ) + e iEquivalently, using excess returns: R i = α i + β i R M + e i The variance of the rate of return can be decomposed into the components: (l) The variance due to the common market factor:22M i σβ (2) The variance due to firm specific unanticipated events: )(σ2i e In this model: σββ),(Cov j i j i r r = The number of parameter estimates is: n = 60 estimates of the mean E (r i ) n = 60 estimates of the sensitivity coefficient β i n = 60 estimates of the firm-specific variance σ2(e i ) 1 estimate of the market mean E (r M ) 1 estimate of the market variance 2M σ Therefore, in total, 182 estimates. The single index model reduces the total number of required estimates from 1,890 to 182. In general, the number of parameter estimates is reduced from: )23( to 232+???? ??+n n n 6. a. The standard deviation of each individual stock is given by: 2/1222)](β[σi M i i e σσ+=Since βA = 0.8, βB = 1.2, σ(e A) = 30%, σ(e B) = 40%, and σM = 22%, we get:σA = (0.82× 222 + 302 )1/2 = 34.78%σB = (1.22× 222 + 402 )1/2 = 47.93%b. The expected rate of return on a portfolio is the weighted average of the expected returns of the individual securities: E(r P ) = w A × E(r A ) + w B × E(r B ) + w f × r f E(r P )= (0.30 × 13%) + (0.45 × 18%) + (0.25 × 8%) = 14% The beta of a portfolio is similarly a weighted average of the betas of the individual securities: βP = w A × βA + w B × βB + w f × β f βP = (0.30 × 0.8) + (0.45 × 1.2) + (0.25 × 0.0) = 0.78 The variance o f this portfolio is: )(σβσ2222P M P P e +=σ where 22σβM P is the systematic component and )(2P e σis the nonsystematic component. Since the residuals (e i ) are uncorrelated, the nonsystematic variance is: 2222222()()()()P A A B B f f e w e w e w e σσσσ=?+?+? = (0.302 × 302 ) + (0.452 × 402 ) + (0.252 × 0) = 405 where σ2(e A ) and σ2(e B ) are the firm-specific (nonsystematic) variances of StocksA and B, and σ2(e f ), the nonsystematic variance of T-bills, is zero. The residual standard deviation of the po rtfolio is thus: σ(e P ) = (405)1/2 = 20.12% The total variance of the portfolio is then: 47.699405)2278.0(σ222=+?=P The total standard deviation is 26.45%.7. a.The two figures depict the stocks’ security characteri stic lines (SCL). Stock A has higher firm-specific risk because the deviations of the observations from the SCL are larger for Stock A than for Stock B. Deviations are measured by the vertical distance of each observation from the SCL.b. Beta is the slope of the SCL, which is the measure ofsystematic risk. The SCL for Stock B is steeper; hence Stock B’s systematic risk is greater.c. The R 2(or squared correlation coefficient) of the SCL is the ratio of the explained variance of the stock’s return to total variance, and the total variance is the sum of the explained variance plus the unexplained variance (the stock’s residual variance): )(σσβσβ222222i M i M i e R += Since the explained variance for Stock B is greater than for Stock A (the explained variance is 22σβM B , which is greater since its beta is higher), and its residual variance 2()B e σ is smaller, its R 2 is higher than Stock A’s.d.Alpha is the intercept of the SCL with the expected return axis. Stock A has a small positive alpha whereas Stock B has a negative alpha; hence, Stock A’s alpha is larger.e.The correlation coefficient is simply the square root of R 2,so Stock B’s correlation with the market is higher.8. a.Firm-specific risk is measured by the residual standard deviation. Thus, stock A has more firm-specific risk: 10.3% > 9.1%b.Market risk is measured by beta, the slope coefficient of the regression. A has a larger beta coefficient: 1.2 > 0.8c.R 2 measures the fraction of total variance of return explained by the market return. A’s R 2 is larger than B’s: 0.576 >0.436d.Rewriting the SCL equation in terms of total return (r ) rather than excess return (R ): ()(1)A f M f A f M r r r r r r r αβαββ-=+?-?=+?-+? The intercept is now equal to: (1)1%(1 1.2)f f r r αβ+?-=+?- Since r f = 6%, the intercept would be: 1%6%(1 1.2)1% 1.2%0.2%+-=-=-。

samplingR：抽样与估计方法 1.0.1说明书

samplingR：抽样与估计方法 1.0.1说明书

