1. What are the limitations of treating b(t, u) as a univariate function of t for each fixed u in the functional quantile regression model?
The limitations of treating b(t, u) as a univariate function of t for each fixed u in the functional quantile regression model are twofold. First, this conventional strategy assumes that the slope function b(t, u) is smooth over both t and u, which is favorable in real applications. However, fitting the regression models for different quantiles separately cannot guarantee that the resulting estimator for b(t, u) is smooth over u. Second, for some observations, the estimation of Q Y (u | X) may not be monotonically increasing in u as it should be, leading to crossing quantiles. These crossing quantiles can further lead to invalid distribution estimation for the response variable. The proposed method addresses these limitations by using bivariate spline basis functions to approximate b(t, u) directly and estimating the corresponding basis coefficients, ensuring smoothness and monotonicity of the estimated quantiles.
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2. How to estimate bivariate slope function?
To estimate the bivariate slope function b(t, u), we propose approximating it by bivariate splines, specifically Bernstein polynomials over triangulations. We use the Bernstein polynomials defined over a triangulation to approximate the slope function on the domain T x A. The approximation is represented as s(t, u) = J j=1 g j b j (t, u) S r d (), where {g j } J j=1 are the corresponding coefficients. Under certain assumptions on X(t), the Mercer's theorem allows us to decompose the functional covariate X(t) into a linear combination of basis functions and coefficients. To approximate c(u), we use univariate B-spline basis functions and truncate the functional observations to reduce dimensionality and smooth them. This approach extends classical linear quantile regression to functional quantile regression, allowing for simultaneous consideration of multiple quantiles of interest.
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3. How is the minimization problem (2.10) formulated in quadratic programming?
The minimization problem (2.10) can be formulated into a quadratic programming problem by considering the minimization of a quadratic function with respect to th, g0, and {wi,r, vi,r} i=1,...,n,r=1,...,n. This approach is based on the idea introduced by Koenker and Bassett (1978) in their classical linear quantile regression model. By transforming the minimization problem into a quadratic programming problem, it becomes easier to solve using optimization techniques. The quadratic programming problem involves finding the optimal values of th, g0, and {wi,r, vi,r} i=1,...,n,r=1,...,n that minimize the quadratic function. This formulation allows for efficient computation and analysis of the minimization problem, making it a valuable tool in statistical modeling and optimization.
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4. How to derive coefficients in Computation17?
To derive coefficients in Computation17, first derive the estimated FPCs {phk(t)} mk=1 and corresponding scores {xi} ni=1. Then compute matrices related to the bivariate spline basis, BT(t,u), Q, BAT, and DAT using numerical integration based on Simpson's rule. These approximations are used to code the quadratic programming problem (2.13) with or without the constraints (2.12) in Statistica Sinica: Newly accepted Paper (accepted author-version subject to English editing).
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