Journal of Econometrics , 37 Weak convergence of the sample distribution function when parameters are estimated. The Annals of Statistics, 1, Taylor and M. Weak convergence of non-stationary multivariate marked processes with applications to martingale testing. Journal of Multivariate Analysis, 98, Generalized spectral tests for the martingale difference hypothesis. Journal of Econometrics, , Testing the martingale difference hypothesis using integrated regression functions.
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An adaptive, rate-optimal test of a parametric model against a nonparametric alternative. Econometrica, 69 3 , On tail index estimation using dependent data. Annals of Statistics, 19, Applied Nonparametric Regression. Cambridge Univ. Comparing nonparametric versus parametric regression fits. The Annals of Statististics, 21, Recent developments in the econometrics of program evaluation..
Journal of Economic Literature, 47 1 , Martingale approach to the goodness of fit tests. Theory of Probability and its Applications, 26, An innovation approach in goodness of fit tests in Rm. The Annals of Statistics, 16, Goodness of fit problem and scanning innovation martingales. Sulla determinazione empirica di una leggi di distribuzione. Giornale dell'Istituta Italiano degli Attuari 4, Nonparametric Model Checks for Time Series. The Annals of Statistics, 27, Nonparametric significance testing. Testing statistical hypotheses. Nonparametric Econometrics: Theory and Practice.
Princeton University Press. A simple consistent bootstrap test for a parametric regression function. Journal of Econometrics, 87, Model checks under random censorship. Statistics and Probability Letters, 32, Male-female wage differentials in urban labor markets. International Economic Review, The method is also applicable to other kinds of constrained nonparametric estimation, including monotone response function estimation.
This work is extended in Banerjee, et al. The first result is the development of second order efficient estimators. The basic idea and an interesting application of the concept is introduced in Dalalyan, et al. The essence of the idea is that a local minimax bound is precisely achieved for certain second order efficient estimators and that this criteria distinguishes among a number of first order efficient estimators. The theory is successfully applied to the shift parameter in a semiparametric Gaussian white noise model through a carefully tuned penalized maximum likelihood estimation process.
The technique appears to be promising for other semiparametric models as well, although there remains much work to be done before achieving this. The general research area of second order semiparametric efficiency has many interesting open questions. The second result involves increasing the accuracy of inference.
The profile sampler results presented in Lee, et al. The main result is that this approach is second order accurate in the presence of a nuisance parameter and also in the presence of regularization. At the time of publication of this work, parallel results were not yet available for the bootstrap, and thus it seemed that a Bayesian approach enjoyed higher order frequentist properties than any known frequentist alternatives. Results for the bootstrap, however, have since then started to move forward, and it appears that the bootstrap may also enjoy second order accuracy of inference.
Nevertheless, there remain yet a number of open questions about second order accuracy of inference, including, for example, how to adaptively select the smoothing parameter in penalized semiparametric maximum likelihood in order to lead to optimal accuracy of inference. Research activity in Bayesian semiparametric methods has continued actively since publication of WKR. There are too many contributions to completely enumerate here, but some highlights include interesting applications in random-effects based meta-analysis Burr and Doss, , isotonic regression Dunson, , reconstruction of past environment based on fossil data Bhattacharya, , and in high-dimensional additive models with applications to electrical utility analysis Panagiotelis and Smith, There have also been interesting developments in the analysis of recurrent events with dependent termination see Sinha, et al.
While most of these contributions are more practical than theoretical, several of them involve the creation of very clever computational methods, and a few include theoretical considerations such as establishing key non-asymptotic properties of the posterior distributions see, e. Unfortunately, very little new asymptotic theory for Bayesian semiparametric methods has been developed beyond Shen Specifically, a Bernstein-von Mises theorem under reasonably general conditions is still lacking, although there are rumors that such a result is forthcoming.
The main difficulty, of course, is that the nonparametric components of the models are very challenging to work with in Bayesian settings, and even consistency can be difficult to establish or verifiably impossible in some cases see, e. However, some progress has been made on methods for establishing consistency and rates of convergence for purely nonparametric models. An important result in this direction is the establishment of consistency of Bayesian density estimates using Dirichlet priors Ghosal and van der Vaart, An important component of this result is that Bayesian methods can achieve the same rate of convergence as the best possible frequentist approaches.
A key extension of this result is the discovery of easily verifiable sufficient conditions under which the Kullback-Leibler property for Dirichlet priors holds for Bayesian density estimation Wu and Ghosal, Conditions for achieving minimaxity in density estimation based on adaptive nonparametric beta mixture priors are given in Rousseau While these results are encouraging, progress is slow and much work remains to be done. An alternative to purely Bayesian approaches is to use approximate Bayes methods. The recent establishment of higher order accuracy of the profile sampler was mentioned previously.
