Approximate inference for location and scale parameters withapplication to failure-time data

A new method (SP/CI) for obtaining approximate confidence intervals for the location, scale, and quantiles based on sample data parameters is presented. The basis of SP/CI is a saddlepoint expansion; the requirements are the maximum likelihood estimate (MLE), a few cross derivatives of the log-likelihood evaluated at the MLE, and quantiles of the log(F) distribution. SP/CI is illustrated numerically on several type-II censored reliability data sets using the extreme value and log-normal models. The accuracy is investigated by numerical comparison with the exact results, with best linear invariant estimator (BLIE), as well as with the results from other approximate methods discussed in the literature, including the asymptotic normality of the MLE, the likelihood ratio (LR) statistic, and corrections to the LR statistic recently developed     

Estimation methods for the mean of the exponential distributionbased on grouped and censored data

For grouped and censored data from an exponential distribution, the method of maximum likelihood (ML) does not in general yield a closed-form estimate of the mean, and therefore, an iterative procedure must be used. Considered are three approximate estimators of the mean: two approximate ML estimators and the midpoint estimator. Their performances are compared by Monte Carlo simulation to those of the ML estimator, in terms of the mean square error and bias. The two approximate ML estimators are reasonable substitutes for the ML estimator, unless the probability of censoring and the number of inspections are small. The effect of inspection schemes on the relative performances of the three approximate methods is investigated     

Point and interval estimation for Gaussian distribution, based on progressively Type-II censored samples

The likelihood equations based on a progressively Type-II censored sample from a Gaussian distribution do not provide explicit solutions in any situation except the complete sample case. This paper examines numerically the bias and mean square error of the MLE, and demonstrates that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic s-normality are unsatisfactory, and particularly so when the effective sample size is small. Therefore, this paper suggests using unconditional simulated percentage points of these pivotal quantities for constructing s-confidence intervals. An approximation of the Gaussian hazard function is used to develop approximate estimators which are explicit and are almost as efficient as the MLE in terms of bias and mean square error; however, the probability coverages of the corresponding pivotal quantities based on asymptotic s-normality are also unsatisfactory. A wide range of sample sizes and progressive censoring schemes are used in this study.     

Parameter estimation for a modified Weibull distribution, for progressively type-II censored samples

In this paper, the estimation of parameters based on a progressively Type-II censored sample from a modified Weibull distribution is studied. The likelihood equations, and the maximum likelihood estimators are derived. The estimators based on a least-squares fit of a multiple linear regression on a Weibull probability paper plot are compared with the MLE via Monte Carlo simulations. The observed Fisher information matrix, as well as the asymptotic variance-covariance matrix of the MLE are derived. Approximate confidence intervals for the parameters are constructed based on the s-normal approximation to the asymptotic distribution of MLE, and log-transformed MLE. The coverage probabilities of the individual s-normal-approximation confidence intervals for the parameters are examined numerically. Some recommendations are made from the results of a Monte Carlo simulation study, and a numerical example is presented to illustrate all of the methods of inference developed here.     

Prediction Intervals for Future Observations from the Inverse Gaussian Distribution

This paper discusses the problem of predicting a future observation from an inverse Gaussian distribution, based on a sample of size n from that same distribution. Two prediction intervals that have been proposed in the literature, one of which is an approximate prediction interval, are compared using Monte Carlo simulations. In many of the simulated cases the approximate prediction interval is superior with respect to larger estimated coverage probabilities and smaller estimated mean lengths. This is true in particular for n at least 15 and for 95% and 99% prediction intervals.     

指数分步场合分组数据下简单步加试验的极大似然估计

对指数分布场合下的寿命分布进行研究,简单步进应力加速寿命试验中,在观察时间间隔不相等的条件下,对所获得的分组数据为寿命数据,给出加速方程中未知参数的极大似然估计（MLE）,从而计算出常应力下产品的平均寿命。     

Best linear unbiased estimator of the Rayleigh scale parameterbased on fairly large censored samples

This paper tabulates: coefficients and relative s-efficiency of the best linear unbiased estimator (BLUE) of the scale parameter of the Rayleigh distribution for type II censored samples of size N=20(5)40, r=0(1)4 (number of observations censored from the left) and s=0(1)4 (number of observations censored from the right); and ranks, coefficients, variances, and relative s-efficiencies of the BLUE of a based on a selected few order statistics (k) for sample size N=20(1)40 and k=2(1)4. These estimators have the minimum variance among the BLUE of a based on the same number of order statistics. As compared to maximum likelihood estimators (MLE) and approximate MLE, the s-efficiency of BLUE of ρ is very high. When estimating the parameter using only a few observations, the k-optimum BLUE of ρ is the only choice, as the MLE of ρ is not available. Therefore, these tables for coefficients of BLUE of ρ based on censored samples and few observations for moderately large samples, have many applications     

