Integrated Mean Square Research Articles

Some comments on Wicksell's problem

In this paper we derive some properties of inverse estimators in the Wicksell unfolding problem, based on a regularized version of the inverse of the preconditioned operator. We focus on the integrated mean square error, data-driven selection of the regularization parameter, and asymptotic efficiency of linear functionals in the Hájek–LeCam sense.

On study of kernel regression function polygons

In the case of the random design nonparametric regression, the regression function estimate is produced practically by joining every two consecutive kernel estimates of regression function values by a straight line segment. Hence, it is of polygon type, and is called the kernel regression function polygon (KRFP) in this paper. The KRFP is analyzed by its asymptotic integrated mean square error (AIMSE). This AIMSE precisely quantifies both effects of the kernel function and of the distance between the points on which kernel estimates of regression function values are calculated on the KRFP. By studying the AIMSE, we have the following findings. First of all, if the distance is of smaller order in magnitude than the bandwidth used by the kernel regression function estimator, then Epanechnikov kernel is still the optimal kernel function for the KRFP. Secondly, if the distance is of the same order in magnitude as the bandwidth, then Epanechnikov kernel is no longer optimal for the KRFP. In this case, using the AIMSE of the KRFP, we obtain the optimal kernel for the KRFP over the class of two-degreepolynomials by numerical calculation. As the distance increases, the computation time of the KRFP decreases. However, the resulting performance of the KRFP deteriorates, since the minimum AIMSE of the KRFP over both the bandwidth and the kernelfunction increases. Finally, if the distance is of larger order in magnitude than the bandwidth, then the uniform kernel is the optimal kernel function for the KRFP.

Coupled sea surface‐atmosphere model: 2. Spectrum of short wind waves

A physical model of the short wind wave spectrum in the wavelength range from a few millimeters to few meters is proposed. The spectrum shape results from the solution of the energy spectral density balance equation. Special attention is paid to the description of the capillary range of the short wave spectrum. It is assumed that in this range the spectrum shape is determined mainly by the mechanism of generation of parasitic capillaries. This is described as the cascade energy transfer from the gravity to the capillary waves. Thus the capillary wave spectrum results through the balance between generation of capillaries and their viscous dissipation. The short gravity wave spectrum results through the balance between wind input and dissipation due to wave breaking. A parameterization of wind input is obtained in part 1 of the present paper. To describe the dissipation due to wave breaking, the approach developed by Phillips [1985] is used. The spectral rate of energy dissipation is presented in the form of a power dependence of the ratio of the saturation spectrum to some threshold level. It is further shown that the threshold level depends on the drift current shift in the water viscous sublayer, which affects the energy losses by wave breaking. To obtain a short wave spectrum which is valid in the whole wavenumber domain, the capillary and the short gravity wave spectra are patched in the vicinity of the wavenumber corresponding to the minimum phase velocity. This short wave spectrum is incorporated into the wind over waves coupled model developed in part 1 of the present paper. The measured statistical properties of the sea surface related to the short waves, such as the spectral shape of omnidirectional and up‐wind spectra, their wind speed dependence and angular spreading, and the wind speed dependence of integral mean square slope and skewness parameters, are well reproduced by the model. Also the model well reproduces the measured wind speed dependence of the drag coefficient and of the coupling parameter.

Stepwise Estimation of Random Processes

The problem of estimating a continuous-time random process from its observations at appropriately designed sampling points is considered. The quality of an estimator is measured by its integrated mean square error (IMSE). Here, sampling points are designed stepwisely to minimize the IMSE and the best linear unbiased estimator (BLUE) is so determined that the earlier calculations do not have to be repeated with addition of one or more new samples. For random processes whose covariance has a sharp corner at the diagonal, it is shown that essentially, an optimal one-step forward sampling location is one of the midpoints of intervals determined by the current and previous sampling points. Both analytical and numerical examples are considered.

The effect of loss functions on empirical Bayes reliability analysis

The aim of the present study is to investigate the sensitivity of empirical Bayes estimates of the reliability function with respect to changing of the loss function. In addition to applying some of the basic analytical results on empirical Bayes reliability obtained with the use of the “popular” squared error loss function, we shall derive some expressions corresponding to empirical Bayes reliability estimates obtained with the Higgins–Tsokos, the Harris and our proposed logarithmic loss functions. The concept of efficiency, along with the notion of integrated mean square error, will be used as a criterion to numerically compare our results.It is shown that empirical Bayes reliability functions are in general sensitive to the choice of the loss function, and that the squared error loss does not always yield the best empirical Bayes reliability estimate.

Convergence en moyenne quadratique de l'estimateur de la régression par splines hybrides

In this work, we consider the estimation of a smooth regression function, belonging to C m [ 0, 1], by hybrid splines. We give the asymptotic behavior of the integrated mean square error by considering two different assumptions on the noise.