Package‘samplingR’June26,2023Type PackageTitle Sampling and Estimation MethodsVersion1.0.1Date2023-06-25Maintainer Javier Estévez<***********************>Description Functions to take samples of data,sample size estimation and getting useful estima-tors such as total,mean,proportion about its population using simple random,stratified,system-atic and cluster sampling.Imports dplyr,methodsLicense GPL-2Encoding UTF-8RoxygenNote7.2.3Suggests knitr,rmarkdownVignetteBuilder knitrNeedsCompilation noAuthor Javier Estévez[aut,cre,cph]Repository CRANDate/Publication2023-06-2611:00:02UTCR topics documented:cluster.estimator (2)cluster.sample (3)cluster.samplesize (4)srs.domainestimator (5)srs.estimator (6)srs.sample (7)srs.samplesize (8)strata.allocation (9)strata.estimator (10)strata.sample (11)12cluster.estimatorstrata.samplesize (12)strata.samplesize.cost (13)syst.all.samples (14)syst.anova (15)syst.estimator (15)syst.intercorr (17)syst.intracorr (17)syst.sample (18)Index20 cluster.estimator Parameter estimation for cluster samplesDescriptionEstimates parameters with optional confidence interval for clustered data of similar cluster size.Usagecluster.estimator(N,data,estimator=c("total","mean","proportion","class total"),replace=FALSE,alpha)ArgumentsN Number of clusters for the populationdata Cluster sampleestimator Estimator to compute.Can be one of"total","mean","proportion","class total".Default is"total".replace Whether the sample to be taken can have repeated instances or not.alpha Optional value to calculate estimation error and build1-alphaDetailsThis function admits both grouped and non-grouped by cluster data.Non-grouped data must have interest variable data in thefirst column and cluster name each indi-vidual belongs to in the last column.Grouped by cluster data must have interest variable data in thefirst column,cluster size in the sec-ond and the cluster name in the last column.Interest values of grouped data must reflect the total value of each cluster.ValueA list containing different interest values:•estimator•variance•sampling.error•estimation.error•confintExamplesd<-cbind(rnorm(500,50,20),rep(c(1:50),10))#Non-grouped datasample<-cluster.sample(d,n=10)#Non-grouped samplesampleg<-aggregate(sample[,1],by=list(Category=sample[,2]),FUN=sum)sampleg<-cbind(sampleg[,2],rep(10,10),sampleg[,1])#Same sample but with grouped data sum(d[,1])cluster.estimator(N=50,data=sample,estimator="total",alpha=0.05)cluster.estimator(N=50,data=sampleg,estimator="total",alpha=0.05)cluster.sample Cluster sample for non-grouped data.DescriptionRetrieves a sample of clusters for data which interest variable values are not grouped by cluster.Usagecluster.sample(data,n,replace=FALSE)Argumentsdata Matrix or data.frame containing the population data infirst column and the clus-ter it belongs to in the last column.n Number of clusters of the returning sample.replace Whether the sample to be taken can have repeated clusters or not.Details#’If your data is grouped by cluster use srs.sample function to retrieve your sample.Remember grouped by cluster data must have interest variable data in thefirst column,cluster size in the second and the cluster name in the last column.Interest values of grouped data must reflect the total value of each cluster.ValueData frame of a clustered sampleExamplesdata<-cbind(rnorm(500,50,20),rep(c(1:50),10))sample<-cluster.sample(data,10);samplecluster.samplesize Sample size estimation on cluster samplingDescriptionCalculates the required sample size in order to achieve an absolute sampling error less or equal to the specified for an specific estimator and an optional confidence interval in cluster sampling. Usagecluster.samplesize(N,data,error,alpha,estimator=c("total","mean","proportion","class total"),replace=FALSE)ArgumentsN Number of clusters in the population.data Dataset.error Sampling error.alpha Significance level to obtain confidence intervals.estimator The estimator to be estimated.Default is"total".replace Whether the samples to be taken can have repeated instances or not.DetailsThis function admits both grouped and non-grouped by cluster data.Non-grouped data must have interest variable data in thefirst column and cluster name each indi-vidual belongs to in the last column.Grouped by cluster data must have interest variable data in thefirst column,cluster size in the sec-ond and the cluster name in the last column.Interest values of grouped data must reflect the total value of each cluster.srs.domainestimator5ValueNumber of clusters to be taken.Examplesd<-cbind(rnorm(500,50,20),rep(c(1:50),10))#Non-grouped datasample<-cluster.sample(d,n=10)#Non-grouped samplesampleg<-aggregate(sample[,1],by=list(Category=sample[,2]),FUN=sum)sampleg<-cbind(sampleg[,2],rep(10,10),sampleg[,1])#Sample sample with grouped data#Cluster size to be taken for estimationcluster.samplesize(N=50,data=sample,error=500,estimator="total",replace=TRUE)newsample<-cluster.sample(d,n=26)#New sample for estimationsum(d[,1])cluster.estimator(N=50,data=newsample,estimator="total",alpha=0.05,replace=TRUE) cluster.estimator(N=50,data=sampleg,estimator="total",alpha=0.05)srs.domainestimator