An Approximation to the Negative Moments of the Positive Binomial Useful in Life Testing

The purpose of this paper is to obtain the mean and variance of the maximum likelihood estimator of the scale parameter of a Weibull distribution where the sample is censored at a fixed time. It will be shown that these moments are functions of the negative moments of the positive binomial distribution. A simple approximation is obtained for the negative moments of the positive binomial, thus giving an approximate expression for the mean and variance of the estimator.     

Some Complete and Censored Sampling Results for the Weibull or Extreme-Value Distribution

A simple, unbiased estimator, based on a censored sample, has been proposed by Rain [1] for the scale parameter of the Extreme-value distribution. This estimator was shown to have high efficiency and to be approximately distributed as a chi-square variable if substantial censoring occurs. Further small sample and asymptotic properties of this estimator are considered in this paper. The estimator is modified so that it is more applicable to the complete sample case and a close chi-square approximation is established for all cases. The estimator is also shown to be related to the maximum likelihood estimator.     

TECHNICAL NOTE: Nonparametric estimation and variate generation for a nonhomogeneous Poisson process from event count data

Given a finite time horizon that has been partitioned into subintervals over which event counts have been accumulated for multiple realizations of a population NonHomogeneous Poisson Process (NHPP), this paper develops point and confidence-interval estimators for the cumulative intensity (or mean value) function of the population process evaluated at each subinterval endpoint. As the number of realizations tends to infinity, each point estimator is strongly consistent and the corresponding confidence-interval estimator is asymptotically exact. If the NHPP has a piecewise constant intensity (rate) function, then the proposed point and confidence-interval estimators for the cumulative intensity function are valid over the entire time horizon and not just at the subinterval endpoints; and in this case algorithms are presented for generating event times from the estimated NHPP. Event count data from a call center illustrate the point and interval estimators.     

Bandwidth selection in kernel density estimation for interval-grouped data

When interval-grouped data are available, the classical Parzen–Rosenblatt kernel density estimator has to be modified to get a computable and useful approach in this context. The new nonparametric grouped data estimator needs of the choice of a smoothing parameter. In this paper, two different bandwidth selectors for this estimator are analyzed. A plug-in bandwidth selector is proposed and its relative rate of convergence obtained. Additionally, a bootstrap algorithm to select the bandwidth in this framework is designed. This method is easy to implement and does not require Monte Carlo. Both proposals are compared through simulations in different scenarios. It is observed that when the sample size is medium or large and grouping is not heavy, both bandwidth selection methods have a similar and good performance. However, when the sample size is large and under heavy grouping scenarios, the bootstrap bandwidth selector leads to better results.     

Uniform integrability of the OLS estimators,and the convergence of their moments

The problem of convergence of moments of a sequence of random variables to the moments of its asymptotic distribution is important in many applications. These include the determination of the optimal training sample size in the cross-validation estimation of the generalization error of computer algorithms, and in the construction of graphical methods for studying dependence patterns between two biomarkers. In this paper, we prove the uniform integrability of the ordinary least squares estimators of a linear regression model, under suitable assumptions on the design matrix and the moments of the errors. Further, we prove the convergence of the moments of the estimators to the corresponding moments of their asymptotic distribution, and study the rate of the moment convergence. The canonical central limit theorem corresponds to the simplest linear regression model. We investigate the rate of the moment convergence in canonical central limit theorem proving a sharp improvement of von Bahr’s (Ann Math Stat 36:808–818, 1965) theorem.     

The LR-Chart: An Alternative to the MR-Chart

An individuals chart is proposed in which the mean level is estimated using the regression smoother LOWESS, and deviations from this mean estimate are plotted. This LOWESS-range (LR) chart yields an estimator of the process variance from stable measurements, which is almost as efficient as the estimator based on the sample standard deviation. The estimator often has lower mean squared-error than a moving-range-based estimator when applied to mean-shifted measurements.     

