Mean Integrated Squared Error Research Articles

The Comparison Between the Bayes Estimator and the Maximum Likelihood Estimator of the Reliability Function for Negative Exponential Distribution

In this paper, the maximum likelihood estimator and the Bayes estimator of the reliability function for negative exponential distribution has been derived, then a Monte –Carlo simulation technique was employed to compare the performance of such estimators. The integral mean square error (IMSE) was used as a criterion for this comparison. The simulation results displayed that the Bayes estimator performed better than the maximum likelihood estimator for different samples sizes.

Generalised gamma kernel density estimation for nonnegative data and its bias reduction

We consider density estimation for nonnegative data using generalised gamma density. What is being emphasised here is that a negative exponent is allowed. We show that, for each positive or negative exponent, (i) generalised gamma kernel density estimator, without bias reduction, has the mean integrated squared error (MISE) of order , as in other boundary-bias-free density estimators from the existing literature, and that (ii) the bias-reduced versions have the MISEs of order , where n is the sample size. We illustrate the finite sample performance of the proposed estimators through the simulations.

On Kernel-Based Intensity Estimation of Spatial Point Patterns on Linear Networks

ABSTRACTWe propose an extension of Diggle’s nonparametric edge-corrected kernel-based intensity estimator to the case of events coming from an inhomogenous point pattern on a linear network. We analyze its statistical properties, showing that it is an unbiased estimator of the first-order intensity; we also provide an expression for the variance, and comment on the appropriate bandwidth selection. Our estimator is compared with the current existing equal-split discontinuous kernel density estimator in terms of the mean integrated squared error (MISE). We then use our estimator on two real datasets. We first revisit street crimes in an area of Chicago, obtaining similar results to previously published ones based on a parametric intensity function. Then, we study network-based spatial events consisting of calls to the Police department reporting anti-social behavior in the city of Castellon (Spain).

Bias reduction in kernel density estimation

ABSTRACTIn this paper, we propose two kernel density estimators based on a bias reduction technique. We study the properties of these estimators and compare them with Parzen–Rosenblatt's density estimator and Mokkadem, A., Pelletier, M., and Slaoui, Y. (2009, ‘The stochastic approximation method for the estimation of a multivariate probability density’, J. Statist. Plann. Inference, 139, 2459–2478) is density estimators. It turns out that, with an adequate choice of the parameters of the two proposed estimators, the rate of convergence of two estimators will be faster than the two classical estimators and the asymptotic MISE (Mean Integrated Squared Error) will be smaller than the two classical estimators. We corroborate these theoretical results through simulations.

Wavelet-based estimators of multivariable mean regression function with long-memory data

ABSTRACTWavelet analysis has been proved to be a powerful statistical technique in the non parametric regression. In this paper, we propose non linear wavelet-based estimators for multivariable mean regression function with long-memory data. We also provide an asymptotic expansion for the mean integrated squared error (MISE) of the function estimators. This MISE expansion still works even when the underlying mean regression function is only piecewise smooth. This paper extends the corresponding results in the literature for single variable to multivariable case.

Limiting bias-reduced Amoroso kernel density estimators for non-negative data

ABSTRACTThe Amoroso kernel density estimator (Igarashi and Kakizawa 2017) for non-negative data is boundary-bias-free and has the mean integrated squared error (MISE) of order O(n− 4/5), where n is the sample size. In this paper, we construct a linear combination of the Amoroso kernel density estimator and its derivative with respect to the smoothing parameter. Also, we propose a related multiplicative estimator. We show that the MISEs of these bias-reduced estimators achieve the convergence rates n− 8/9, if the underlying density is four times continuously differentiable. We illustrate the finite sample performance of the proposed estimators, through the simulations.

