1 Motivation Suppose IXt}is a strictly stationary process with marginal probability density func- tion g()and pairwise joint probability density function fi(,y),and a random sample X of size T is observed.Then, .How to estimate the marginal pdf g(r)of [X:? .How to estimate the pairwise joint pdf fi(r,y)of (Xt,X)? How to estimate the autoregression function rj()=E(XX-j=z)? How to estimate the spectral density h(w)of [X}? .How to estimate the generalized spectral density f(w,u,v)of [X)? .How to estimate the bispectral density b(w1,w2)? How to estimate a nonlinear autoregressive conditional heteroskedastic model Xi=u(X:-1,...,Xi-p)+o(X:-1,...,Xi-q)Et,e}~i.i.d.(0,1) where u()and o()are unknown functions of the past information.Under certain regularity conditions,u()is the conditional mean of Xt given I1=[X-1,X-2,...} and o2()is the conditional variance of Xt given It-1. How to estimate a semi-nonparametric functional coefficient autoregressive process X=∑agX-X-+ E(el-1)=0a.s, =1 where ai()is unknown,and d>0 is a time lag parameter? How to estimate a nonparametric additive autoregressive process Xi= ∑，(X-）+et, E(et It-1)=0 a.s., j=1 where the ()functions are unknown? How to estimate a locally linear time-varying regression model Yi=XiB(t/T)+Et, where B(.)is an unknown smooth deterministic function of time?1 Motivation Suppose fXtg is a strictly stationary process with marginal probability density func￾tion g(x) and pairwise joint probability density function fj (x; y); and a random sample fXtg T t=1 of size T is observed. Then,  How to estimate the marginal pdf g(x) of fXtg?  How to estimate the pairwise joint pdf fj (x; y) of (Xt ; Xt￾j )?  How to estimate the autoregression function rj (x) = E(Xt jXt￾j = x)?  How to estimate the spectral density h(!) of fXtg?  How to estimate the generalized spectral density f(!; u; v) of fXtg?  How to estimate the bispectral density b(!1; !2)?  How to estimate a nonlinear autoregressive conditional heteroskedastic model Xt = (Xt￾1; :::; Xt￾p) + (Xt￾1; :::; Xt￾q)"t ; f"tg  i:i:d:(0; 1); where (
) and (
) are unknown functions of the past information. Under certain regularity conditions, (
) is the conditional mean of Xt given It￾1 = fXt￾1; Xt￾2; :::g and  2 (
) is the conditional variance of Xt given It￾1.  How to estimate a semi-nonparametric functional coe¢ cient autoregressive process Xt = X p j=1 j (Xt￾d)Xt￾j + "t ; E("t jIt￾1) = 0 a.s., where j (
) is unknown, and d > 0 is a time lag parameter?  How to estimate a nonparametric additive autoregressive process Xt = X p j=1 j (Xt￾j ) + "t ; E("t jIt￾1) = 0 a.s., where the j (
) functions are unknown?  How to estimate a locally linear time-varying regression model Yt = X 0 t(t=T) + "t ; where (
) is an unknown smooth deterministic function of time? 3