# Semiparametric Inference

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### A semiparametric inference to regression analysis with missing covariates in survey data

Semiparametric inference based on an efficient score function has become more popular recently. The semiparametric efficient estimator of Robins, Rot- nitzky and Zhao (1994) and Robins, Hsieh and Newey (1995) achieves the semi- parametric information bound. Zhao, Lipsitz and Lew (1996) proposed a joint estimating equation approach for missing covariates by modeling the response mechanism. Wang and Paik (2006) and Didelez (2002) provided comparison of the aforementioned semiparametric efficient estimators. In the context of the missing covariate problem, the marginal distribution of the covariates can be viewed as a nuisance parameter. If the nuisance parameter is infinite-dimensional but the regression model itself is parametric, the joint model becomes semipara- metric. Zhang and Rockette (2005) considered the problem with a single covariate and obtained a semiparametric efficient estimator of the regression parameters but did not discuss an extension to complex survey sampling.

### Semiparametric inference for transformation models via empirical likelihood

Note that in Table 1 , we find that for a very small sample size n = 40 and very heavy censoring rate 40% both the NA method and EL method perform worse. The reason is that estimators of regression parameters are asymptotically biased for the estimating equations. To ensure better finite sample performance and also the consistency of the proposed estimator, a tail restriction is usually needed. Fine et al. [ 14 ] investigated this important problem for the linear transformation model, and proposed a modification of the estimation procedures [ 10 ] for regression parameters. In this paper, we did not apply the tail restriction [ 14 ] to the estimation equation for EL inference. Thus, it may deteriorate the performance of the proposed EL method for very heavy censoring, see Table 1 . It is worthwhile to investigate transformation models combining empirical likelihood and tail restriction. In the future, we will study this interesting and important issue and hope to improve the performance for very heavy censoring rates substantially.

### Semiparametric inference based on a class of zero-altered distributions

In modeling count data collected from manufacturing processes, eco- nomic series, disease outbreaks and ecological surveys, there are usually a relatively large or small number of zeros compared to positive counts. Such low or high frequencies of zero counts often require the use of un- der or over dispersed probability models for the underlying data generating mechanism. The commonly used models such as generalized or zero-inflated Poisson distributions can usually account for only the over dispersion, but such distributions are often found to be inadequate in modeling underdis- persion because of the need for awkward parameter or support restrictions. This article introduces a flexible class of semiparametric zero-altered models which account for both under and over dispersion and includes other famil- iar models such as those mentioned above as special cases. Consistency and asymptotic normality of the dispersion parameter are derived under general conditions. Numerical support for the performance of the proposed method of inference is presented for the case of common discrete distributions.

### Enhancing Statistician Power: Flexible Covariate-Adjusted Semiparametric Inference for Randomized Studies with Multivariate Outcomes

The semiparametric locally efficient estimator of marginal treatment effects for cor- related outcomes was applied to data from AIDS Clinical Trial Group Study 398 (ACTG 398) {Hammer et al. (2002)}. ACTG 398 was a multicenter, double-blind trial, in which 481 HIV-infected patients were randomized to one of four arms, A) saquinavir, B) indi- navir, C) nelfinavir, or D) placebo based on their past protease inhibitor (PI) treatment. Patients were only randomized to drugs to which they had no prior exposure. Random- ized treatments were given to all participants in combination with antiretroviral therapy. Subjects’ CD4 was measured at weeks 0 (baseline), 4, 8, and every 8 weeks thereafter until 48 weeks or dropout. GEE estimators were applied to compare the nelfinavir and placebo arms among patients who were eligible for both according to the stratified randomization scheme. Additional baseline covariates were age, sex, past PI use, past non-nucleoside reverse transcriptase inhibitor (NNRTI) exposure, weight, Karnofsky score, intravenous drug use, and race/ethnicity. Weeks 4-32 of followup were included for analysis, with CD4 measurements at week 4 and beyond included as outcomes and week 0 CD4 in- cluded as a baseline covariate. Data were approximately 90% complete through week 32. In evaluating the effect of treatment on CD4, the best fitting marginal model was E(Y ij |A i ) = β 0 + β 1 A i + β 2 t ij , where t ij indicates the week of the j th measurement on the

### Semiparametric inference in correlated long memory signal plus noise models

z t and the spectral density of the observable z t shares the divergency of f y (λ) at the origin with the same memory parameter d > 0. This spectral property entitles the estimation of the memory parameter of the latent signal using semiparametric or local techniques originally proposed for fully observable long memory series, which only consider spectral behaviour around frequency zero. However, the added noise affects the properties of these estimators, inducing a large bias which limits the efficiency by compelling the use of frequencies very close to the origin. This effect has been analyzed by Deo and Hurvich (2001) and Arteche (2004) for the log-periodogram regression and the local Whittle estimators respectively. To reduce this bias Sun and Phillips (2003), Hurvich et al. (2005) and Arteche (2006) propose modifications of both estimators that include the added noise in the estimation procedures. Sun and Phillips (2003) and Arteche (2006) consider only the case of independent signal and noise in a log periodogram regression context. Hurvich et al. (2005) extend the local Whittle estimator by including explicitly in the estimation procedure both the added white noise and the potential correlation between signal and noise by incorporating terms that account for the spectral density of u t and the non null cross spectral density of y t and