국소 선형 복합 분위수 회귀에서의 평활계수 선택

국소복합분위수 회귀모형을 활용한 비모수적 함수 추정방법이 높은 효율성과 더불어 활발히 연구되고 있다. 이러한 추정과정에 커널을 사용한 자료 평활방법이 대표적으로 사용되고 있으며, 그 성능은 커널보다는 평활계수의 선택 크게 의존한다. 한편, 회귀함수 추정방법의 성능을 평가하는 기준으로는 통상적으로 <TEX>$L_2$</TEX>-노름이 사용되어 평균제곱오차 또는 평균적분제곱오차를 최소화하는 평활계수의 선택에 대한 많은 연구가 진행되어 왔다. 본 논문에서는 국소선형 복합 분위수 회귀방법을 활용한 비모수 회귀모형 추정량의 성능을 결정하는 평활계수 선택의 최적성에 관해 연구하였다. 특히, 여러 장점을 가졌으나 수리적 어려움으로 연구가 미흡한 평균절대오차 및 평균적분절대오차를 최적의 기준으로 삼아 최적의 평활계수를 구하고 그 유일성에 관해 연구하였다. 나아가 기존의 평가기준인 평균제곱오차 및 평균적분제곱오차를 사용한 선택과의 관계를 파악하고 그 성능을 비교하였다. 이러한 과정에서 다양한 상황에서의 모의실험을 통해 제안한 방법의 특성을 규명하였다. Local composite quantile regression is a useful non-parametric regression method widely used for its high efficiency. Data smoothing methods using kernel are typically used in the estimation process with performances that rely largely on the smoothing parameter rather than the kernel. However, <TEX>$L_2$</TEX>-norm is generally used as criterion to estimate the performance of the regression function. In addition, many studies have been conducted on the selection of smoothing parameters that minimize mean square error (MSE) or mean integrated square error (MISE). In this paper, we explored the optimality of selecting smoothing parameters that determine the performance of non-parametric regression models using local linear composite quantile regression. As evaluation criteria for the choice of smoothing parameter, we used mean absolute error (MAE) and mean integrated absolute error (MIAE), which have not been researched extensively due to mathematical difficulties. We proved the uniqueness of the optimal smoothing parameter based on MAE and MIAE. Furthermore, we compared the optimal smoothing parameter based on the proposed criteria (MAE and MIAE) with existing criteria (MSE and MISE). In this process, the properties of the proposed method were investigated through simulation studies in various situations.

Statistical Inference for Renewal Processes

AbstractWe consider non‐parametric estimation for interarrival times density of a renewal process. For continuous time observation, a projection estimator in the orthonormal Laguerre basis is built. Nonstandard decompositions lead to bounds on the mean integrated squared error (MISE), from which rates of convergence on Sobolev–Laguerre spaces are deduced, when the length of the observation interval gets large. The more realistic setting of discrete time observation is more difficult to handle. A first strategy consists in neglecting the discretization error. A more precise strategy aims at taking into account the convolution structure of the data. Under a simplifying ‘dead‐zone’ condition, the corresponding MISE is given for any sampling step. In the three cases, an automatic model selection procedure is described and gives the best MISE, up to a logarithmic term. The results are illustrated through a simulation study.

Wavelet estimation of density for censored data with censoring indicator missing at random

ABSTRACTIn this paper, we define the nonlinear wavelet estimator of density for the right censoring model with the censoring indicator missing at random (MAR), and develop its asymptotic expression for mean integrated squared error (MISE). Unlike for kernel estimator, the MISE expression of the estimator is not affected by the presence of discontinuities in the curve. Meanwhile, asymptotic normality of the estimator is established. The proposed estimator can reduce to the estimator defined by Li [Non-linear wavelet-based density estimators under random censorship. J Statist Plann Inference. 2003;117(1):35–58] when the censoring indicator MAR does not occur and a bandwidth in non-parametric estimation is close to zero. Also, we define another two nonlinear wavelet estimators of the density. A simulation is done to show the performance of the three proposed estimators.

