亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

In recent years, power analysis has become widely used in applied sciences, with the increasing importance of the replicability issue. When distribution-free methods, such as Partial Least Squares (PLS)-based approaches, are considered, formulating power analysis turns out to be challenging. In this study, we introduce the methodological framework of a new procedure for performing power analysis when PLS-based methods are used. Data are simulated by the Monte Carlo method, assuming the null hypothesis of no effect is false and exploiting the latent structure estimated by PLS in the pilot data. In this way, the complex correlation data structure is explicitly considered in power analysis and sample size estimation. The paper offers insights into selecting statistical tests for the power analysis procedure, comparing accuracy-based tests and those based on continuous parameters estimated by PLS. Simulated and real datasets are investigated to show how the method works in practice.

In this thesis, we study problems at the interface of analysis and discrete mathematics. We discuss analogues of well known Hardy-type inequalities and Rearrangement inequalities on the lattice graphs $\mathbb{Z}^d$, with a particular focus on behaviour of sharp constants and optimizers.In the first half of the thesis, we analyse Hardy inequalities on $\mathbb{Z}^d$, first for $d=1$ and then for $d \geq 3$. We prove a sharp weighted Hardy inequality on integers with power weights of the form $n^\alpha$. This is done via two different methods, namely super-solution and Fourier method. We also use Fourier method to prove a weighted Hardy type inequality for higher order operators. After discussing the one dimensional case, we study the Hardy inequality in higher dimensions ($d \geq 3$). In particular, we compute the asymptotic behaviour of the sharp constant in the discrete Hardy inequality, as $d \rightarrow \infty$. This is done by converting the inequality into a continuous Hardy-type inequality on a torus for functions having zero average. These continuous inequalities are new and interesting in themselves. In the second half, we focus our attention on analogues of Rearrangement inequalities on lattice graphs. We begin by analysing the situation in dimension one. We define various notions of rearrangements and prove the corresponding Polya-Szeg\H{o} inequality. These inequalities are also applied to prove some weighted Hardy inequalities on integers. Finally, we study Rearrangement inequalities (Polya-Szeg\H{o}) on general graphs, with a particular focus on lattice graphs $\mathbb{Z}^d$, for $d \geq 2$. We develop a framework to study these inequalities, using which we derive concrete results in dimension two. In particular, these results develop connections between Polya-Szeg\H{o} inequality and various isoperimetric inequalities on graphs.

Generative diffusion models have achieved spectacular performance in many areas of generative modeling. While the fundamental ideas behind these models come from non-equilibrium physics, variational inference and stochastic calculus, in this paper we show that many aspects of these models can be understood using the tools of equilibrium statistical mechanics. Using this reformulation, we show that generative diffusion models undergo second-order phase transitions corresponding to symmetry breaking phenomena. We show that these phase-transitions are always in a mean-field universality class, as they are the result of a self-consistency condition in the generative dynamics. We argue that the critical instability that arises from the phase transitions lies at the heart of their generative capabilities, which are characterized by a set of mean field critical exponents. Furthermore, using the statistical physics of disordered systems, we show that memorization can be understood as a form of critical condensation corresponding to a disordered phase transition. Finally, we show that the dynamic equation of the generative process can be interpreted as a stochastic adiabatic transformation that minimizes the free energy while keeping the system in thermal equilibrium.

The ParaDiag family of algorithms solves differential equations by using preconditioners that can be inverted in parallel through diagonalization. In the context of optimal control of linear parabolic PDEs, the state-of-the-art ParaDiag method is limited to solving self-adjoint problems with a tracking objective. We propose three improvements to the ParaDiag method: the use of alpha-circulant matrices to construct an alternative preconditioner, a generalization of the algorithm for solving non-self-adjoint equations, and the formulation of an algorithm for terminal-cost objectives. We present novel analytic results about the eigenvalues of the preconditioned systems for all discussed ParaDiag algorithms in the case of self-adjoint equations, which proves the favorable properties the alpha-circulant preconditioner. We use these results to perform a theoretical parallel-scaling analysis of ParaDiag for self-adjoint problems. Numerical tests confirm our findings and suggest that the self-adjoint behavior, which is backed by theory, generalizes to the non-self-adjoint case. We provide a sequential, open-source reference solver in Matlab for all discussed algorithms.