### Likelihood inference on semiparametric models with generated regressors

Although our assumptions on the bandwidths h and b are relatively mild, their optimal selec- tion rules are substantial open problems. In the existing literature on two-step semiparametric inference, most papers employ the MSE optimal or cross validation bandwidths for the first stage nonparametric estimation; see, e.g., BEV, Zhu and Xue (2006), Zhu, et al. (2010), and Xue and Xue (2011). In our simulation study below, we also choose the bandwidths h and b based on the MSE optimal rate for estimation of µ and ϕ, respectively, multiplied by several constants to check their robustness. However, it is not obvious whether the optimal bandwidths for nonparametric first stage estimation have desirable properties for inference on the parametric component β of interest. Indeed such literature on bandwidth selection for semiparametric inference is very thin. One promising way is to establish a higher order approximation for the coverage error (or size distortion) by our GEL statistic ℓ(β ), and to choose the bandwidths to minimize the coverage error (see, Nishiyama and Robinson, 2000, and Linton, 2002). Such higher order analysis is complicated even for the two-step inference, and we leave it for future research. 10

### Bayesian semiparametric GARCH models

In comparison to the semiparametric GARCH model with localized bandwidths, the t- GARCH model underestimates the VaR by an amount that is between \$0.168 and \$0.356 for a \$100 investment when the semiparametric model is favored against its competitor. Even though the semiparametric GARCH model is not favored against the t -GARCH model for FTSE, DAX and AORD return series, the t -GARCH model still underestimates the VaR by an amount and between \$0.13 and \$0.20 for a \$100 investment. This is not surprising. Due to the fallout of high volatilities that originated from the USA stock market during the global financial crisis, the frequency of observed deep downs during this period was higher than that during non-crisis periods in any mature stock market. Consequently, the left tail of the return density is thicker than the right tail. However, the symmetric Student t density fails to capture the asymmetric thickness between the two tails of a return density.

### "Implementing Nonparametric and Semiparametric Estimators"

The chapter begins by illustrating how flexible modeling methods have been applied in empirical research, drawing on recent examples of applications from labor economics, consumer demand estimation and treatment effects models. Then, key concepts in semiparametric and nonparametric modeling are introduced that do not have counterparts in parametric modeling, such as the so-called curse of dimensionality, the notion of models with an infinite number of parameters, the criteria used to define optimal convergence rates, and “dimension-free” estimators. After defining these new concepts, a large literature on nonparametric estimation is reviewed and a unifying framework presented for thinking about how different approaches relate to one another. Local polynomial estimators are discussed in detail and their distribution theory is developed. The chapter then shows how nonparametric estimators form the building blocks for many semiparametric estimators, such as estimators for average derivatives, index models, partially linear models, and additively separable models. Semiparametric methods offer a middle ground between fully nonparametric and parametric approaches. Their main advantage is that they typically achieve faster rates of convergence than fully nonparametric approaches. In many cases, they converge at the parametric rate.

### Semiparametric Estimation of Fractional Cointegration

We …nd it convenient to treat our case of nonparametric autocorrelation in the frequency domain. This prompts consideration of two alternative methods of estimating . One involves a ratio of weighted periodogram averages either across all frequencies in the Nyquist band, or only over those within a shrinking neighbourhood of zero frequency. The weighting is inverse with respect to smoothed esti- mates of f . Because of the concentration of spectral mass around zero frequency, where f changes little, computationally simpler estimates, with the same asymptotic properties, replace the weights by multiplicative factors based on an estimate of f (0). Both types of estimate are described in the follow- ing section. Regularity conditions and asymptotic properties are presented in Section 3. The conditions include some unprimitive ones on the estimates of , and f , and these are checked in Section 4 for particular estimates; this is an especially delicate issue in our semiparametric setting. Section 5 contains a Monte Carlo study of …nite-sample behaviour. All proofs are relegated to Appendices.