Interval-censored unimodal kernel density estimation via data sharpening

ABSTRACTIn this paper, we extend Choi and Hall's [Data sharpening as a prelude to density estimation. Biometrika. 1999;86(4):941–947] data sharpening algorithm for kernel density estimation to interval-censored data. Data sharpening has several advantages, including bias and mean integrated squared error (MISE) reduction as well as increased robustness to bandwidth misspecification. Several interval metrics are explored for use with the kernel function in the data sharpening transformation. A simulation study based on randomly generated data is conducted to assess and compare the performance of each interval metric. It is found that the bias is reduced by sharpening, often with little effect on the variance, thus maintaining or reducing overall MISE. Applications involving time to onset of HIV and running distances subject to measurement error are used for illustration.

Adaptive estimation for stochastic damping Hamiltonian systems under partial observation

The paper considers a process Zt=(Xt,Yt) where Xt is the position of a particle and Yt its velocity, driven by a hypoelliptic bi-dimensional stochastic differential equation. Under adequate conditions, the process is stationary and geometrically β-mixing. In this context, we propose an adaptive non-parametric kernel estimator of the stationary density p of Z, based on n discrete time observations with time step δ. Two observation schemes are considered: in the first one, Z is the observed process, in the second one, only X is measured. Estimators are proposed in both settings and upper risk bounds of the mean integrated squared error (MISE) are proved and discussed in each case, the second one being more difficult than the first one. We propose a data driven bandwidth selection procedure based on the Goldenshluger and Lespki (2011) method. In both cases of complete and partial observations, we can prove a bound on the MISE asserting the adaptivity of the estimator. In practice, we take advantage of a very recent improvement of the Goldenshluger and Lespki (2011) method provided by Lacour et al. (2016), which is computationally efficient and easy to calibrate. We obtain convincing simulation results in both observation contexts.

Convex relaxation for IMSE optimal design in random-field models

The construction of optimal designs for random-field interpolation models via convex design theory is considered. The definition of an Integrated Mean-Squared Error (IMSE) criterion yields a particular Karhunen–Loève expansion of the underlying random field. After spectral truncation, the model can be interpreted as a Bayesian (or regularised) linear model based on eigenfunctions of this Karhunen–Loève expansion, and can be further approximated by a linear model involving orthogonal observation errors. Using the continuous relaxation of approximate design theory, the search of an IMSE optimal design can then be turned into a Bayesian A-optimal design problem, which can be efficiently solved by convex optimisation. A careful analysis of this approach is presented, also including the situation where the model contains a linear parametric trend, which requires specific treatments. Several approaches are proposed, one of them enforcing orthogonality between the trend functions and the complementary random field. Convex optimisation, based on a quadrature approximation of the IMSE criterion and a discretisation of the design space, yields an optimal design in the form of a probability measure with finite support. A greedy extraction procedure of the exchange type is proposed for the selection of observation locations within this support, the size of the extracted design being controlled by the level of spectral truncation. The performance of the approach is investigated on a series of examples indicating that designs with high IMSE efficiency are easily obtained.

Approximation of IMSE-optimal Designs via Quadrature Rules and Spectral Decomposition

We address the problem of computing integrated mean-squared error (IMSE) optimal designs for interpolation of random fields with known mean and covariance. We assume that the mean squared error is integrated through a discrete measure and restrict the design space to its support. We show that the IMSE and its approximation by spectral truncation can be easily evaluated, which makes their global minimization affordable. Numerical experiments are carried out that illustrate the effectiveness of the approach.

The selection of the number of terms in an orthogonal series cumulative function estimator

Let h(.) be a continuous, strictly positive probability density function over an interval [a, b] and H(.) its associated cumulative distribution function (cdf). Given a sample set \(X_{1},\ldots ,X_{n}\) of independent identically distributed variables, we want to estimate H(.) from this sample set. The present work has two goals. The first one is to propose an estimator of a cdf based on an orthogonal trigonometric series and to give its statistical and asymptotic proprieties (bias, variance, mean square error, mean integrated squared error, convergence of the bias, convergence of variance, convergence of the mean squared error, convergence of the mean integrated squared error, uniform convergence in probability and the rate of convergence of the mean integrated squared error). The second is to introduce a new method for the selection of a “smoothing parameter”. The comparison by simulation between this method and Kronmal–Tarter’s method, shows that the new method is more performant in the sense of the mean integrated square error.