Inference for a Simple Step-Stress Model With Type-II Censoring, and Weibull Distributed Lifetimes

The simple step-stress model under Type-II censoring based on Weibull lifetimes, which provides a more flexible model than the exponential model, is considered in this paper. For this model, the maximum likelihood estimates (MLE) of its parameters, as well as the corresponding observed Fisher Information Matrix, are derived. The likelihood equations do not lead to closed-form expressions for the MLE, and they need to be solved by using an iterative procedure, such as the Newton-Raphson method. We also present a simplified estimator, which is easier to compute, and hence is suitable to use as an initial estimate in the iterative process for the determination of the MLE. We then evaluate the bias, and mean square error of these estimates; and provide asymptotic, and bootstrap confidence intervals for the parameters of the Weibull simple step-stress model. Finally, the results are illustrated with some examples.      

Estimation of distortion product otoacoustic emissions

Distortion product otoacoustic emissions (DPOAE's) are acoustic signals generated from the cochlea when stimulated by dual tone stimulus. The measurement of the DPOAE's is useful in objectively assessing the hearing functionality of humans. The parameter estimation of the DPOAE's has been based on the fast Fourier transform (FFT) technique, which may not guarantee optimum performance in general. In this paper, we investigate the optimal maximum-likelihood estimator (MLE) for the DPOAE's. Experiments showed that the performance of the MLE is superior to those of the conventional FFT estimators. Furthermore, it is proven that the unwindowed FFT estimator is essentially the MLE when the stimulus tones are constrained to exhibit complete cycle within the data frame.     

On the validity of hypothesis testing for feasibility of image reconstructions

Feasible images have been defined as those images that could have generated the original data by the statistical process that governs the measurement. In the case of emission tomography, the statistical process of emission is Poisson and it is known that feasible images resulting from the maximum likelihood estimator (MLE) and Bayesian methods with entropy priors can be of high quality. Tests for feasibility have been described that are based on one critical assumption: the image that is being tested is independent of the data, even though the reconstruction algorithm has used those data in order to obtain the image. This fact could render the procedure unacceptable unless it is shown that its effects on the results of the tests are negligible. Experimental evidence is presented showing that images reconstructed by the MLE and stopped before convergence do indeed behave as if independent of the data. The results justify the use of hypothesis testing in practice, although they leave the problem of analytical proof still open.     

Target Localization Based on Angle of Arrivals

Mobile location using angle of arrival （AOA） measurements has received considerable attention. This paper presents an approximation of maximum likelihood estimator （MLE） for localizing a source based on AOA measurements. By introducing an intermediate variable, the nonlinear equations relating AOA estimates can be transformed into a set of equations which are linear in the unknown parameters. It is an approximate realization of the MLE. Simulations show that the proposed algorithm outperforms the previous contribution.     

Estimating the effective number of looks in interferometric SAR data

The probability density function (pdf) of the multi-look interferometric phase between two complex synthetic aperture radar (SAR) images is parameterized by the number of looks and the complex correlation coefficient. In practice, adjacent pixels in a real SAR interferogram, are statistically dependent due to filtering, and hence, the number of looks is usually smaller than the number of samples averaged. It has been shown that compensation with an effective number of looks, rather than an intractable rederivation of the pdf, can account for the statistical dependence. This paper addresses the challenge of how to determine a suitable value for the effective number of looks. It is shown that an optimum value can be found via a maximum-likelihood estimator (MLE) based on the interferometric phase pdf. However, since such an MLE is computationally intensive and numerically unstable, an estimator based on the method of moments (MoM) possessing similar fidelity is proposed. MoM is fast and robust and can be used in operational applications, such as determining constant false alarm rate (CFAR) detection thresholds for moving-target detection in SAR along-track interferometry.     

Estimation of the geometric survival distribution

This paper discusses both point and interval estimation of the survivor function S₀=Pr{X⩾x₀} for the geometric distribution. When the number of devices n⩾50, the performance of the maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) of S₀ are essentially equivalent with respect to the relative mean-square-error (RMSE) to S₀. However, when the failure probability per time unit p⩾0.50, and n⩽30, the UMVUE is preferable to the MLE with respect to the RMSE. For interval estimation of S₀ with no censoring, 4 asymptotic interval-estimators are derived from large-sample theory, and one from the exact distribution of the negative binomial. When p⩽0.2 and n⩾30, all 5 interval-estimators perform reasonably well with respect to coverage probability. Since using the interval estimator derived from the exact distribution can assure “coverage probability”⩾“desired confidence”, this estimator is probably preferable to the other asymptotic ones when p⩾0.50, and n⩽10. Finally, consider right-censoring, in which the failure-time that occurs after a fixed follow-up time period, is censored. We extend the interval estimator using the asymptotic properties of the MLE to account for right censoring. Monte Carlo simulation is used to evaluate the performance of this interval estimator; the censoring effect on efficiency is discussed for a variety of situations     

