Model averaging estimation for generalized partially linear varying‐coefficient models

As an important part of modern health care, medical imaging data, which can be regarded as densely sampled functional data, have been widely used for diagnosis, screening, treatment, and prognosis, such as finding breast cancer through mammograms. The aim of this paper is to propose a functional linear regression model for using functional (or imaging) predictors to predict clinical outcomes (e.g., disease status), while addressing missing clinical outcomes. We introduce an exponential tilting semiparametric model to account for the nonignorable missing data mechanism. We develop a set of estimating equations and its associated computational methods for both parameter estimation and the selection of the tuning parameters. We also propose a bootstrap resampling procedure for carrying out statistical inference. Under some regularity conditions, we systematically establish the asymptotic properties (e.g., consistency and convergence rate) of the estimates calculated from the proposed estimating equations. Simulation studies and a real data analysis are used to illustrate the finite sample performance of the proposed methods.

Fragmentary data is becoming more and more popular in many areas which brings big challenges to researchers and data analysts. Most existing methods dealing with fragmentary data consider a continuous response while in many applications the response variable is discrete. In this paper, we propose a model averaging method for generalized linear models in fragmentary data prediction. The candidate models are fitted based on different combinations of covariate availability and sample size. The optimal weight is selected by minimizing the Kullback–Leibler loss in the completed cases and its asymptotic optimality is established. Empirical evidences from a simulation study and a real data analysis about Alzheimer disease are presented.

In this paper, we study optimal model averaging estimators of regression coefficients in a multinomial logit model, which is commonly used in many scientific fields. A Kullback–Leibler (KL) loss-based weight choice criterion is developed to determine averaging weights. Under some regularity conditions, we prove that the resulting model averaging estimators are asymptotically optimal. When the true model is one of the candidate models, the averaged estimators are consistent. Simulation studies suggest the superiority of the proposed method over commonly used model selection criterions, model averaging methods, as well as some other related methods in terms of the KL loss and mean squared forecast error. Finally, the website phishing data is used to illustrate the proposed method.

Outlier detection is imperative in biomedical data analysis to achieve reliable knowledge discovery. In this paper, a new outlier detection method based on Kullback-Leibler (KL) divergence is presented. The original concept of KL divergence was designed as a measure of distance between two distributions. Stemming from that, we extend it to biological sample outlier detection by forming sample sets composed of nearest neighbors. To handle the non-linearity during the KL divergence calculation and to tackle with the singularity problem due to small sample size, we map the original data into a higher feature space and apply kernel functions without resorting to a mapping function. A sample possessing the largest KL divergence is detected as an outlier. The proposed method is tested with one synthetic data, two public gene expression data sets, and our own mass spectrometry data generated for prostate cancer study.

Wind power curve (WPC) is an important index of wind turbines, and it plays an important role in wind power prediction and condition monitoring of wind turbines. Motivated by model parameter estimation of logistic function in WPC modelling, aimed at the problem of selecting initial value of model parameter estimation and local optimum result, based on the combination of genetic algorithm and least square estimation method, a genetic least square estimation (GLSE) method of parameter estimation is proposed, and the global optimum estimation result can be obtained. Six evaluation indices including the root mean square error, the coefficient of determination R2, the mean absolute error, the mean absolute percentage error, the improved Akaike information criterion and the Bayesian information criterion are used to select the optimal power curve model in the different candidate models, and avoid the model’s over-fitting. Finally, to predict the annual energy production and output power of wind turbines, a two-component Weibull mixture distribution wind speed model and five-parameter logistic function power curve model are applied in a wind farm of Jiangsu Province, China. The results show that the GLSE approach proposed in this paper is feasible and effective in WPC modelling and wind power prediction, which can improve the accuracy of model parameter estimation, and five-parameter logistic function can be preferred compared with high-order polynomial and four-parameter logistic function when the fitting accuracy is close.

This paper considers a Kullback-Leibler distance (KLD) which is asymptotically equivalent to the KLD by Goutis and Robert [1] when the reference model (in comparison to a competing fitted model) is correctly specified and that certain regularity conditions hold true (ref. Akaike [2]). We derive the asymptotic property of this Goutis-Robert-Akaike KLD under certain regularity conditions. We also examine the impact of this asymptotic property when the regularity conditions are partially satisfied. Furthermore, the connection between the Goutis-Robert-Akaike KLD and a weighted posterior predictive p-value (WPPP) is established. Finally, both the Goutis-Robert-Akaike KLD and WPPP are applied to compare models using various simulated examples as well as two cohort studies of diabetes.

Among several semiparametric models, the Cox proportional hazard model is widely used to assess the association between covariates and the time-to-event when the observed time-to-event is interval-censored. Often, covariates are measured with error. To handle this covariate uncertainty in the Cox proportional hazard model with the interval-censored data, flexible approaches have been proposed. To fill a gap and broaden the scope of statistical applications to analyze time-to-event data with different models, in this paper, a general approach is proposed for fitting the semiparametric linear transformation model to interval-censored data when a covariate is measured with error. The semiparametric linear transformation model is a broad class of models that includes the proportional hazard model and the proportional odds model as special cases. The proposed method relies on a set of estimating equations to estimate the regression parameters and the infinite-dimensional parameter. For handling interval censoring and covariate measurement error, a flexible imputation technique is used. Finite sample performance of the proposed method is judged via simulation studies. Finally, the suggested method is applied to analyze a real data set from an AIDS clinical trial.

