We consider conditional estimation in two-stage sample size adjustable designs and the following bias. More specifically, we consider a design which permits raising the sample size when interim results look rather promising, and, which keeps the originally planned sample size when results look very promising. The estimation procedures reported comprise the unconditional maximum likelihood, the conditionally unbiased Rao-Blackwell estimator, the conditional median unbiased estimator, and the conditional maximum likelihood with and without bias correction. We compare these estimators based on analytical results and by a simulation study. We show in a real clinical trial setting how they can be applied.
In learning-phase clinical trials in drug development, adaptive designs can be efficient and highly informative when used appropriately. In this article, we extend the multiple comparison procedures with modeling techniques (MCP-Mod) procedure with generalized multiple contrast tests (GMCTs) to two-stage adaptive designs for establishing proof-of-concept. The results of an interim analysis of first-stage data are used to adapt the candidate dose-response models and the dosages studied in the second stage. GMCTs are used in both stages to obtain stage-wise p-values, which are then combined to d…
Response-adaptive allocation designs refer to a class of designs where the probability an observation is assigned to a treatment is changed throughout an experiment based on the accrued responses. Such procedures result in random treatment sample sizes. Most of the current literature considers unconditional inference procedures in the analysis of response-adaptive allocation designs. The focus of this work is inference conditional on the observed treatment sample sizes. The inverse of information is a description of the large sample variance of the parameter estimates. A simple form for the co…
This paper offers a new method for estimation and forecasting of the volatility of financial time series when the stationarity assumption is violated. Our general local parametric approach particularly applies to general varying-coefficient parametric models, such as GARCH, whose coefficients may arbitrarily vary with time. Global parametric, smooth transition, and change-point models are special cases. The method is based on an adaptive pointwise selection of the largest interval of homogeneity with a given right-end point by a local change-point analysis. We construct locally adaptive estima…
We present a Bayesian nonparametric model for conditional distribution estimation using Bayesian additive regression trees (BART). The generative model we use is based on rejection sampling from a base model. Typical of BART models, our model is flexible, has a default prior specification, and is computationally convenient. To address the distinguished role of the response in the BART model we propose, we further introduce an approach to targeted smoothing which is possibly of independent interest for BART models. We study the proposed model theoretically and provide sufficient conditions for …
This paper considers a semiparametric generalized autoregressive conditional heteroskedasticity (S-GARCH) model. For this model, we first estimate the time-varying long run component for unconditional variance by the kernel estimator, and then estimate the non-time-varying parameters in GARCH-type short run component by the quasi maximum likelihood estimator (QMLE). We show that the QMLE is asymptotically normal with the parametric convergence rate. Next, we construct a Lagrange multiplier test for linear parameter constraint and a portmanteau test for model checking, and obtain their asymptot…
Shrinkage algorithms are of great importance in almost every area of statistics due to the increasing impact of big data. Especially time series analysis benefits from efficient and rapid estimation techniques such as the lasso. However, currently lasso type estimators for autoregressive time series models still focus on models with homoscedastic residuals. Therefore, an iteratively reweighted adaptive lasso algorithm for the estimation of time series models under conditional heteroscedasticity is presented in a high-dimensional setting. The asymptotic behaviour of the resulting estimator is a…
Sequential Kernel-based Conditional Independence Testing via Adaptive Betting
Department of Computer Science, University of British Columbia, Vancouver, Canada（不列颠哥伦比亚大学计算机科学系，温哥华，加拿大）；Alberta Machine Intelligence Institute, Edmonton, Canada（阿尔伯塔机器智能研究所，爱德蒙顿，加拿大）
The problem of nonparametric estimation of the conditional density of a response, given a vector of explanatory variables, is classical and of prominent importance in many prediction problems since the conditional density provides a more comprehensive description of the association between the response and the predictor than, for instance, does the regression function. The problem has applications across different fields like economy, actuarial sciences and medicine. We investigate empirical Bayes estimation of conditional densities establishing that an automatic data-driven selection of the p…
Occupancy models involve both the probability a site is occupied and the probability occupancy is detected. The homogeneous occupancy model, where the occupancy and detection probabilities are the same at each site, admits an orthogonal parameter transformation that yields a two-stage process to calculate the maximum likelihood estimates so that it is not necessary to simultaneously estimate the occupancy and detection probabilities. The two-stage approach is examined here for the heterogeneous occupancy model where the occupancy and detection probabilities now depend on covariates that may va…
We introduce a nonparametric prior on the conditional distribution of a (univariate or multivariate) response given a set of predictors. The prior is constructed in the form of a two-stage generative procedure, which in the first stage recursively partitions the predictor space, and then in the second stage generates the conditional distribution by a multi-scale nonparametric density model on each predictor partition block generated in the first stage. This design allows adaptive smoothing on both the predictor space and the response space, and it results in the full posterior conjugacy of the…
Adaptive subgroup enrichment design is an efficient design framework that allows accelerated development for investigational treatments while also having flexibility in population selection within the course of the trial. The adaptive decision at the interim analysis is commonly made based on the conditional probability of trial success. However, one of the critical challenges for such adaptive designs is immature data for interim decisions, particularly in the targeted subgroup with a limited sample size at the first stage of the trial. In this paper, we improve the interim decision making by…
We investigate a confirmatory two stage adaptive procedure for assessing average bioequivalence and provide some insights to its theoretical properties. Effectively, we perform Two One-Sided Tests (TOST) to reach overall decision about each of the two traditional null-hypotheses involved in declaring average bioequivalence. The tests are performed as combination tests separately for each hypothesis based on the corresponding pair of stagewise p-values. Features of the procedure include a built in futility, sample size reassessment, and the ability to simultaneously assess average bioequivalenc…
We consider the estimation of average and counterfactual treatment effects, under two settings: back-door adjustment and front-door adjustment. The goal in both cases is to recover the treatment effect without having an access to a hidden confounder. This objective is attained by first estimating the conditional mean of the desired outcome variable given relevant covariates (the "first stage" regression), and then taking the (conditional) expectation of this function as a "second stage" procedure. We propose to compute these conditional expectations directly using a regression function to the …
Density regression provides a flexible strategy for modeling the distribution of a response variable $Y$ given predictors $\mathbf{X}=(X_1,\ldots,X_p)$ by letting that the conditional density of $Y$ given $\mathbf{X}$ as a completely unknown function and allowing its shape to change with the value of $\mathbf{X}$. The number of predictors $p$ may be very large, possibly much larger than the number of observations $n$, but the conditional density is assumed to depend only on a much smaller number of predictors, which are unknown. In addition to estimation, the goal is also to select the importa…
This paper develops maximum score estimation of preference parameters in the binary choice model under uncertainty in which the decision rule is affected by conditional expectations. The preference parameters are estimated in two stages: we estimate conditional expectations nonparametrically in the first stage and then the preference parameters in the second stage based on Manski (1975, 1985)'s maximum score estimator using the choice data and first stage estimates. The paper establishes consistency and derives rate of convergence of the two-stage maximum score estimator. Moreover, the paper a…