Simple Random Sample parameter estimation of domains.DescriptionFunction to make estimations of diferent parameters on a given domain based on a Simple Random Sample.Usagesrs.domainestimator(Nh,data,estimator=c("total","mean","proportion","class total"),domain,replace=FALSE,alpha)ArgumentsNh Number of instances of the data set domain.data Sample of the data.It must constain a column with the data to estimate and a second column with the domain of each instance.estimator One of"total","mean".Default is"total".domain Domain of the sample from which parameter estimation will be done.replace Whether the sample to be taken can have repeated instances or not.alpha Optional value to calculate estimation error and build1-alpha confidence inter-val.6srs.estimatorDetailsData columns must be arranged with interest values on thefirst column and domain values on the last column.Domain parameter can be either numeric or character and must be equal to one of the values of the domain column of data.ValueA list containing different interest values:•estimator•variance•sampling.error•estimation.error•confintReferencesPérez,C.(1999)Técnicas de muestreo estadístico.Teoría,práctica y aplicaciones informáticas.193-195Examplesdata<-cbind(rnorm(500,50,20),rep(c(1:2),250))sample<-data[srs.sample(500,100),]sum(data[which(data[,-1]==1),1])srs.domainestimator(Nh=250,data=sample,estimator="total",domain=1)srs.estimator Simple Ramdom Sampling parameter estimation.DescriptionFunction to make estimations of diferent parameters based on a Simple Random Sample.Usagesrs.estimator(N,data,estimator=c("total","mean","proportion","class total"),replace=FALSE,alpha)srs.sample7 ArgumentsN Number of instances of the data set.data Sample of the data.It must only contain a single column of the data to estimate.estimator Estimator to compute.Can be one of"total","mean","proportion","class total".Default is"total".replace Whether the sample has been taken with replacement or not.alpha Optional value to calculate estimation error and build1-alpha confidence inter-val.ValueA list containing different interest values:•estimator•variance•sampling.error•estimation.error•confintExamplesdata<-rnorm(200,100,20)sample<-data[srs.sample(200,50)]tau<-sum(data);tausrs.estimator(200,sample,"total",alpha=0.05)mu<-mean(data);musrs.estimator(200,sample,"mean",alpha=0.05)srs.sample Simple Random SampleDescriptionWith this function you receive a simple random sample consisting on a list of the instances indexUsagesrs.sample(N,n,replace=FALSE,data)8srs.samplesize ArgumentsN Number of instances of the data set.n Number of instances of the returning sample.replace Whether the sample to be taken can have repeated instances or not.data Optional matrix or data.frame containing the population data.If specified an object of same class as data will be returned with sample instances.ValueList of size n with numbers from1to N indicating the index of the data set’s instances to be taken.Examplessrs.sample(10,3)data<-matrix(data=c(1:24),nrow=8)N<-dim(data)[1]sample<-srs.sample(N,3,data=data)samplesrs.samplesize Simple Random Sample size.DescriptionCalculates the required sample size in order to achieve a relative or absolute sampling error less or equal to the specified for an specific estimator and an optional confidence interval in simple random sampling.Usagesrs.samplesize(N,var,error,alpha,estimator=c("total","mean","proportion","class total"),p,mean,replace=FALSE,relative=FALSE)strata.allocation9 ArgumentsN Number of instances of the data set.var Estimated quasivariance.error Sampling erroralpha Significance level to obtain confidence intervals.estimator One of"total","proportion","mean","class total".Default is"total"p Estimated proportion.If estimator is not"proportion"or"class total"it will be ignored.mean Estimated mean.If relative=FALSE it will be ignored.replace Whether the sample to be taken can have repeated instances or not.relative Whether the specified error is relative or not.DetailsIf the sample size result is not a whole number the number returned is the next whole number so srs.samplesize>=n is satisfied.To estimate sample size of estimators"total"and"mean"estimated quasivariance must be provided.If the error is relative then estimated mean must also be provided.To estimate sample size of estimator"proportion"and"class total"estimated proportion must be provided.If p is not specified sample size will be estimated based on worst-case scenario of p=0.5.N must be always be provided for calculations.ValueNumber of instances of the sample to be