### doi:http://dx.doi.org/10.5705/ss.2012.230 A NOTE ON A NONPARAMETRIC REGRESSION TEST THROUGH PENALIZED SPLINES

"... Abstract: We examine a test of a nonparametric regression function based on pe-nalized spline smoothing. We show that, similarly to a penalized spline estimator, the asymptotic power of the penalized spline test falls into a small-K or a large-K scenarios characterized by the number of knots K and t ..."

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Abstract: We examine a test of a nonparametric regression function based on pe-nalized spline smoothing. We show that, similarly to a penalized spline estimator, the asymptotic power of the penalized spline test falls into a small-K or a large-K scenarios characterized by the number of knots K and the smoothing parameter. However, the optimal rate of K and the smoothing parameter maximizing power for testing is different from the optimal rate minimizing the mean squared error for estimation. Our investigation reveals that compared to estimation, some under-smoothing may be desirable for the testing problems. Furthermore, we compare the proposed test with the likelihood ratio test (LRT). We show that when the true function is more complicated, containing multiple modes, the test proposed here may have greater power than LRT. Finally, we investigate the properties of the test through simulations and apply it to two data examples. Key words and phrases: Goodness of fit, likelihood ratio test, nonparametric re-gression, partial linear model, spectral decomposition. 1.

### Computing the equivalent number of parameters of fixed-interval smoothers

"... The problem of reconstructing an unknown signal from n noisy samples can be ad-dressed by means of nonparametric estimation techniques such as Tikhonov regular-ization, Bayesian regression and state-space fixed-interval smoothing. The practical use of these approaches calls for the tuning of a regul ..."

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The problem of reconstructing an unknown signal from n noisy samples can be ad-dressed by means of nonparametric estimation techniques such as Tikhonov regular-ization, Bayesian regression and state-space fixed-interval smoothing. The practical use of these approaches calls for the tuning of a regularization parameter that con-trols the amount of smoothing they introduce. The leading tuning criteria, includ-ing Generalized Cross Validation and Maximum Likelihood, involve the repeated computation of the so-called equivalent number of parameters q(γ), a normalized measure of the flexibility of the nonparametric estimator. The paper develops new state-space formulas for the computation of q(γ) in O(n) operations. The results are specialized to the case of uniform sampling yielding closed-form expressions of q(γ) for both linear splines and first-order deconvolution.

### ∫ 1

, 2015

"... Model (non-parametric regression) Consider n observations from the non-parametric regression model Yi = f(xi) + σi, i = 1,..., n. • The function f belongs to the Sobolev space Wβ(M), β ≥ 1/2. • The observation errors 1,..., n are i.i.d. standard Gaussian, and σ2> 0. • Parameters f, β, and σ2 are ..."

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Model (non-parametric regression) Consider n observations from the non-parametric regression model Yi = f(xi) + σi, i = 1,..., n. • The function f belongs to the Sobolev space Wβ(M), β ≥ 1/2. • The observation errors 1,..., n are i.i.d. standard Gaussian, and σ2> 0. • Parameters f, β, and σ2 are unknown and of interest. We work under the frequentist assumption: the data Y = (Y1,..., Yn) follow the model above for some ”true”parameters f and σ. 2 / 16 The estimators (smoothing splines) The minimiser of the penalised least squares criterium 1 n n∑ i=1 Yi − f(xi)

### Relationship Pattern of Poverty and Unemployement in Indonesia with Bayesian Spline Approach 1

"... Poverty is one of fundamental problems which become major concern of Indonesia Government. World Poverty Commission said that unemployment is one of the main causes of poverty. A lot of literatures state that there is a strong correlation between unemployment and poverty, but to prove it empirically ..."

