The estimation of the conditional average treatment effect holds critical importance in causal inference and is widely applied across numerous domains. Conditional average treatment effect facilitates the revelation of treatment effect heterogeneity among individuals with distinct characteristics. This paper examines four estimators for conditional average treatment effect under covariate-adaptive randomization: unadjusted, stratified, regression-adjusted, and calibration-based estimator, and establishes their theoretical properties. In terms of asymptotic variance, the regression-adjusted estimator has a variance no greater than that of the stratified estimator, which in turn has a variance no greater than the unadjusted estimator. When the covariate dimension diverges, we enhance multivariate adaptive regression splines by incorporating linear combinations of covariates to achieve sufficient dimension reduction. This approach constructs the conditional average treatment effect estimators while reducing dependence on model specification. The proposed sufficient dimension reduction estimator outperforms comparable dimension-reduction alternatives. Both numerical simulations and empirical results validate these theoretical conclusions.
This paper aims to study the effect of rotation in incompressible magnetohydrodynamics (MHD) flow with partial dissipation. We look at the 3D incompressible MHD equations within a rotating frame with Coriolis force of order 1 ϵ . Due to the dispersive behavior, the global solution with large initial data to the partial dissipation MHD system is established when the rotation was fast enough.
In this paper, we consider parameter estimation of the stochastic fractional heat equation ∂∂ tu(t, x) = Δ_αu(t, x) + σẆ(t, x), t ≥ 0, x ∈ℝ with u(0, x) = 0, where W(t, x) is a space-time white noise and Δ_α = -(- Δ)^α/2 denotes the fractional Laplacian with α ∈ (1,2). By using the quasi-likelihood method and by observing the process at discrete time instants ti = ih, i = 0, 1, …, n, we derive an estimator of the volatility parameter σ and analyze its asymptotic behaviors provided h = h(n) → 0 and there exists γ ≥ 0 such that nh1+γ → R ≠ 0 as n → ∞.
The dependence of eigenvalues of second-order Sturm-Liouville problems containing Herglotz-type eigenparameter-dependent transmission conditions on the parameters of the problems is derived. For this purpose, we first transform the problems into linear operator problems, then we prove the self-adjointness, certain basic eigenvalue properties and the Green’s function of such problems, respectively. Finally, the continuity and differentiability theorems of the problems are given, especially the differential expressions of the eigenvalues for each parameters which determine the problems are obtained under different type of derivatives, including the parameters in the Herglotz-type eigenparameter-dependent transmission conditions.
We investigate the optimization problem concerning service rate and service price in a customer intensive service system, where the server provides service of uncertain value to rationally inattentive customers. We demonstrate the stability and uniqueness of customers’ equilibrium unconditional joining probability and analyze the impact of customer intensity and unit information cost on probability, optimal price of the server and welfare-optimal price. First, the equilibrium probability is monotonically increasing (decreasing) in customer intensity when service rate is low (high). Under the influence of price, the equilibrium probability concerning unit information cost may be non-monotonic. Additionally, there exists a service rate that maximizes the equilibrium probability. Second, when the service rate is low (high), the optimal price for the server increases (decreases) with customer intensity. Furthermore, the server should enhance the difficulty of information acquisition to achieve higher revenue. Finally, similar to the optimal price of the server, the impact of customer intensity on the welfare-optimal price also depends on the service rate, and the welfare-optimal price increases with unit information cost. Moreover, both the server and social planner should maintain the service rate at a moderate level.