具有多级自适应滤波器的有限元密度估计器

基于可能相关并具有某种遍厉性的样本数据,该文设计了一个具有多级自适应滤波器的有限元估计器去估计相应的(平稳)密度函数。数值例了表明了估计器的有效性。同时也讨论了一个有关估计器稳定性的性质。     

Weighting effect on the discrete time Fourier transform of noisysignals

The effect of weighting on the uncertainty of the discrete time Fourier transform (DTFT) samples of a signal corrupted by additive noise is investigated. Making very weak assumptions, it is shown how the adopted window sequence and the autocovariance function of the noise affect the second-order stochastic moments of the frequency-domain data. The relationship obtained extends the results reported in the literature and is useful in many frequency-domain estimation problems. It is shown how the knowledge of the second-order moments of the transform has allowed the application of the least squares technique for the estimation of the parameters of a multifrequency signal in the frequency-domain. The estimator obtained is very useful when high-accuracy results are required under real-time constraints. The procedure exhibits a better accuracy than similar frequency-domain methods proposed in the literature     

The Variance of an Estimator in Variables Sampling

Let p be the proportion of a normal population in some tolerance region. The equivalence of two different forms of the minimum variance unbiased estimator of p given in the literature is indicated, and a third form is derived and used to obtain the variance of the estimator. For sample sizes of four and six the variance of the estimator is given explicitly and finally estimation of the variance for larger sample sizes is considered.     

The effects of timing jitter in sampling systems

Timing jitter generally causes a bias (systematic error) in the amplitude estimates of sampled waveforms. Equations are developed for computing the bias in both the time and frequency domains. Two principle estimators are considered: the sample mean and the so-called Markov estimator used in some equivalent-time sampling systems. Examples are given using both real and simulated data. It is shown that the bias that results from using the sample mean as an estimator can be approximated in the frequency domain by a simple filter function. The Markov estimator is shown to asymptotically converge to the population median. It is therefore an unbiased estimator for monotonic waveforms sampled with jitter distributions having a median of zero     

Some Percentile Estimators for Weibull Parameters

A percentile estimator for the shape parameter of the Weibull distribution, based on the 17th and 97th sample percentiles, is proposed which is asymptotically about 66% efficient when compared with the MLE (maximum likelihood estimator). A two-observation percentile estimator, based on the 40th and 82nd sample percentiles, for the scale parameter when the shape parameter is unknown is asymptotically about 82y0 efficient when compared with the MLE. The 24th and 93rd sample percentiles yield asymptotically about 41ye jointly efficient percentile estimators for both the scale and shape parameters in a class of two-observation percentile estimators when compared with their MLEs. Some other simple percentile estimators for these parameters are also briefly discussed. Finally, asymptotic properties of these estimators are investigated and their application in statistical inference problems is mentioned.     

Estimation of Particle Size Distribution Based on Observed Weights of Groups of Particles

Particulate matter is collected on some sampling medium. The particles of interest are collected by some means, and weighed as a group. Weights are observed for a number of replicate samples. These observed weights may include background contribution from particles existing on the sampling medium prior to its use. Given these data, plus similar date to establish the background, the problem is to estimate the average number of particles per sample, and their weight distribution. The estimation is accomplished by equating sample moments to population moments. The first four moments of the population are found for an arbitrary weight distribution function possessing finite first four moments. Specific estimates are found in the event the weight distribution is exponential in form, and approximate sampling variances of these estimates are derived. A numerical example is included.     

Moments,Variances, and Covariances of Sines and Cosines of Arguments which are Subject to Random Error

A simple, unbiased estimator, based on a censored sample, is proposed for the scale parameter of the extreme-value distribution. The exact distribution of the estimator is determined for the cases in which only the first two or only the first three ordered observations are available. The asymptotic distribution is derived, and an approximate distribution for small sample size is also provided. Interval estimation for the scale parameter is developed and a conservative interval estimate for reliability is also obtained.     

Use of median-based estimator to construct Phase II exponential chart

The Shewhart-type exponential control chart is a popular and extensively used among all time-between-events control charts for its simplicity. When the parameter is unknown, Phase II control limits are constructed, and the success of its implementation depends to an extent on the estimated value of the parameter, obtained from Phase I dataset. However, when the Phase I data are contaminated with spurious observations/outliers, the performance of the chart is suspected to deviate from what is normally expected. Traditionally, maximum likelihood estimator (MLE) and minimum variance unbiased estimator (MVUE) are used to estimate the unknown process parameter. Both of estimators are the functions of sample mean. In this paper, the median-based estimator (MBE) that is a function of sample median is used to construct Phase II control limits. Moreover, performance of the proposed chart is examined when Phase I sample consists of contaminated observations/outliers. It is found that the proposed chart outperforms the existing charts whether the sample is contaminated or not.     

Kernel estimators of extreme level curves

We address the estimation of extreme level curves of heavy-tailed distributions. This problem is equivalent to estimating quantiles when covariate information is available and when their order converges to one as the sample size increases. We show that, under some conditions, these so-called “extreme conditional quantiles” can still be estimated through a kernel estimator of the conditional survival function. Sufficient conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed estimators. Making use of this result, some kernel estimators of the conditional tail-index are introduced and a Weissman type estimator is derived, permitting to estimate extreme conditional quantiles of arbitrary large order. These results are illustrated through simulated and real datasets.     