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Poverty is one of fundamental problems which become major concern of Indonesia Government. World Poverty Commission said that unemployment is one of the main causes of poverty. A lot of literatures state that there is a strong correlation between unemployment and poverty, but to prove it empirically, was not easy. To see the relationship pattern between poverty and unemployment in Indonesia, it can be used spline nonparametric regression model. Spline estimator in nonparametric regression can be obtained by Bayessian approach by using prior Gaussian improper and in order to choose the optimal smoothing parameter, Generalized Cross Validation (GCV) method is choosen. Relationship model of poverty and unemployment in Indonesia obtained in the form of a quadratic spline model with two optimal knots where percentage of poverty is in quadratic curve and rise in the stage when open unemployment rate is less than 3.87, and will be declined when the open unemployment rate moved between 3.87 and 4.24. But after the open unemployment rate reached 4.24, the percentage of poverty re-patterned quadratically but decreased slowly. So, for the case in Indonesia, unidirectional relationship between poverty and unemployment in the region occurred only partially, while some are actually spinning.

### Minimax Estimation of Linear Functionals, Particularly in Nonparametric Regression and Positron Emission Tomography

"... Often it is required to estimate a linear functional of a function f... ..."

### INRIA- Sierra project-team

"... We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels lead-ing to infinite-dimensional feature spaces, a common practical limiting difficulty is the necessity of ..."

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We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels lead-ing to infinite-dimensional feature spaces, a common practical limiting difficulty is the necessity of computing the kernel matrix, which most frequently leads to algorithms with running time at least quadratic in the number of observations n, i.e., O(n2). Low-rank approximations of the kernel matrix are often considered as they allow the reduction of running time complexities to O(p2n), where p is the rank of the approximation. The practicality of such methods thus depends on the required rank p. In this paper, we show that in the context of kernel ridge regression, for approx-imations based on a random subset of columns of the original kernel matrix, the rank p may be chosen to be linear in the degrees of freedom associated with the problem, a quantity which is classically used in the statistical analysis of such methods, and is often seen as the implicit num-ber of parameters of non-parametric estimators. This result enables simple algorithms that have sub-quadratic running time complexity, but provably exhibit the same predictive performance than existing algorithms, for any given problem instance, and not only for worst-case situations.

### ON RELAXED BOUNDARY SMOOTHING SPLINES FOR NONPARAMETRIC REGRESSION

"... Abstract. We study the relaxed boundary splines of Oehlert (1992) for the nonparametric regression problem with deterministic quasi-uniform interior designs. For integer m> 1, we show that the relaxed boundary splines of order 2m realize the optimal convergence rates for the expected L2 error whe ..."

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Abstract. We study the relaxed boundary splines of Oehlert (1992) for the nonparametric regression problem with deterministic quasi-uniform interior designs. For integer m> 1, we show that the relaxed boundary splines of order 2m realize the optimal convergence rates for the expected L2 error when the regression function belongs to a Sobolev space of order k with m 6 k 6 2m. The analysis is based on the identification of the relevant weighted Sobolev spaces as reproducing kernel Hilbert spaces (but not in the usual Bayesian view of smoothing splines).

### SMOOTHNESS PRIORS SOBOLEV SPACES SPLINE FUNCTIONS STATIONARY PROCESSES

"... We give an account of the Pinsker bound describing the exact asymptotics of the minimax risk in a class of nonparametric smoothing problems. The parameter spaces are Sobolev classes or ellipsoids, and the loss is of squared L2-type. The result from 1980 turned out to be a major step in the theory of ..."

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We give an account of the Pinsker bound describing the exact asymptotics of the minimax risk in a class of nonparametric smoothing problems. The parameter spaces are Sobolev classes or ellipsoids, and the loss is of squared L2-type. The result from 1980 turned out to be a major step in the theory of nonparametric function estimation. Keywords:

### Summary

, 2007

"... In this paper we study the class of penalized regression spline estimators, which enjoy similarities to both regression splines (without penalty and with less knots than data points) and smoothing splines (with knots equal to the data points and a penalty controlling the roughness of the fit). Depen ..."

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In this paper we study the class of penalized regression spline estimators, which enjoy similarities to both regression splines (without penalty and with less knots than data points) and smoothing splines (with knots equal to the data points and a penalty controlling the roughness of the fit). Depending on an assumption on the number of knots, sample size and penalty, we show that the theoretical properties of penalized regression spline estimators are either similar to those of regression splines or to those of smoothing splines, with a clear breakpoint distinguishing the cases. We prove that using less knots results in better asymptotic rates than when using a large number of knots. We obtain expressions for bias and variance and asymptotic rates for the number of knots and penalty parameter. 1