arXiv subjects

Taku Moriyama

Publications and source records attributed to Taku Moriyama.

9 recordsLinked to original sources

arXiv2026

On large bandwidth matrix values kernel smoothed estimators for multi-index models

The kernel smoothing with large bandwidth values causes oversmoothing or underfitting in general. However, when irrelevant variables are included, the corresponding large bandwidth values are known to have an effect of shrinking them. This study investigates asymptotic properties of the kernel conditional density estimator and the regression estimator with large bandwidth matrix elements for cases of multi-index model. It is clarified that the optimal convergence rate of the estimators depends on not the number of the variables but the effective dimension without eliminating the irrelevant variables. Thus, the kernel conditional density estimator and regression estimator are demonstrated to equip the reduction of the curse of dimensionality by nature. Finite sample performances are investigated by a numerical study, and the bandwidth selection is discussed. Finally a case study on the Boston housing data is provided.

math.ST↗

arXiv2025

A semiparametric probability distribution estimator of sample maximums

This study proposes a computationally efficient semiparametric distribution estimator, which is a slight modification of the naive mixture proposed by Schuster and Yakowitz (1985) and Olkin and Spiegelman (1987). The proposed method is applied to probability distribution estimation of a sample maximum. Two approaches for the sample maximum distribution estimation, one based on extreme value theory and the other on nonparametric smoothing, exist; however, theoretical and numerical properties of the two approaches are known to heavily depend on the case and greatly differ. This study demonstrates that the semiparametric mixture distribution estimators have good properties of both approaches. The cross-validation method is proposed for the mixing ratio selection for the proposed mixture distribution estimator. The result of simulation experiments and three case studies are reported.

math.ST↗

arXiv2024

On tail inference in iid settings with nonnegative extreme value index

In extreme value inference it is a fundamental problem how the target value is required to be extreme by the extreme value theory. In iid settings this study both theoretically and numerically compares tail estimators, which are based on either or both of the extreme value theory and the nonparametric smoothing. This study considers tail probability estimation and mean excess function estimation. This study assumes that the extreme value index of the underlying distribution is nonnegative. Specifically, the Hall class or the Weibull class of distributions is supposed in order to obtain the convergence rates of the estimators. This study investigates the nonparametric kernel type estimators, the fitting estimators to the generalized Pareto distribution and the plug-in estimators of the Hall distribution, which was proposed by Hall and Weissman (1997). In simulation studies the mean squared errors of the estimators in some finite sample cases are compared.

math.ST↗

arXiv2024

Parametric and nonparametric probability distribution estimators of sample maximum

Extreme value theory has constructed asymptotic properties of the sample maximum. This study concerns probability distribution estimation of the sample maximum. The traditional approach is parametric fitting to the limiting distribution -- the generalized extreme value distribution; however, the model in non-limiting cases is misspecified to a certain extent. We propose a plug-in type of nonparametric estimator that does not need model specification. Asymptotic properties of the distribution estimator are derived. The simulation study numerically investigates the relative performance in finite-sample cases. This study assumes that the underlying distribution of the original sample belongs to one of the Hall class, the Weibull class or the bounded class, whose types of the limiting distributions are all different: the Frechet, Gumbel or Weibull. It is proven that the convergence rate of the parametric fitting estimator depends on both the extreme value index and the second-order parameter and gets slower as the extreme value index tends to zero. On the other hand, the rate of the nonparametric estimator is proven to be independent of the extreme value index under certain conditions. The numerical performances of the parametric fitting estimator and the nonparametric estimator are compared, which shows that the nonparametric estimator performs better, especially for the extreme value index close to zero. Finally, we report two real case studies: the Potomac River peak stream flow (cfs) data and the Danish Fire Insurance data.

math.ST↗

arXiv2022

Asymptotic properties of parametric and nonparametric probability density estimators of sample maximum

Asymptotic properties of three estimators of probability density function of sample maximum $f_{(m)}:=mfF^{m-1}$ are derived, where $m$ is a function of sample size $n$. One of the estimators is the parametrically fitted by the approximating generalized extreme value density function. However, the parametric fitting is misspecified in finite $m$ cases. The misspecification comes from mainly the following two: the difference $m$ and the selected block size $k$, and the poor approximation $f_{(m)}$ to the generalized extreme value density which depends on the magnitude of $m$ and the extreme index $γ$. The convergence rate of the approximation gets slower as $γ$ tends to zero. As alternatives two nonparametric density estimators are proposed which are free from the misspecification. The first is a plug-in type of kernel density estimator and the second is a block-maxima-based kernel density estimator. Theoretical study clarifies the asymptotic convergence rate of the plug-in type estimator is faster than the block-maxima-based estimator when $γ> -1$. A numerical comparative study on the bandwidth selection shows the performances of a plug-in approach and cross-validation approach depend on $γ$ and are totally comparable. Numerical study demonstrates that the plug-in nonparametric estimator with the estimated bandwidth by either approach overtakes the parametrically fitting estimator especially for distributions with $γ$ close to zero as $m$ gets large.

math.ST↗

arXiv2017

A new method of joint nonparametric estimation of probability density and its support

In this paper we propose a new method of joint nonparametric estimation of probability density and its support. As is well known, nonparametric kernel density estimator has "boundary bias problem" when the support of the population density is not the whole real line. To avoid the unknown boundary effects, our estimator detects the boundary, and eliminates the boundary-bias of the estimator simultaneously. Moreover, we refer an extension to a simple multivariate case, and propose an improved estimator free from the unknown boundary bias.

math.ST↗

arXiv2017

Smoothed nonparametric two-sample tests

We propose new smoothed median and the Wilcoxon's rank sum test. As is pointed out by Maesono et al.(2016), some nonparametric discrete tests have a problem with their significance probability. Because of this problem, the selection of the median and the Wilcoxon's test can be biased too, however, we show new smoothed tests are free from the problem. Significance probabilities and local asymptotic powers of the new tests are studied, and we show that they inherit good properties of the discrete tests.

math.ST↗

arXiv2016

A new kernel estimator of hazard ratio and its asymptotic mean squared error

The hazard function is a ratio of a density and survival function, and it is a basic tool of the survival analysis. In this paper we propose a kernel estimator of the hazard ratio function, which are based on a modification of Ćwik and Mielniczuk's method. We study nonparametric estimators of the hazard function and compare those estimators by means of asymptotic mean squared error ($AMSE$). We obtain asymptotic bias and variance of the new estimator, and compare them with a naive estimator. The asymptotic variance of the new estimator is always smaller than the naive estimator's, so we also discuss an improvement of $AMSE$ using Terrell and Scott's bias reduction method. The new modified estimator ensures the non-negativity, and we demonstrate the numerical improvement.

math.ST↗

arXiv2016

Smoothed nonparametric tests and their properties

In this paper we propose new smoothed sign and Wilcoxon's signed rank tests, which are based on a kernel estimator of the underlying distribution function of data. We discuss approximations of $p$-values and asymptotic properties of these tests. The new smoothed tests are equivalent to the ordinary sign and Wilcoxon's tests in the sense of the Pitman's asymptotic relative efficiency, and the differences of the ordinary and the new tests converge to zero in probability. Under the null hypothesis, the main terms of the asymptotic expectations and variances of the tests do not depend on the underlying distribution. Though the smoothed tests are not distribution-free, we can obtain Edgeworth expansions with residual term $o(n^{-1})$, which do not depend on the underlying distribution.

math.ST↗