### Semiparametric multivariate volatility models

Estimation of multivariate volatility models is usually carried out by quasi max- imum likelihood (QMLE), for which consistency and asymptotic normality have been proven under quite general conditions. However, there may be a substan- tial efficiency loss of QMLE if the true innovation distribution is not multinormal. We suggest a nonparametric estimation of the multivariate innovation distribution, based on consistent parameter estimates obtained by QMLE. We show that under standard regularity conditions the semiparametric efficiency bound can be attained. Without reparametrizing the conditional covariance matrix (which depends on the particular model used), adaptive estimation is not possible. However, in some cases the efficiency loss of semiparametric estimation with respect to full information maximum likelihood decreases as the dimension increases. In practice, one would like to restrict the class of possible density functions to avoid the curse of dimen- sionality. One way of doing so is to impose the constraint that the density belongs to the class of spherical distributions, for which we also derive the semiparametric efficiency bound and an estimator that attains this bound. A simulation experiment demonstrates the efficiency gain of the proposed estimator compared with QMLE.

### Semiparametric Bayesian Quantile Regression.

In this chapter, we propose a semiparametric Bayesian method for quantile regression with random effects. We approximate the likelihood using the LIGPD method and pro- pose a Metropolis-within-Gibbs algorithm to update fixed and random effects. The pro- posed algorithm avoids the quantile crossing problem, and yields the joint posterior dis- tribution of quantile coefficients at multiple quantiles. Through simulation studies, we demonstrate that by approximating the likelihood through information-sharing across quantiles, the proposed method leads to more efficient multiple-quantile estimation than existing methods for quantile regression with random effects in finite samples.

### Bayesian Semiparametric Regression

We consider Bayesian estimation of restricted conditional moment models with linear regression as a particular example. The standard practice in the Bayesian literature for semiparametric models is to use flexible families of distributions for the errors and assume that the errors are independent from covariates. However, a model with flexible covariate dependent error distributions should be preferred for the following reasons: consistent estimation of the parameters of interest even if errors and covariates are dependent; possibly superior prediction intervals and more efficient estimation of the parameters under heteroscedasticity. To address these issues, we develop a Bayesian semiparametric model with flexible predictor dependent error densities and with mean restricted by a conditional moment condition. Sufficient conditions to achieve posterior consistency of the regression parameters and conditional error densities are provided. In experiments, the proposed method compares favorably with classical and alternative Bayesian estimation methods for the estimation of the regression coefficients.

### Semiparametric Fractional Cointegration Analysis

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### Inequality and Growth: A Semiparametric Investigation

In the absence of very strong assumptions regarding political processes, technology, preferences, endowments, the convexity of the factors of production (e.g. capital), and the complete[r]

### A semiparametric spatial dynamic model

Stimulated by the Boston house price data, in this paper, we propose a semiparametric spatial dynamic model, which extends the ordinary spatial au- toregressive models to accommodate the effects of some covariates associated with the house price. A profile likelihood based estimation procedure is pro- posed. The asymptotic normality of the proposed estimators are derived. We also investigate how to identify the parametric/nonparametric components in the proposed semiparametric model. We show how many unknown parame- ters an unknown bivariate function amounts to, and propose an AIC/BIC of nonparametric version for model selection. Simulation studies are conducted to examine the performance of the proposed methods. The simulation results show our methods work very well. We finally apply the proposed methods to analyze the Boston house price data, which leads to some interesting findings.

### Semiparametric identification in duration models

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### Semiparametric estimation of conditional copulas

The manner in which two random variables influence one another often depends on covariates. A way to model this dependence is via a conditional copula function. This paper contributes to the study of semiparametric estimation of conditional copulas by starting from a parametric copula function in which the parameter varies with a covariate, and leaving the marginals unspecified. Consequently, the unknown parts in the model are the parameter function and the unknown marginals. The authors use a local pseudo-likelihood with nonparametrically estimated marginals approximating the unknown parameter function locally by a polynomial. Under this general setting, they prove the consistency of the estimators of the parameter function as well as its derivatives; they also establish asymptotic normality. Furthermore, they derive an expression for the theoretical optimal bandwidth and discuss practical bandwidth selection. They illustrate the performance of the estimation procedure with data-driven bandwidth selection via a simulation study and a real-data case.

### Estimation in semiparametric spatial regression

Estimation in semiparametric spatial regression Gao, Jiti and Lu, Zudi and Tjostheim, Dag The University of Adelaide, London School of Economics, The University of Bergen... marginal add[r]

### Flexible semiparametric mixed models

There is an extensive body of literature on the linear mixed model, early highlights being Hen- derson (1953), Laird & Ware (1982) and Harville (1977). Nice overviews including more recent work are found in Verbeke & Molenberghs (2001), McCulloch & Searle (2001). In common linear mixed models the influence of covariates is restricted to a strictly parametric form. While in regression models much work has been done to extend the strict parametric form to more flexible forms of semi- and nonparametric regression, much less has been done to develop flexible mixed model. For overviews on semiparametric regression models see Hastie & Tibshirani (1990), Green & Silverman (1994) and Schimek (2000).