taken.Examplesdata<-rnorm(200,100,20)n<-srs.samplesize(200,var(data),estimator="total",error=400,alpha=0.05);nsample<-data[srs.sample(200,n)]srs.estimator(200,sample,"total",alpha=0.05)$sampling.errorstrata.allocation Strata allocation given a sample sizeDescriptionFunction to allocate the number of samples to be taken for each strata given the total sample size and the strata.allocation method.The number of allocations returned will be equal to the length of the parameters.10strata.estimator Usagestrata.allocation(Nh,n,var,alloc=c("unif","prop","min","optim"),C,cini,ch)ArgumentsNh Vector of population strata sizes.n Sample sizevar Vector of strata variances.alloc The allocation method to be used.Default is"unif".C Total study cost.cini Overhead study cost.ch Vector of costs to take an individual from a strata for the sample.Detailsalloc="optim"is the only that requires cost function data.Total study and overhead study costs are optional.If given allocation will be done so total study cost is not surpassed.sValueVector of strata sample sizes.Examplesstrata.allocation(Nh=rep(125,4),n=100,alloc="unif")#25,25,25,25strata.allocation(Nh=c(100,50,25),n=100,alloc="prop")strata.estimator Parameter estimation of stratified dataDescriptionFunction to make estimations of diferent parameters based on a stratified sample.Usagestrata.estimator(N,Nh,data,estimator=c("total","mean","proportion","class total"),replace=FALSE,alpha)ArgumentsN Population size.Nh Size of each population strata.data Stratified sample.estimator Estimator to compute.Can be one of"total","mean","proportion","class total".Default is"total".replace Whether the sample to be taken can have repeated instances or not.alpha Optional value to calculate estimation error and build1-alphaDetailsNh length must be equal to number of strata in data.data is meant to be a returned object of strata.sample function.ValueA list containing different interest values:•estimator•variance•sampling.error•estimation.error•confintstrata.sample Stratified sampleDescriptionWith this function you receive a sample of each strata within your data with specified size for each strata.Usagestrata.sample(data,n,replace=FALSE)Argumentsdata Population data consisting of a number of columns of data and a last column specifying the strata each instance belongs to.n Numeric array of sample sizes for each strata to be taken.replace Whether the sample to be taken can have repeated instances or not.Detailsn length must be equal to number of strata in data.On return list each strata sample can be accessed calling object$strataname where strataname are values of the last column of the original data.ValueA list containing one strata sample per index.Examplesdata<-cbind(rnorm(500,50,20),rep(c("clase1","clase2","clase3","clase4"),125)) strata.sample(data=data,n=c(10,20,30,40))strata.samplesize Sample size estimation on stratified samplingDescriptionCalculates the required sample size in order to achieve an absolute or relative sampling error less or equal to the specified for an specific estimator and an optional confidence interval in stratified sampling.Usagestrata.samplesize(Nh,var,error,alpha,estimator=c("total","mean","proportion","class total"),alloc=c("prop","min","optim"),ch,p,mean,replace=FALSE,relative=FALSE)ArgumentsNh Vector of population strata sizes.var Vector of estimated strata variances.error Sampling error.alpha Significance level to obtain confidence intervals.estimator The estimator to be estimated.Default is"total".alloc The allocation to be used when taking samples.Default is"prop".ch Vector of cost per strata to select an individual for the sample.p Estimated population proportion.If estimator is not"proportion"or"class total"it will be ignored.mean Estimated population mean.If relative=FALSE it will be ignored.replace Whether the samples to be taken can have repeated instances or not.relative Whether the specified error is relative or not.DetailsWith"proportion"and"class total"estimators variance vector must contain var return values equal to Nhp∗(1−p)values.