Bandwidth Selection in Deconvolution Kernel Distribution Estimators Defined by Stochastic Approximation Method with Laplace Errors

In this paper we consider the kernel estimators of a distribution function defined by the stochastic approximation algorithm when the observation are contamined by measurement errors. It is well known that this estimators depends heavily on the choice of a smoothing parameter called the bandwidth. We propose a specific second generation plug-in method of the deconvolution kernel distribution estimators defined by the stochastic approximation algorithm. We show that, using the proposed bandwidth selection and the stepsize which minimize the MISE (Mean Integrated Squared Error), the proposed estimator will be better than the classical one for small sample setting when the error variance is controlled by the noise to signal ratio. We corroborate these theoretical results through simulations and a real dataset.

A new method for higher-order linear phase FIR digital filter using shifted Chebyshev polynomials

In this work, a new method for the design of linear phase finite impulse response (FIR) filters using shifted Chebyshev polynomial is proposed. In this method, magnitude response of FIR filter is approximated with the help of shifted Chebyshev polynomials. The number of polynomials used for approximation depends upon the order of filter. Design problem of filter is constructed as minimization of integral mean-square error between the ideal response and actual response through differentiating it with respect to its coefficients, which leads to a system of linear equations. The simulation results included in this paper show the efficiency of proposed method. It is also evident from the results that the proposed method is suitable for higher filter taps.

Consistent Smooth Bootstrap Kernel Intensity Estimation for Inhomogeneous Spatial Poisson Point Processes

AbstractNon‐parametric estimation and bootstrap techniques play an important role in many areas of Statistics. In the point process context, kernel intensity estimation has been limited to exploratory analysis because of its inconsistency, and some consistent alternatives have been proposed. Furthermore, most authors have considered kernel intensity estimators with scalar bandwidths, which can be very restrictive. This work focuses on a consistent kernel intensity estimator with unconstrained bandwidth matrix. We propose a smooth bootstrap for inhomogeneous spatial point processes. The consistency of the bootstrap mean integrated squared error (MISE) as an estimator of the MISE of the consistent kernel intensity estimator proves the validity of the resampling procedure. Finally, we propose a plug‐in bandwidth selection procedure based on the bootstrap MISE and compare its performance with several methods currently used through both as a simulation study and an application to the spatial pattern of wildfires registered in Galicia (Spain) during 2006.

Comparison study to bandwidth selection in binomial kernel estimation using Bayesian approaches

Recently, several works have shown that the Bayesian approaches to global bandwidth and variable (local and adaptive) bandwidths selection in binomial kernel estimation for probability mass functions (pmfs) outperform the common classical methods, such as mean integrated squared error (MISE) and cross-validation (CV) methods. In this article, we first review the global, local, and adaptive binomial kernel estimator combined with the Bayesian approaches for selecting the bandwidths. Then we compare them by using several count data sets with different designs, in particular for small and moderate sample sizes. All the Bayesian bandwidth selection approaches are also applied to a real count data sets. In general, in terms of integrated squared error (ISE) and execution times, the local Bayesian approach outperforms the other Bayesian approaches.

Nonlinear wavelet density estimation with data missing at random when covariates are present

In this paper, we construct the nonlinear wavelet estimator of a density with data missing at random when covariables are present, and provide an asymptotic expression for the mean integrated squared error (MISE) of the estimator. Unlike for kernel estimators, the MISE expression of the wavelet-based estimator still holds when the density function is piecewise smooth. Also, the asymptotic normality of the estimator is established.

Second order asymptotical efficiency for a Poisson process

We consider the problem of the estimation of the mean function of an inhomogeneous Poisson process when its intensity function is periodic. For the mean integrated squared error (MISE) there is a classical lower bound for all estimators and the empirical mean function attains that lower bound, thus it is asymptotically efficient. Following the ideas of the work by Golubev and Levit, we compare asymptotically efficient estimators and propose an estimator which is second order asymptotically efficient. Second order efficiency is done over Sobolev ellipsoids, following the idea of Pinsker.