On the other hand, when joint inference for both the parametric and nonparametric component is desired as in Kim and Lee, , for example , the piggyback bootstrap of Dixon, et al. The main idea is to use the profile sampler Lee, et al. The random weights are re-drawn for each j. It is unclear to what extent these approximations can be utilized for Bayesian inference, and more work, both philosophically and technically, is needed.
There has been significant progress in the theory of semiparametric transformation and frailty models beyond the results presented in WKR. For transformation models, many of the recent developments have focused on the setting where the residual distribution is known, or known up to a one-dimensional family. Fairly general results for efficient inference for both univariate and multivariate transformation models based on frailties is given in Zeng and Lin This article also contains a very interesting and extensive series of discussion articles from many experts in semiparametrics as well as a rejoinder by the authors.
Asymptotically valid inference and efficiency for normal transformation models for bivariate survival data is developed in Li, et al. Asymptotic normality, but without efficiency, for semiparametric shared frailty survival analysis in case-control studies is given in Gorfine, et al. These papers represent only a sampling of the many results in semiparametric transformation and frailty models for right-censored data. There has been significantly less work done for interval censored data.
The author is only aware of one paper for transformation models for interval censored data. That paper is Ma and Kosorok b who develop efficient inference for partly linear transformation models for current status data. An interesting issue is that the slow convergence rate for the baseline hazard seems to interfere with the convergence rate for the nonparametric additive regression functions. This is connected to the general issue of how to handle multiple nuisance parameters with potentially different convergence rates.
There remain many interesting open questions concerning transformation and frailty models for interval censored data. The Cox model for right-censored data with a change-point in the regression was studied first by Pons who was able to utilize the partial-profile likelihood to avoid nuisance parameter hazard function estimation.
A recent discovery is that the standard argmax theorem for M-estimators is not valid for the change-point parameter and a different approach is needed see Section In Kosorok and Song , the results in Pons are extended for general transformation models, and a test of the existence of a change-point was also developed.
A major difficulty for general transformation models is that the nuisance parameter can no longer be ignored and thus extreme care must be taken in the convergence rate calculations. There are a number of other interesting results for semiparametric change-point models which we will not fully enumerate here.
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A particularly interesting example is given in Lee and Seo who develop efficient estimation for a semiparametric binary response model with econometric applications. They also develop a test of the change-point similar to the test developed in Kosorok and Song. Many open theoretical questions for semiparametric change-point models remain, including what happens under model misspecification. It remains to be seen how this situation changes in the presence of infinite-dimensional nuisance parameters. Empirical processes techniques are necessary for most non-trivial theoretical research questions in semiparametrics.
This includes the development and use of Glivenko-Cantelli theorems to establish uniform consistency of suitably standardized objective functions for both existence and consistency of estimators.
Establishing this can be quite challenging: see, for example, the proofs of Theorems 1 and 3 of Kosorok, et al. Empirical processes are also needed for determining rates of convergence of estimators. This is often quite challenging, especially when more than one rate is involved. See both Chapter 3. Donsker theorems are usually needed for establishing weak convergence of both finite and infinite dimensional parameters. Both Z-estimation see Chapter 3. For a challenging example that uses most of these techniques, see the previously mentioned Kosorok and Song paper.
Another very powerful tool from empirical processes that is quite useful for semiparametric models is the empirical process bootstrap and related results. For a discussion of general empirical process bootstrap theory, see Chapter 3. Monte Carlo approaches such as the bootstrap are especially crucial for inference for infinite-dimensional parameters, such as the baseline integrated hazard in right-censored Cox regression and similar settings where there exists a tight limiting distribution.
In these settings, the infinite-dimensional aspects prevent using the estimated covariance function directly for inference as can be done in finite-dimensional settings, and confidence bands need to be constructed from Monte Carlo realizations of the approximate limiting distribution.
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Note that the weighted bootstrap see Section 2. For semiparametric models with all root- n consistent parameters, the validity of the bootstrap follows fairly directly from consistency and weak convergence validation. Establishing such validity for the parametric root- n consistent parameter in a semiparametric model with slower-than-root- n consistent nuisance parameters is much more difficult. An early version of such theory which requires arduous entropy calculations is given in Ma and Kosorok a. More recently, Cheng and Huang have developed very general bootstrap theory for this setting that is more straight-forward.
For parametric components that are not root- n consistent, the bootstrap is known to not be valid in some settings see, for example, Kosorok, b , and Sen, et al. The foregoing is only a subset of the many empirical process tools involved in semiparametric research, and, moreover, many of the most challenging questions in semiparametric inference will require the development of new theory and results in empirical processes. Expertise in empirical processes is thus absolutely essential for most of the many open research problems in semiparametrics.
Statistical applications in high dimensional data are quite challenging for a number of reasons, including the fact that assumptions generally need to be stronger than in lower dimensional settings before statistical procedures will work. For high dimensional data, the statistical trade-offs between data mining and inference are particularly acute.