Uniqueness of maximum likelihood estimators of the 2-parameterWeibull distribution

We present a simple statistic, calculated from either complete failure data or from right-censored data of type-I or -II. It is useful for understanding the behavior of the parameter maximum likelihood estimates (MLE) of a 2-parameter Weibull distribution. The statistic is based on the logarithms of the failure data and can be interpreted as a measure of variation in the data. This statistic provides: (a) simple lower bounds on the parameter MLE, and (b) a quick approximation for parameter estimates that can serve as starting points for iterative MLE routines; it can be used to show that the MLE for the 2-parameter Weibull distribution are unique     

Interval estimation from censored and masked system-failure data

The author addresses confidence interval (CI) estimation in a competing risk (or multiple failure mode) framework where sample data are singly time-censored on the right and partially masked. A three-component series system with exponentially-distributed component-failure times is considered in order to represent cases involving full as well as partial masking. The approximate CIs considered are based on: asymptotic-normal theory for maximum likelihood estimators; cube-root transformation of the exponential distribution rate parameter; and inverted likelihood ratio tests. The small-sample coverage properties of these approximate CIs are estimated via computer simulation. These results also apply to models where component-failure times are Weibull distributed with known shape parameters     

Mean-squared error and threshold SNR prediction of maximum-likelihood signal parameter estimation with estimated colored noise covariances

An interval error-based method (MIE) of predicting mean squared error (MSE) performance of maximum-likelihood estimators (MLEs) is extended to the case of signal parameter estimation requiring intermediate estimation of an unknown colored noise covariance matrix; an intermediate step central to adaptive array detection and parameter estimation. The successful application of MIE requires good approximations of two quantities: 1) interval error probabilities and 2) asymptotic (SNRrarrinfin) local MSE performance of the MLE. Exact general expressions for the pairwise error probabilities that include the effects of signal model mismatch are derived herein, that in conjunction with the Union Bound provide accurate prediction of the required interval error probabilities. The Crameacuter-Rao Bound (CRB) often provides adequate prediction of the asymptotic local MSE performance of MLE. The signal parameters, however, are decoupled from the colored noise parameters in the Fisher Information Matrix for the deterministic signal model, rendering the CRB incapable of reflecting loss due to colored noise covariance estimation. A new modification of the CRB involving a complex central beta random variable different from, but analogous to the Reed, Mallett, and Brennan beta loss factor provides a working solution to this problem, facilitating MSE prediction well into the threshold region with remarkable accuracy     

Statistical Estimation of Mean Signal Strength in a Rayleigh-Fading Environment

A commonly used model for signal fading in many types of communication channels is that the amplitude of the received signal at a given time is a Rayleigh-distributed random variable. In this paper we show how classical statistical techniques may be applied to the problem of estimating the Rayleigh distribution parameter (i.e., the mean), given samples from the distribution. In particular, we first consider the problem of estimating the population mean, given a sequence of independent samples. We derive an unbiased maximum-likelihood estimator. We show that this estimator is unique, and since it is based on a sufficient statistic, it is therefore "best" in the Blackwell-Rao sense of minimizing expected loss. Using this estimator, we then develop confidence intervals whose length can be used as a guide in selecting the required sample size. We then consider the same estimation problem when the signal samples are obtained from the output of a logarithmic receiver. We derive an interval estimator which does not require taking the antilogs of the log samples, and which is not appreciably worse than the "best" estimator.     

Statistical estimation of mean signal strength in a Rayleigh-fading environment

A commonly used model for signal fading in many types of communication channels is that the amplitude of the received signal at a given time is a Rayleigh-distributed random variable. In this paper we show how classical statistical techniques may be applied to the problem of estimating the Rayleigh distribution parameter (i.e., the mean), given samples from the distribution. In particular, we first consider the problem of estimating the population mean, given a sequence of independent samples. We derive an unbiased maximum-likelihood estimator. We show that this estimator is unique, and since it is based on a sufficient statistic, it is therefore "best" in the Blackwell-Rao sense of minimizing expected loss. Using this estimator, we then develop confidence intervals whose length can be used as a guide in selecting the required sample size. We then consider the same estimation problem when the signal samples are obtained from the output of a logarithmic receiver. We derive an interval estimator which does not require taking the antilogs of the log samples, and which is not appreciably worse than the "best" estimator.     

电子产品平均寿命的区间估计法

利用随机变量函数的分布,研究并证明了电子产品平均寿命的区间估计方法,并从置信区间的本质意义出发．通过极值理论给出了电子产品平均寿命的最小区间的估计方法。最后,用实例说明这种最小区间估计方法比传统方法更具优越性。     

Estimation of P[Y

Kundu  D. Gupta  R.D. 《Reliability, IEEE Transactions on》2006,55(2):270-280