Nonlinear Tikhonov regularization in Hilbert scales with balancing principle tuning parameter in statistical inverse problems

In this article, we study the balancing principle for Tikhonov regularization in Hilbert scales for deterministic and statistical nonlinear inverse problems. While the rates of convergence in deterministic setting is order optimal, they prove to be order optimal up to a logarithmic term in the stochastic framework. The two-step approach allows us to consider a data-driven algorithm in a general error model for which an exponential behaviour of the tail of the estimator chosen in the first step is valid. Finally, we compute the overall rate of convergence for a Hammerstein operator equation and for a parameter identification problem. Moreover, we illustrate these rates for the last application after we study some large sample properties of the local polynomial estimator in a general stochastic framework.     

Paramagnetic Effect in Nano-opal-lead Structure

We report magnetic studies of the paramagnetic effect observed in the superconducting nano-structured opal-lead system. Positive magnetization is clearly observed when the sample is cooled in field. The paramagnetic effect is strongly dependent on the applied field and the cooling rate. The results suggest that the paramagnetic moment is due to flux trapping and the competition between the positive and negative moments due to the temperature dependence of the penetration depth.     

Bias-corrected and robust estimation of the bivariate stable tail dependence function

The stable tail dependence function gives a full characterisation of the extremal dependence between two or more random variables. In this paper, we propose an estimator for this function which is robust against outliers in the sample. The estimator is derived from a bivariate second-order tail model together with a proper transformation of the bivariate observations, and its asymptotic properties are studied under some suitable regularity conditions. Our estimation procedure depends on two parameters: \(\alpha \), which controls the trade-off between efficiency and robustness of the estimator, and a second-order parameter \(\tau \), which can be replaced by a fixed value or by an estimate. In case where \(\tau \) has been replaced by the true value or by an external consistent estimator, our robust estimator is asymptotically unbiased, whereas in case where \(\tau \) is mis-specified, one loses this property, but still our estimator performs quite well with respect to bias. The finite sample performance of our robust and bias-corrected estimator of the stable tail dependence function is examined on a simulation study involving uncontaminated and contaminated samples. In particular, its behavior is illustrated for different values of the pair \((\alpha , \tau )\) and is compared with alternative estimators from the extreme value literature.     

A new time-domain narrowband velocity estimation technique for Doppler ultrasound flow imaging. II. Comparative performance assessment

For pt.I see ibid., vol.45, no.4, pp.939-54 (1998). The statistical performance of the new 2-D narrowband time-domain root-MUSIC blood velocity estimator described previously is evaluated using both simulated and flow phantom wideband (50% fractional bandwidth) ultrasonic data. Comparisons are made with the standard 1-D Kasai estimator and two other wideband strategies: the time domain correlator and the wideband point maximum likelihood estimator. A special case of the root-MUSIC, the "spatial" Kasai, is also considered. Simulation and flow phantom results indicate that the root-MUSIC blood velocity estimator displays a superior ability to reconstruct spatial blood velocity information under a wide range of operating conditions. The root-MUSIC mode velocity estimator can be extended to effectively remove the clutter component from the sample volume data. A bimodal velocity estimator is formed by processing the signal subspace spanned by the eigenvectors corresponding to the two largest eigenvalues of the Doppler correlation matrix. To test this scheme, in vivo common carotid flow complex Doppler data was obtained from a commercially available color flow imaging system. Velocity estimates were made using a reduced form of this data corresponding to higher frame rates. The extended root-MUSIC approach was found to produce superior results when compared to both 1- and 2-D Kasai-type estimators that used initialized clutter filters. The results obtained using simulated, flow phantom, and in vivo data suggest that increased sensitivity as well as effective clutter suppression can be achieved using the root-MUSIC technique, and this may be particularly important for wideband high frame rate imaging applications.     

On Some Permissible Estimators of the Location Parameter of the Weibull and Certain Other Distributions

In this paper an estimator of the location parameter of the Weibull distribution is proposed which is independent of its scale and shape parameters. Several properties of this estimator are established which suggest a proper choice of three ordered sample observations insuring a permissible estimate of the location parameter. This result is valid for every distribution which has the location parameter acting as the origin or threshold parameter. Asymptotic properties of such an estimator of the location parameter of the Weibull distribution is discussed. Finally, the paper contains a brief discussion on a percentile estimator of the location parameter of the Weibull distribution and includes some numerical illustration.