A trivial example happens in linear regression when the number of predictors exceeds the number of observations: in these setting, model prediction can easily be perfect for the data used for estimation but disastrous for predicting future observations. The importance of the scientific questions raised earlier concerning the hope of semiparametrics for relaxing assumptions is both acute and complex when the dimension increases with the sample size.
Empirical process methods, such as maximal inequalities see Chapter 8.
A very interesting paper which discusses many of these issues is Donoho There have been a number of interesting, recent developments for using semiparametrics in high dimensional data, especially for microarrays. The earliest such example the author is aware of is Huang, et al. Additional developments for the semi-linear approach can be found in Huang, et al.
A robust semiparametric approach based on least-absolute deviation regression is developed in Ma, et al. Another interesting example is the univariate shrinkage approach based on the Lasso for Cox regression when the number of covariates p exceeds the sample size n Tibshirani, Because the Lasso is used, model selection is accomplished along with estimation. There are a number of other related results, but the applications and theoretical results for semiparametric models in high dimensions is in its infancy, and much work remains to be done.
This area could rapidly become one of the most active and vibrant research areas in all of statistics. A very crucial gap exists between semiparametric methodology and the routine use of semiparametric methods in practice. This gap is largely due to the limited availability of practical software for semiparametrics. Right-censored survival analysis is a fortunate exception to this since semiparametric survival analysis is widely used and revered in medical research.
This broad use is in part attributable to the availability of semiparametric survival analysis software in widely available packages such as SAS Cary, North Carolina, U. The author is not aware of any other examples where semiparametric methods are so broadly used and accepted in practice. The challenges that need to be overcome before practical software becomes more widely available for semiparametric methodology are three fold:.
The importance of developing effective software needs to be dramatically better recognized among semiparametric methods researchers, and there needs to be greater encouragement given to researchers interested in software implementation. Often the presence of tuning parameters in semiparametric inference algorithms precludes the development of usefully general algorithms which are capable of automatically selecting the tuning parameters, without requiring human intervention for each new setting.
In an increasing number of cases, the computational complexity is so great that expertise from computer science or operations research is needed before real-time computation is feasible. Thus, collaborations with computing specialists should be more frequently sought out and encouraged in semiparametric research. Hopefully it is clear by this point in the paper that semiparametrics is an interesting and important research area with tremendous growth potential in statistics, biostatistics and econometrics. Thus it is important to ensure that graduate education at universities is strengthened to accommodate this need.
There are several important points to consider in going forward with this strengthening process: First, semiparametrics is by now a classical area of statistics and should have a prominent place in the core graduate curriculum in statistics, biostatistics and econometrics. Second, it is not enough to just present a few examples in a few courses as is currently often done. Third, empirical process techniques are absolutely necessary for almost all cutting-edge research in this area and thus training should be available to all doctoral students in statistics, biostatistics and econometrics.
Fourth, some programs may need to increase their analytical theory training in the kind of analysis useful in empirical process and semiparametric research, especially functional analysis, although much of this could be accomplished as part of a course on empirical processes. Part of this included an upgrading of the two-semester core theory sequence of the Ph. For students interested in more technically advanced statistical research, such as semiparametrics and empirical processes, the core theory sequence is followed by a one semester advanced asymptotic statistics course at the level of van der Vaart followed by a one semester course on empirical processes and semiparametric inference at the level of K.
A similar two-course advanced sequence was implemented at UW in the recent past.
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The experience to date is that this advanced sequence is very successful and students emerge from this training very well prepared to do technically challenging research in semiparametrics. Empirical processes and some functional analysis is introduced in Chapters 18— Semiparametric inference and some additional functional analysis is presented in Chapter The K text is significantly more developed in the empirical processes and semiparametrics areas and will enable students to prepare for research in semiparametrics and also enable utilization of the classic texts in modern empirical processes, VW, and in modern semiparametric theory, BKRW.
Two other recent reference texts that can be useful for students interested in empirical process work are Pollard and van de Geer , although these books are somewhat narrow for use as course texts. The modeling and estimation of the conditional volatility associated with a stochastic process usually have been based on parametric ARCH-type and stochastic volatility models. These time series models are very powerful in representing the dynamic stochastic properties of the data generating process only if the parametric functions are correctly specified. The nonparametric approach acquires importance as a complementary and flexible method to explore these properties without imposing particular functional forms on the conditional moments of process.
Additionally, we estimate the conditional mean function under this approach. Key words: Nonparametric regression, Local polynomial regression, Nonlinear time series, Variance function estimation, Autoregressive conditional heteroscedasticity, Time series analysis. Ait-Sahalia, Y. Andersen, T. Baillie, R. Bollerslev, T. Bossaerts, P. Karlsruher Okonometrie-Workshop. Universit at Karlsruhe. Brock, W. Cabrera, J. Chen, R. Cleveland, W.