Approximate Solution And Convergence Of Parameter Estimation Of A Linear Model With Linear Inequality Constraints

In this paper we consider parameters estimation question of math model, which contains two linear fixed or mixed models with the same parameters with linear inequality constraints:Where ζ is a fixed effects vector, ξ_i is a random effects vector for i = 1,2,..., κ. Aζ = c is a linear inequality constraint,Y_i is an n-vector of observation, X,_i,U_i are n_i x p and n_i x q(n_i > p + q) matrix of fixed known values of rank p and rank q,and e_i is an n_i -vector of disturbances.These are double linear fixed and mixed models with linear inequality constraints.When A =I,b = 0,k== 1, letWhere T is subject to normal distribution. EΥ — O,CσνΥ = Σ =diag andfor E￡ = d > 0,note Y = U(￡ - d) 4- e,T = ( ] ,so the model (4.2.1) is stillchanged to:f Y = X< + Ud + U￡ - Ud + e,1 C>O,d>O,or( Y = (X.:U)r 4- Y, (5)r > o,whereY is subjected to normal distribution with independence,i?Y = 0,CavT = c^UU' 4- ( ^ ° ) (6)is satisfied.Now, (3) and (4) will be give a study in detail. First.the solution of (3) is given,while error variance matrix is known,rH o -gt-t (7)where Hy?denote the corresponding T*'s elements the matrix in H .IV' =(H'r-HrO"lHV'L. (8)Now assume Xr? denote the corresponding T*'s elements the matrix in X . then Hj- = E~aXr? ■ we have substituting Hr? = S~5Xr?,L = T,"^Y into (8),thus Fyis still expressed as:meanwhile, let F is the least square estimation of (3) with no restrictions, however7 V? ?we have■where T* denote the corresponding T* elements set in 'Ei 3 . next the solution of (4): 0, i￡T-T\ whereT1 = {1,2, ? ? ? ,p + q}. U is a vertical matrix,(X:U)(X:U)' > 0, I* C T1While error variance matrix is unknown,we give iterative sequence and approximate solution of problem (3)Step 0 About X and Y,giVing a group of data- y = j J and x = I },V y2;\ x2}using the method of article [6] gaining T" ,and computing:and the ordinary value for af :Step 1 substituting cr2]^^^, computing:IV(o) - (x'rE-1xT.)-1x'i.Xl-ly) and output:(^ 0, ieT-T*. let n = 1;Step 2 computing:and output;(cr2L(n-j,cr22(n))'.then using v2i(n)>a22(n) to replace o\,a\, computingand output:Step 3 let n = n -I-1 ,to Step 2.For the above algorithm,we can have two sequence:furthermore,get estimated sequence and approximate solution of F, and (of, #2)'-Similarly, we can have AIM algorithm about mixed model and gain two estimated sequence, and approximate. solution,too:) in=l' U* l(n)? ^ 2(n )To prove the convergence that the two estimated sequence in probability l,we give convergent condition:LEMMAl'9' Assume that /(x;\fr, Q(r,U0) is positive definite for Uo;? in RP x R%,Q(T,U) is semi-convex function for T,U .THEOREM 2 Note U = (a\,^f)', arid^ HS-^y^ - x.f.)))2 + iis-*xt(f t - r)ip4 ni_p4.|:r*let vector function/2(U) = (X'E^XJ-^'E"^,here S~l = dio^(cr^ ,--,ct^ ,0*2 ,--, ^ convergent in probability 1,which is originated from AIM algorithm.Furthermore theorem from 5-8 proved that sequence {r^/n>}, {(v2un), (r22(n))'} is convergent in probability l,too. they can be obtained using the methods as above-mentioned.