(Nh−1)ValueNumber of instances of the sample to be taken.Examplesstrata.samplesize(c(120,100,110,50),c(458,313,407,364),error=5,alpha=0.05,"mean","prop")strata.samplesize.costStrata sample size by costs functionDescriptionThis function returns the total sample size given a costs function consisting on thefixed total study cost,overhead study cost and a vector of costs by strata.can be given so the allocation is calculated to not exceed the total study cost.14syst.all.samplesUsagestrata.samplesize.cost(Nh,var,C,cini,ch,alloc=c("unif","prop","optim"))ArgumentsNh Vector of population strata sizes.var Vector of strata variance values.C Total study cost.cini Overhead study cost.ch Vector of costs to take an individual from a strata for the sample.alloc The allocation method to be used.Default is"unif".DetailsStrata variance values are only necessary for optim allocation.ValueSample size.Examplesstrata.samplesize.cost(Nh=c(100,500,200),C=1000,cini=70,ch=c(9,5,12),alloc="prop") syst.all.samples Systematic samplesDescriptionReturns all possible systematic samples of size nUsagesyst.all.samples(data,n)Argumentsdata Population datan Sample sizesyst.anova15 ValueList with a sample per entranceExamplesdata<-c(1,3,5,2,4,6,2,7,3)syst.all.samples(data,3)syst.anova Analysis of variance of population dataDescriptionAnalysis of variance of population dataUsagesyst.anova(data,n)Argumentsdata Population datan Sample sizeValueSummaryExamplesdata<-c(1,3,5,2,4,6,2,7,3)syst.anova(data,3)syst.estimator Parameter estimation on a systematic sampleDescriptionParameter estimation on a systematic sample16syst.estimatorUsagesyst.estimator(N,sample,estimator=c("total","mean","proportion","class total"),method=c("srs","strata","syst"),alpha,data,t)ArgumentsN Population sizesample Vector containing the systematic sampleestimator Estimator to compute.Can be one of"total","mean","proportion","class total".Default is"total".method Method of variance estimation.Can be one of"srs","strata","syst".alpha Optional value to calculate estimation error and build1-alpha confidence inter-val.data Population data.t Number of systematic samples to take with interpenetrating samples method. DetailsVariance estimation has no direct formula in systematic sampling,thus estimation method must be done.Refer to syst.intracorr and syst.intercorr functions details for more information."syst"method uses interpenetrating samples method in which t systematic samples of size=nt aretaken to estimate.nt must be even.By choosing the start at random for all the samples they can be considered random taken.With this method population data and t must be given.ValueA list containing different interest values:•estimator•variance•sampling.error•estimation.error•confintExamplesdata<-c(1,3,5,2,4,6,2,7,3)sample<-syst.sample(9,3,data)syst.estimator(N=9,sample,"mean","srs",0.05)syst.intercorr17 syst.intercorr Stratified correlation coefficientDescriptionStratified correlation coefficientUsagesyst.intercorr(N,n,data)ArgumentsN Population sizen Sample sizedata Population dataDetailsThis value serves as a comparison between systematic and stratified sampling precision.At value=1the systematic precision is minimum.At value=0both sampling methods precision areequal.At value=−1n−1systematic precision is maximum.Summarising at values between1and0stratified sampling estimation has more precision than systematic,so method="strata"should be set at syst.estimator.The other way method="syst"of interpenetrating samples method is better.ValueCorrelation coefficientExamplesdata<-c(1,3,5,2,4,6,2,7,3)syst.intercorr(9,3,data)#0.09022556syst.intracorr Intraclass correlation coefficientDescriptionIntraclass correlation coefficientUsagesyst.intracorr(N,n,data)ArgumentsN Population sizen Sample sizedata Population dataDetailsThis value serves as a comparison between systematic and simple random sampling precision.At value=1the systematic precision is minimum.At value=0both sampling methods precision areequal.At value=−1n−1systematic precision is maximum.Summarising at values between1and0simple random sampling estimation has more precision than systematic,so method="srs"should be set at syst.estimator.The other way method="syst"of interpenetrating samples method is better.ValueIntraclass correlationExamplesdata<-c(1,3,5,2,4,6,2,7,3)syst.intracorr(9,3,data)#0.34375example1syst.sample Systematic sampling sampleDescriptionRetrieves a Nnsystematic sampleUsagesyst.sample(N,n,data)ArgumentsN Population size.n Sample sizedata Optional data of the population.DetailsIf Nn is not an even number a1infloor(Nn)sample will be taken.ValueVector of size n with numbers from1to N indicating the index samples to be taken.If data is provided then the instances will be returned.Examplesdata<-runif(40)syst.sample(40,8,data)Indexcluster.estimator,2cluster.sample,3cluster.samplesize,4srs.domainestimator,5srs.estimator,6srs.sample,3,7srs.samplesize,8strata.allocation,9strata.estimator,10strata.sample,11,11strata.samplesize,12strata.samplesize.cost,13syst.all.samples,14syst.anova,15syst.estimator,15,17,18syst.intercorr,16,17syst.intracorr,16,17syst.sample,18var,1320。