Frequency polygons for weakly dependent processes

The purpose of this paper is to investigate the frequency polygon as a density estimator for stationary strong mixing processes. Optimal bin widths which asymptotically minimize integrated mean square errors (IMSE) are derived. Under weak conditions, frequency polygons achieve the same rate of convergence to zero of the IMSE as kernel estimators. They can also attain the optimal uniform rate of convergence ( (n −1 logn) 1 3 under general conditions. Frequency polygons thus appear to be very good density estimators with respect to both criteria of IMSE and uniform convergence.

Exact convergence rate of the Euler-Maruyama scheme, with application to sampling design

We consider regular partitions of the interval [0,T] into n subintervals based on a sampling density h. If xt is the solution of the original stochastic differential equation and is the approximate solution using the Euler-Maruyama scheme corresponding to a time discretization partition π, we find the limit as a functional of the sampling density h and the parameters of the stochastic differential equation. We then find the best sampling density and thus the best way to discretize regularly the interval for integrated mean square approximation error. The linear case is also considered in detail. We also apply the same idea to obtain optimal asymptotically efficient schemes.

NONPARAMETRIC ESTIMATION OF THE DERIVATIVES OF A DENSITY BY THE METHOD OF WAVELETS

A method of estimation of the derivatives of a probability density using wavelet systems is proposed. Precise order for the integrated mean square of the proposed estimator is obtained.

Mixture Experiments : All-Variance and Minimum-Bias Designs

The problem considered is that of finding designs in case of inadequacy in the model of a mixture experiment. Box and Draper (1959) proposed for comparing designs, the Integrated Mean Square Error (IMSE) criterion which is composed of integrated variance and integrated bias. A class of augmented axial designs [Formula: see text] is proposed and is used to find the all-variance and all-bias designs for the first and second degree model.

Density estimation for time series by histograms

Histogram estimates of the density of time series are studied. optimal bin widths which asymptotically minimize integrated mean square errors are derived. Uniform strong consistency of the estimates and their rates of convergence are obtained. Histogram estimates can achieve the uniform optimal rate of convergence (n−1logn)13 for appropriate choices of the bin widths. The time series are assumed to satisfy some strong mixing or absolute regularity conditions.

Noise variability of the hard X-ray transient Nova Persei

view Abstract Citations (9) References (19) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Noise Variability of the Hard X-Ray Transient Nova Persei Denis, M. ; Olive, J. -F. ; Mandrou, P. ; Roques, J. P. ; Ballet, J. ; Goldwurm, A. ; Laurent, Ph. ; Cordier, B. ; Vikhlinin, A. ; Churazov, E. ; Gilfanov, M. ; Sunyaev, R. ; Dyachkov, A. ; Khavenson, N. ; Kremnev, R. ; Kovtunenko, V. Abstract The hard X-ray transient Nova Persei (GRO J0422+32) was observed with the SIGMA telescope continuously for 2 months, soon after its discovery in 1992 August. The temporal analysis of the 40-300 keV emission of this source showed a very high level of noise variability (approximately 20%). Such variability has previously only been observed at hard X-ray/soft gamma-ray energies from an accreting X-ray source, in the case of the black hole candidate Cygnus X-1. The Fourier power distribution in the 2 x 10-4 -1.25 x 10-1 Hz range is flat up to a break frequency (approximately 0.03 Hz) and then steepens to a power law with an index of approximately -0.8 around 0.1 Hz. These results can be interpreted in terms of the shot noise model. The photons in the shots represent a significant fraction (approximately 20%) of the source hard X-ray flux. The derived shot duration range from approximately to a few seconds, depending on the shot shape. The shot shape and duration do not show any correlation with the hard X-ray source flux, as the spectral shape did not apparently evolve during the outburst, so probably they are only weakly correlated with the accretion rate. While the source flux decreased uniformly from around 3 to 1 Crab unit in the period 1992 August 15 until September 25, the fractional integrated root mean square variability increased from about 15% to 30%. We have studied this correlation with the help of a 'source total flux-source noise flux' diagram which reveals a regular but rather complex dependence with at least three different regimes. Publication: The Astrophysical Journal Supplement Series Pub Date: June 1994 DOI: 10.1086/191996 Bibcode: 1994ApJS...92..459D Keywords: Black Holes (Astronomy); Gamma Ray Astronomy; Novae; X Ray Sources; Accretion Disks; Flux (Rate); Fourier Analysis; Power Spectra; Transient Oscillations; Astronomy full text sources ADS | data products SIMBAD (2) HEASARC (1)

Function estimation using partitions of unity

We present a new nonparametric function estimate based on partitions of unity. These density estimates are applicable not only to subsets of R n but to arbitrary metric spaces. They can be used for estimating conditional expectations, and for constructing both nonparametric energy density and nonparametric probability density estimates. We show that under reasonable conditions that the nonparametric density estimates converge in mean square error and under slightly more restrictive conditions that they converge in integrated mean square error. A computer graphics application is presented.

