Semiparametric Adjustment For Nonparametric And Semiparametric Models With Independent And Dependent Data

Traditionally, there are two approaches for estimation of functions in statistical analysis. One is called the parametric approach, the other, nonparametrie.The nonparametric approach has attractive flexibility, however, in the multivariate setting, its underlying function cannot be estimated with reasonable accuracy due to the so-called "curse of dimensionality". To avoid this problem, there have been increasing interests and activities in the general area of semiparametric approach, such as additive modelling, partially linear modelling, etc.In chapter 1, we introduce some fundamental nonparametric and semiparametric models.For nonparametrie models, a representative method is kernel smoothing which includes kernel density estimation and kernel regression estimation. In the context of kernel regression, traditional approaches have included the Nadaraya-Watson estimator , Gasser-Muller estimator and local polynomial estimator.For semiparametric models, we introduce some typical forms such as additive modeling, partially linear modeling, generalized additive modeling, generalized partially linear modeling and generalized partially linear partially additive modeling.The estimation of the common probability density function of a univariate random sample is a fundamental problem in statistics. Let X₁, ... , X_n be independently and identically distributed with density f. To estimate f, we can use parametric approaches such as maximum likelihood estimation, or use nonparametrie approaches such as kerned density estimation.The nonparametrie model has attractive flexibility; however, the parametric model is difficult to discount because a well estimated structure by the parametric rnodel is easy to understand. This motivates some authors (Hjort and Glad, 1995, Hjort and Jones, 1996, Naito, 2004) to propose a semiparametric approach with multiplicative adjustment which includes both the parametric approach and the nonparametric approach. In the proposed approach, the parametric density estimator g(x, (?)) is utilized, but it is seen as a crude guess of the true density f(x). This initial parametric approximation is adjusted via multiplication by an adjustment factorξ=ξ(x) which can be determined by nonparametric approaches using some criteria. Hjort and Glad (1995) proposed a density estimator based on the naive estimator ofξ. Hjort and Jones (1996) suggested and investigated two versions of multiplicative density estimator. Naito (2004) proposed a local L₂-fitting criterion with index a, including the above estimators proposed by Hjort and Glad (1995) and Hjort and Jones (1996) as special cases. Theoretical comparison reveals that the estimators in this class are better than, or at least competitive with, the traditional kernel estimator in a broad class of densities.However, the focus of all of the above mentioned papers is on i.i.d. observations. The statistical properties of this semiparametric approach for dependent data, have so far not been studied. The need for nonlinear time series modeling, estimating time trend, constructing predictive intervals, understanding divergence of nonlinear time eries compels us to consider the dependent data case.In chapter 2, we extend this semiparametric approach to time series context. We discuss its asymptotic theory and give some simulation studies. We find that, under certain mixing conditions, the results are very much like those for independent samples: the bias is unaffected by dependence and the asymptotic: variance is the same as in the i.i.d. case. However, we also find that the result for the asymptotic: variance heavily depends on the strength of the dependence among the samples. Let z, = K_h(X_t - x)g₀(X_t)^1-a, thenThe first term on the right hand side is equal to the variance based on independent data. The second term reflects the extra variability due to the dependence of the X_t's. The strength of the dependence can be measured through the size of (?) Cov(z₁,z₁₊₁). For anα-mixing process, a standard result isSoVar(?)α(x) = 1/ThR(K)f(x) + o(1/Th).However, if without certain condition, the size of the second term on the right side of (*) can be significantly larger and can dominate the usual leading variance term, leading to a worse rate of convergence of (?)_α(x).In chapter 3, we consider the. univariate regression problem of estimating the conditional mean functionm(x) = E(Y|X = x),when we observe a stationary sequence (X₁, Y₁), ... , (X_T, Y_T).In the i.i.d. case, Glad (1998) presented a new approach to regression function estimation in which a nonparametric regression estimator was guided by a parameter pilot estimate with the aim of reducing the bias. He introduced new classes of para-metrically guided kernel weighted local polynomial estimators, and derived formulas for asymptotic expectation and variance, hence approximated mean squared error and mean integrated squared error. It was shown that the new classes of estimators have the very same large sample variance as the estimators in the standard nonparametric setting, while there is substantial room