1. How to estimate mean function in random regression?
To estimate the mean function in random regression, we can construct a consistent estimator using Theorem 1. Consider N independent copies of Model (1) with unknown a.s. continuous stochastic processes f(t), f1(t), ..., fn(t). The mean function can be estimated by introducing the notation f*N,n,e(t) := 1Nj=1f*n,ej(t) and applying the conditions mentioned in the theorem. By using the given sequences eeNn and NNn, we can prove the uniform consistency of the estimator M*N,n(t1, t2) := 1Nj=1f*n,ej(t1)f*n,ej(t2), t1, t2 [0,1]k. This estimator is used to estimate the unknown mixed second-moment function Ef(t1)f(t2). The proof is based on the same arguments as those in proving Theorem 2, and therefore is omitted. In summary, under the specified conditions, the estimator Cov*n(t1, t2) := M*N,n(t1, t2) - f*N,n,e(t1)f*N,n,e(t2) is uniformly consistent for the covariance function of the random regression field f(t).
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2. What are sufficient conditions for asymptotic normality of estimators?
Sufficient conditions for asymptotic normality of estimators f*n,e(t) include non-dependence of design {Xi} on n, E(X2j|Fj-1) = s^2a.s for all j, and max j E(X2j1(X2j > a/hn))|Fj-1 p-0 for all a>0. These conditions ensure that the estimator follows a normal distribution with mean f(t) and variance s^2, denoted as N(0, s^2). The theorem is derived from Corollary 3.1 in Hall and Heyde (1980).
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3. How do simulation examples compare estimators?
Simulation examples compare estimators by using algorithms like Voronoi partitioning and recursive partitioning by coordinate-wise medians. In the provided section, the estimators f * n,e (t) and fn,e (t) are compared using 1000 simulation runs with 5000 design points. The optimal e is calculated using 10-fold cross-validation to minimize the average mean-square error. The mean-square error (MSE) and maximal absolute error (MaxE) are computed for the validation set and the true values of the target function f on a 100 x 100 uniform lattice. The results are presented as median (1-st quartile, 3-rd quartile) and compared using the paired Wilcoxon test. The simulation examples demonstrate the advantages of the new estimator in nonuniform design point densities.
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4. How does the ULCV estimator perform compared to NW estimator?
In the given example, the ULCV estimator outperformed the NW estimator in terms of Mean Squared Error (MSE) and Maximum Error (MaxE) accuracy measures. The MSE for the ULCV estimator was 0.2661 (0.2584, 0.2742) while for the NW estimator it was 0.2734 (0.2650, 0.2819). The p-value for the MSE comparison was less than 0.0001, indicating a statistically significant difference. Similarly, the MaxE for the ULCV estimator was 0.7878 (0.7013, 0.9230) compared to 1.1998 (1.0911, 1.3250) for the NW estimator, with a p-value less than 0.0001. This demonstrates that the ULCV estimator was more accurate and reliable in estimating the function f(x, y) in this example.
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