In this article, we study the Fekete problem in segmental and combined nodal-segmental univariate polynomial interpolation by investigating sets of segments, or segments combined with nodes, such that the Vandermonde determinant for the respective polynomial interpolation problem is maximized. For particular families of segments, we will be able to find explicit solutions of the corresponding maximization problem. The quality of the Fekete segments depends hereby strongly on the utilized normalization of the segmental information in the Vandermonde matrix. To measure the quality of the Fekete segments in interpolation, we analyse the asymptotic behaviour of the generalized Lebesgue constant linked to the interpolation problem. For particular sets of Fekete segments we will get, similar to the nodal case, a favourable logarithmic growth of this constant.

In logistic regression modeling, Firth's modified estimator is widely used to address the issue of data separation, which results in the nonexistence of the maximum likelihood estimate. Firth's modified estimator can be formulated as a penalized maximum likelihood estimator in which Jeffreys' prior is adopted as the penalty term. Despite its widespread use in practice, the formal verification of the corresponding estimate's existence has not been established. In this study, we establish the existence theorem of Firth's modified estimate in binomial logistic regression models, assuming only the full column rankness of the design matrix. We also discuss other binomial regression models obtained through alternating link functions and prove the existence of similar penalized maximum likelihood estimates for such models.

For regression model selection under the maximum likelihood framework, we study the likelihood ratio confidence region for the regression parameter vector of a full regression model. We show that, when the confidence level increases with the sample size at a certain speed, with probability tending to one, the confidence region contains only vectors representing models having all active variables, including the parameter vector of the true model. This result leads to a consistent model selection criterion with a sparse maximum likelihood interpretation and certain advantages over popular information criteria. It also provides a large-sample characterization of models of maximum likelihood at different model sizes which shows that, for selection consistency, it suffices to consider only this small set of models.

Confidence intervals based on the central limit theorem (CLT) are a cornerstone of classical statistics. Despite being only asymptotically valid, they are ubiquitous because they permit statistical inference under weak assumptions and can often be applied to problems even when nonasymptotic inference is impossible. This paper introduces time-uniform analogues of such asymptotic confidence intervals, adding to the literature on confidence sequences (CS) -- sequences of confidence intervals that are uniformly valid over time -- which provide valid inference at arbitrary stopping times and incur no penalties for "peeking" at the data, unlike classical confidence intervals which require the sample size to be fixed in advance. Existing CSs in the literature are nonasymptotic, enjoying finite-sample guarantees but not the aforementioned broad applicability of asymptotic confidence intervals. This work provides a definition for "asymptotic CSs" and a general recipe for deriving them. Asymptotic CSs forgo nonasymptotic validity for CLT-like versatility and (asymptotic) time-uniform guarantees. While the CLT approximates the distribution of a sample average by that of a Gaussian for a fixed sample size, we use strong invariance principles (stemming from the seminal 1960s work of Strassen) to uniformly approximate the entire sample average process by an implicit Gaussian process. As an illustration, we derive asymptotic CSs for the average treatment effect in observational studies (for which nonasymptotic bounds are essentially impossible to derive even in the fixed-time regime) as well as randomized experiments, enabling causal inference in sequential environments.

Principal stratification is a popular framework for causal inference in the presence of an intermediate outcome. While the principal average treatment effects have traditionally been the default target of inference, it may not be sufficient when the interest lies in the relative favorability of one potential outcome over the other within the principal stratum. We thus introduce the principal generalized causal effect estimands, which extend the principal average causal effects to accommodate nonlinear contrast functions. Under principal ignorability, we expand the theoretical results in Jiang et. al. (2022) to a much wider class of causal estimands in the presence of a binary intermediate variable. We develop identification formulas and derive the efficient influence functions of the generalized estimands for principal stratification analyses. These efficient influence functions motivate a set of multiply robust estimators and lay the ground for obtaining efficient debiased machine learning estimators via cross-fitting based on $U$-statistics. The proposed methods are illustrated through simulations and the analysis of a data example.