This study examines an adaptive log-density estimation method with an $$\ell _1$$ -type penalty. The proposed estimator is guaranteed to be a valid density in the sense that it is positive and integrates to one. The smoothness of the estimator is controlled in a data-adaptive way via $$\ell _1$$ penalization. The advantages of the penalized log-density estimator are discussed with an emphasis on wavelet estimators. Theoretical properties of the estimator are studied when the quality of fit is measured by the Kullback–Leibler divergence (relative entropy). A nonasymptotic oracle inequality is obtained assuming a near orthogonality condition on the given dictionary. Based on the oracle inequality, selection consistency and minimax adaptivity are proved under some regularity conditions. The proposed method is implemented with a coordinate descent algorithm. Numerical illustrations based on the periodized Meyer wavelets are performed to demonstrate the finite sample performance of the proposed estimator.

In this paper, we propose a novel and robust procedure for model identification in semiparametric additive models based on rank regression and spline approximation. Under some mild conditions, we establish the theoretical properties of the identified nonparametric functions and the linear parameters. Furthermore, we demonstrate that the proposed rank estimate has a great efficiency gain across a wide spectrum of non-normal error distributions and almost not lose any efficiency for the normal error compared with that of least square estimate. Even in the worst case scenarios, the asymptotic relative efficiency of the proposed rank estimate versus least squares estimate, which is show to have an expression closely related to that of the signed-rank Wilcoxon test in comparison with the t-test, has a lower bound equal to 0.864. Finally, an efficient algorithm is presented for computation and the selections of tuning parameters are discussed. Some simulation studies and a real data analysis are conducted to illustrate the finite sample performance of the proposed method.

In this paper, we present a new robust estimation procedure for semi-functional linear regression models by using exponential squared loss. The outstanding advantage of the proposed method is the resulting estimators are more efficient than the least squares estimators in the presence of outliers or heavy-tail error distributions. The slope function and functional predictor variable are approximated by functional principal component basis functions. Under some regularity conditions, we obtain the optimal convergence rate of slope function, and the asymptotic normality of parameter vector and variance estimator. Finally, we investigate the finite sample performance of the proposed method through a simulation study and real data analysis.

The seamless-L0 (SELO) penalty is a smooth function that very closely resembles the L0 penalty, which has been demonstrated theoretically and practically to be effective in nonconvex penalization for variable selection. In this paper, the authors first generalize the SELO penalty to a class of penalties retaining good features of SELO, and then develop variable selection and parameter estimation in Cox models using the proposed generalized SELO (GSELO) penalized log partial likelihood (PPL) approach. The authors show that the GSELO-PPL procedure possesses the oracle property with a diverging number of predictors under certain mild, interpretable regularity conditions. The entire path of GSELO-PPL estimates can be efficiently computed through a smoothing quasi-Newton (SQN) with continuation algorithm. The authors propose a consistent modified BIC (MBIC) tuning parameter selector for GSELO-PPL, and show that under some regularity conditions, the GSELOPPL- MBIC procedure consistently identifies the true model. Simulation studies and real data analysis are conducted to evaluate the finite sample performance of the proposed method.

Kullback–Leibler life time testing

ABSTRACTIn this study, we consider an entropy-type goodness-of-fit (GOF) test based on integrated distribution functions. We first construct the test for the simple vs. simple hypothesis and then extend it to the composite hypothesis case. It is shown that under regularity conditions, the null limiting distribution of the proposed test is a function of a Brownian bridge. A bootstrap method is also considered and is shown to be weakly consistent. A simulation study and real data analysis are conducted for illustration.

We describe and develop a close relationship between two problems that have customarily been regarded as distinct: that of maximizing entropy, and that of minimizing worst-case expected loss. Using a formulation grounded in the equilibrium theory of zero-sum games between Decision Maker and Nature, these two problems are shown to be dual to each other, the solution to each providing that to the other. Although Topsøe described this connection for the Shannon entropy over 20 years ago, it does not appear to be widely known even in that important special case. We here generalize this theory to apply to arbitrary decision problems and loss functions. We indicate how an appropriate generalized definition of entropy can be associated with such a problem, and we show that, subject to certain regularity conditions, the above-mentioned duality continues to apply in this extended context. This simultaneously provides a possible rationale for maximizing entropy and a tool for finding robust Bayes acts. We also describe the essential identity between the problem of maximizing entropy and that of minimizing a related discrepancy or divergence between distributions. This leads to an extension, to arbitrary discrepancies, of a well-known minimax theorem for the case of Kullback–Leibler divergence (the “redundancy-capacity theorem” of information theory). For the important case of families of distributions having certain mean values specified, we develop simple sufficient conditions and methods for identifying the desired solutions. We use this theory to introduce a new concept of “generalized exponential family” linked to the specific decision problem under consideration, and we demonstrate that this shares many of the properties of standard exponential families. Finally, we show that the existence of an equilibrium in our game can be rephrased in terms of a “Pythagorean property” of the related divergence, thus generalizing previously announced results for Kullback–Leibler and Bregman divergences.

In this article, a general goodness of fit test is developed by using a new estimate of Kullback-Leibler (KL) information based on Type-II censored data. The proposed test is consistent, and the test statistic is nonnegative, just like KL information. Then, the test statistic is used to test for exponentiality based on Type-II censored data. Through a simulation study, power values of the proposed test are compared with some prominent existing tests. A real-life data analysis is finally presented for illustrative purpose.