Sieve MLE For Errors In Variables Model With Interval Censoring

Consider semiparametric errors in variables model:We make some explanations and assumptions for this model as follow:1. Parametersβ= (β₁,…,β_d)T∈A,A is a bounded closed set in R^d, g is anunknown Borel function. We nameβas parametric vector,and g as nonparametricvector .2. Explained variable X^o is not observed,but we can observe X which containserror e. Xo is bounded,then exists M₁ > 0, we have P(||X^o|| > M₁) = 0,T can beobserved and T∈[0,1].3. The distribution of Random error e is F_e,and is known. density function feis continuous and di?erence, and (?)x∈R^d,f_e(x) > 0 ,Ee = 0, e is independent of(β,g) and (X^o,T). e is a bounded random variable, then exist a bounded set C (?) R^d,s.t. e∈C.We can also know that exist M₂ > 0, s.t P(||e|| > M₂) = 0.4. Assume the joint density function of (X^o,T) isφo(x,t), and it is independentof (β,g). Thus we can get the joint density function of (X,T),φ(x,t) is independentof (β,g).5. The respond variable Y is not observed, we can only get its I interval censoredobservations, we have a bounded random variable Z,then it makesδ= I{Y Z} and Z are observed. In this paper Z is independent of (X^o,T), Z is independent of e, thenZ is independent of (X,T).λ(z) which is the density function of Z is independent of(β,g).6. The distribution of random errorεis Fε,and known.The density function fεcontinuous and di?erence, (?)x∈R,fε(x) > 0,Eε= 0, random errorεis independentof (X^o,T,Z), andεis independent of e, we know thatεis independent of (X,T,Z).Assume fεis a single apex function, its both sides are quickly decreasing, even f_ε(x)have a up bound.7. I interval censoring is di?erent from right censoring, in the right censoring,ifvariable Y is smaller than Z , then we can observe Y , else if Y is larger than Z,we can't observe Y , but in I interval censoring, we can't observe Y no matter whatcondition is. we only can observe Z, and the relation between Y and Z.8. Because X^o is bounded, e is also bounded, we can know X is also bounded.Noteθ= (β,g)^T, we call it as parameter of the model.θ₀ = (β₀,g0)^T is reallyvalue of the model. Notewhere r = 1 or 2, m0 and M0 are known. Assume parametric space isΘ= {θ:θ∈A (?) B} = A (?) B.Model(1.1) can be transformed toNote U =ε-β^Te, The distribution of U is F_β,ε,e,its density function is f_β,ε,e.Lemma 2.1 The joint density function of (X,T) isandφ(x,t) is independent of (β,g).Lemma 2.2 The density function of U is f_U(u) = C_fε(u +β^Tv)f_e(v)dv. Lemma 2.3Theorem 2.1 Note W = (X,T,Z,δ)^T. The density function of W isThe control measureμis a product measure which is consist of Lebesgue measure inR^d+2 and count measure in {0,1}.Note its logarithm likelihood function is l(w,θ) = log Q(w,θ).Pθis a probability distribution of W under condition that model parameter isθ,E0 is expectation under P_θ0.By using the concavity of log x and Jensen inequation, we getθ0 is the max pointof E_0l(θ,W).Because the parametric space ofΘis infinite dimensional, we can approach it byusing a series of finite dimensional spaces. We estimateθin these finite dimensionalspaces, and get a series of{θn}, we call this method as Sieve. Concretely speaking:1. Assumeρis a distance onΘ.(or"false distance", see[30]).2. Ln(θ,Xn) is a function of observation X_n andθwhen we have n samples. Itcan re(?)ect goodness of fit between model and parameter when the parameter isθ. Wecall it as empirical criterion function.3.Θ₁,Θ₂,···,Θn,···is a series of sets, they are used to approach toΘ. Howto approach,?θ∈Θ, (?)Ï€_nθ∈Θ_n, s.t when n→∞,ρ(Ï€_nθ,θ)→0, reference [31] call{Θ_n} is Sieve spaces,θn which satisfy following conditions is Sieve MLE.whereηn ?→0,(n→∞)。The distance can be various,for example L₂ distance,Hellinger distance. The log-arithm likelihood function and quadratic punish function can be regarded as criterionfunction. Sieve space can be built by triangle series,subsection linear function,B-splineet. Even we choose quadratic punish function as criterion function, the estimation isalso called as Sieve MLE, and don't call it as least square Sieve estimation(see[30]).η_ndon't e(?)ect asymptotic property of estimation. AssumeW_i = (X_i,T_i,Z_i,δ_i)^T.(i = 1,2,···,n) are i.i.d samples, note W_n = (W₁,···,W_n)^T,P_n is a empirical probability measure.We construct the Sieve MLE ofθ.forθ_i = (β_i,g_i)T∈Θ,i = 1,2, define distanceρthis is a kind of L2 distance.thenwhenδ= 1,whenδ= 0in a word:Lemma 3.1 Use marks which appear in above paragraphs, whenξis betweenξ1andξ₂, we getand exist two positive constantsNamely is uniform bounded on x. Theorem 3.1 The L2 distance defined in this paper is equal to Hellinger distance.The Hellinger distance isNextly we construct Sieve spacesAssume 0 = t0 < t₁ <···< t_m = 1 is a kind of partition in interval [0,1],usuallywe select m through sample n, generally we choose m = O(n^k), 0 < k < 1, and makethe partition satisfy where C is constant.Apparently Gm is subsection linear function which is built by jointing (t₀,b₀),(t₁,b₁),···,(t_m,b_m)in turn, b is coe?cient of Gm. NoteTheorem 3.2 For allθ= (β,g)^T∈Θ, sample n and relevant m, selectThus we can chooseΘn = A (?) B as Sieve space ofΘ, select empirical criterionfunction in (1.4)Note Sieve MLE ofθ₀ isθ_n = (β_n, g_n)T. Thenwhere Because f_ε(u +β^Tv) is continuous onβ, we know fU|X(u) is continuous onβ, further-more we know FU|X(u) is continuous onβ, and z -β^Tx- g(t) is continuous on (β,g),so Ln(θ,W_n) is continuous onθ.BecauseΘn is bounded closed set, from theorem (3.2) we knowΘn has inner point.In a word,θ_n must exist.Lemma 4.1 Define two functional collectionaG = {g(·)}, F = {f(g(·)) : g∈G,f satisfies following Lipschitz condition}Lipschitz condition: (?)f(t1),f(t2)∈F,exist constant C, s.tFor random probability measure P, theε? covering numbersN(ε,F,L₂(P)) and N(ε,G ,L₂(P)) of classes F,G (see[25]P₂₅,P₃₁,see reference) haverelations as follow :Remark: This idea comes from reference [32][33], furthermore it can extend tocovering numbers under di(?)erent measure,as long as conclusion which is similar to(1.10) follows. For example we have two measure P and Q, fromwe getwhere constant C can relate to n.Lemma 4.2 N(ε,Bn,L∞) Lemma 4.2 Lemma 4.3 FOr a elass of funetionA。={l(0,·):0ä»»0。},its eovering number Lemma 4.4 whenE(X一E(X}T))^o2>O,assume its the least eigenvalue inλ₁.We can detaeh Darametrie veetorβand nonDarametrie veetor 0 fromo(0.0。,.we get Theorem 4.1 Use conditions and marks in 1-3 sections, we getρ(θ_n,θ₀)→0, a.s.P_θ0. If satisfy following conditionsfurthermore we getTheorem 5.1 Use conditions and marks in 1-3 sections, we getif we select we can get the best convergence speed satisfy condition (1.11), furthermore we get...