Sampling designs for estimation of a random process

A random process X( t), tϵ[0,1], is sampled at a finite number of appropriately designed points. On the basis of these observations, we estimate the values of the process at the unsampled points and we measure the performance by an integrated mean square error. We consider the case where the process has a known, or partially or entirely unknown mean, i.e., when it can be modeled as X( t) = m( t) + N( t), where m( t) is nonrandom and N( t) is random with zero mean and known covariance function. Specifically, we consider (1) the case where m( t) is known, (2) the semiparametric case where m(t) = β 1ƒ 1(t)+⋯+β qƒf q(t) , the β i 's are unknown coefficients and the ƒ i 's are known regression functions, and (3) the nonparametric case where m( t) is unknown. Here f i ( t) and m( t) are of comparable smoothness with the purely random part N( t), and N( t) has no quadratic mean derivative. Asymptotically optimal sampling designs are found for cases (1), (2) and (3) when the best linear unbiased estimator (BLUE) of X( t) is used (a nearly BLUE in case (3)), as well as when the simple nonparametric linear interpolator of X( t) is used. Also it is shown that the mean has no effect asymptotically, and several examples are considered both analytically and numerically.

A New Class of Kernels for Nonparametric Curve Estimation

We introduce a new class of variable kernels which depend on the smoothing parameter b through a simple scaling operation, and which have good integrated mean square error (IMSE) convergence properties. These kernels deform "automatically" near the boundary, eliminating boundary bias. Computational formulas are given for all orders of kernel in terms of exponentially damped sines and cosines. The kernel is a computationally convenient approximation to a certain Green's function, with the resulting kernel estimate closely related to a smoothing spline estimate.

A new version of the gasser-mueller estimator

In the case of the fixed design nonparametric regression, the kernel estimator proposed by Gasser and Mueller (1979, 1984) is one of the most widely used regression smoothers. However, if some design points are closer to their neighbors than others are, then there is a drawback to the Gasser-Mueller estimator (GME). The drawback is that the variance of the GME will be increased by the too close design points. In the case of the jittered equidistant design, this effect is precisely quantified through the asymptotic mean square error (AMSE) of the GME. In this case, to overcome the effect, a new version of the GME is proposed. The AMSE of the new version of the GME is the same as that of the GME in the case of the equidistant design. Hence, in the jittered equidistant design case, the new version of the GME is of better performance than the GME, in the sense of minimum integrated mean square error. Simulations demonstrate that the asymptotic effects hold for reasonable sample sizes.

Nonparametric probability density estimation: Improvements to the histogram for laboratory data

The histogram has long been used in the clinical laboratory for the depiction and manipulation of frequency data. We present recent results of refinements to the usual histogram procedures along with modern alternative methods of estimating frequency distributions, including the kernel and discrete maximum penalized likelihood estimation (DMPLE) approaches. We compared these nonparametric methods on 15 different types of simulated distributions, and on several sets (>1000 subjects/set) of real data, including alanine aminotransferase, aspartate aminotransferase, and lactate dehydrogenase levels. Each frequency curve estimation technique was evaluated by measuring the integrated mean square error between each technique's prediction and the true underlying distribution, using Monte Carlo techniques on sample sets with size 49 and 119. The kernel methos was the clear method of choice, both in performance (best in 22 36 cases) and in practical usage.

Nonparametric probability density estimation using normalized b–splines

We present a new nonparametric density estimate based on normalized tensor B–Splines. We show under the expected conditions that the non- parametric density estimate converges in mean square error and integrated mean square error. Results of simulations are also presented.

A Risk Bound in Sobolev Class Regression

For nonparametric regression estimation, when the unknown function belongs to a Sobolev smoothness class, sharp risk bounds for integrated mean square error have been found recently which improve on optimal rates of convergence results. The key to these has been the fact that under normality of the errors, the minimax linear estimator is asymptotically minimax in the class of all estimators. We extend this result to the nonnormal case, when the noise distribution is unknown. The pertaining lower asymptotic risk bound is established, based on an analogy with a location model in the independent identically distributed case. Attainment of the bound and its relation to adaptive optimal smoothing are discussed.

Smoothing Periodic Curves by a Method of Regularization

The present work develops an extension of the theory of polynomial smoothing splines for the statistical problem of estimating a periodic curve based on discrete and noisy observations of the curve itself or of a convolution functional of the curve. These estimates, called periodic $\alpha $-splines, are still of the method of regularization type, with a penalty involving Fourier coefficients. The questions of asymptotic rate of convergence of the integrated mean square error and the relationships with kernel estimates in the case of equispaced design points are addressed.