for reducing the bias if the chosen parametric pilot function belongs to a wide neighborhood around the true regression line.In this chapter, we show that this approach can be extended to the case of mixing dependent data, forρ-mixing andα-mixing processes. The asymptotic theory and simulation study are discussed. Although the result for the stationary processes with certain mixing conditions is the same as that in the i.i.d., the proof is different: for independent data, we can obtain the bias and variance expression via conditioning on the design matrix X. However, for dependent observations, conditioning on X would mean conditioning on nearly the entire series. Hence we derive the asymptotic bias and variance using the asymptotic normality rather than conditional expectation.In chapter 4. we propose an adaptive semiparametric estimation for the nonparametric component of the following fixed design partially linear model:Y_i = x_i^τβ+ g(t_i) +ε_i, i=1,... ,n where {x_i} and {t_i} are respectively p-dimensional and scalar explanatory variables, data {Y_i} are observed at (x_i,t_i),βis an unknown p-dimensional parameter, g(t) is an unknown smooth function for t∈[0,1], andε₁, ... ,ε_n are independent and identically distributed with mean zero and varianceσ². Without loss of generality, assume that the design points t₁, ... ,t_n satisfy 0≤t₁ < t₂ ...< t_n≤1.Let (?)_n(t) be the estimator of g(t) by the usual estimation procedure. Under some regularity conditions, the mean squared error (MSE) of (?)_n(t) satisfiesMSE((?)_n(t)) =σ²/nh + O(h⁴) + o(n^-1h^-1) + o(h⁴).This suggests that the optimal choice for h is proportional to n^-1/5 and then MSE ((?)_n (t)) is proportional to n^-4/5.Note that n^-4/5 is the standard convergence rate of nonparametric estimation. However, although we do not know which form g(t) has, we are very interested in finding an estimation procedure to guarantee that the estimator of g(t) has the following properties: if g(t) is in fact a parametric function, then the estimator has parametric convergence rate; otherwise, the estimator has the nonparametric convergence rate. In other words, the estimation procedure should adapt to the model function g(t).In this chapter, we will utilize a semiparametric adjustment approach to solve the above problem. The resulting estimator can achieve a satisfactory convergence rate, even the parametric convergence rate, when regression function space has "good" properties in the sense that the underlying functions are sufficiently smooth or are already parametric.It is worth pointing out that, in our estimation procedure, we need not to have any knowledge about the function g(t). The convergence rate of the MSE of the new estimator is of optimal nonparametric rate O(n^-4/5) generally and can achieve the parametric rate O(n^-1) on some conditions. However, for the usual nonparametric estimator, its convergence rate is at most O(n^-4/5), and in this sense, the new estimator is better than the traditional nonparametric method.Furthermore, in order to determine the adjustment factor, we use a local L₂-fitting criteria similar to the idea of Hjort and Jones (1996), however, the method involved is different- To estimate the density function, the method of Hjort and Jones (1996) is based on the empirical version. However, in the regression case, the empirical version is unavailable, so we propose a new method, i.e. we use some approximations to the integrals in the adjustment factor. In this chapter, we also extend this method to random design regression model.In chapter 5, we are concerned with the estimation of the generator of backward stochastic differential equations (BSDEs) using additive models. This type of equation appears in numerous problems in finance. The generator g plays an important role in this area. However, the statistical analysis for the generator has not been explored systematically . We consider a kind of Markovian FBSDE (EL Karoui, Peng and Quenez, 1997),Assume that we are given the data {(X_ti,Y_ti), i = 0, ... , n) sampled at. discrete time points, t₀ < ... < t_n, and for simplicity, assume that the time points are equally spaced, letâ–³= t_i+1-t_i. From theoretical analysis and some examples, we propose that the form of g satisfies an additive structure, so we can estimate it by the method of additive models.For the estimation of additive models, there exist several methods, such as back-fitting, marginal integration estimator, Horowitz and Mammen's two-stage estimator and Lin, Cui and Zhu's adaptive two-stage estimator. In this chapter, we use a method similar to the adaptive two-stage estimation method (Lin. Cui and Zhu, 2006). But. there are some differences. One difference is that, to determine the adjustment factor in the second-stage, we use the local polynomial fitting instead of the L₂-fitting criteria in Lin, Cui and Zhu (2006), which will utilize the advantages of local polynomial estimator ; Another difference is that, in the BSDEs case what we observed is time series ata, while the method proposed by Lin, Cui and Zhu (2006) is on i.i.d. observations. so we must address the problem of dependency to derive the asymptotic behavior.