Estimation of Weibull Parameters With Competing-Mode Censoring

Existing results are reviewed for the maximum likelihood (ML) estimation of the parameters of a 2-parameter Weibull life distribution for the case where the data are censored by failures due to an arbitrary number of independent 2-parameter Weibull failure modes. For the case where all distributions have a common but unknown shape parameter the joint ML estimators are derived for i) a general percentile of the j-th distribution, ii) the common shape parameter, and iii) the proportion of failures due to failure mode j. Exact interval estimates of the common shape parameter are constructable in terms of the ML estimates obtained by using i) the data without regard to failure mode, and ii) existing tables of the percentage points of a certain pivotal function. Exact interval estimates for a general percentile of failure-mode-j distribution are calculable when the failure proportion due to failure-mode-j is known; otherwise a joint s-confidence region for the percentile and failure proportion is calculable. It is shown that sudden death endurance test results can be analyzed as a special case of competing-mode censoring. Tabular values for the construction of interval estimates for the 10-th percentile of the failure-mode-j distribution are given for 17 combinations of sample size (from 5 to 30) and number of failures.     

ML parameter estimation for Markov random fields with applicationsto Bayesian tomography

Markov random fields (MRFs) have been widely used to model images in Bayesian frameworks for image reconstruction and restoration. Typically, these MRF models have parameters that allow the prior model to be adjusted for best performance. However, optimal estimation of these parameters (sometimes referred to as hyperparameters) is difficult in practice for two reasons: (i) direct parameter estimation for MRFs is known to be mathematically and numerically challenging; (ii) parameters can not be directly estimated because the true image cross section is unavailable. We propose a computationally efficient scheme to address both these difficulties for a general class of MRF models, and we derive specific methods of parameter estimation for the MRF model known as generalized Gaussian MRF (GGMRF). We derive methods of direct estimation of scale and shape parameters for a general continuously valued MRF. For the GGMRF case, we show that the ML estimate of the scale parameter, sigma, has a simple closed-form solution, and we present an efficient scheme for computing the ML estimate of the shape parameter, p, by an off-line numerical computation of the dependence of the partition function on p. We present a fast algorithm for computing ML parameter estimates when the true image is unavailable. To do this, we use the expectation maximization (EM) algorithm. We develop a fast simulation method to replace the E-step, and a method to improve the parameter estimates when the simulations are terminated prior to convergence. Experimental results indicate that our fast algorithms substantially reduce the computation and result in good scale estimates for real tomographic data sets.     

Maximum likelihood array processing for stochastic coherent sources

Maximum likelihood (ML) estimation in array signal processing for the stochastic noncoherent signal case is well documented in the literature. We focus on the equally relevant case of stochastic coherent signals. Explicit large-sample realizations are derived for the ML estimates of the noise power and the (singular) signal covariance matrix. The asymptotic properties of the estimates are examined, and some numerical examples are provided. In addition, we show the surprising fact that the ML estimates of the signal parameters obtained by ignoring the information that the sources are coherent coincide in large samples with the ML estimates obtained by exploiting the coherent source information. Thus, the ML signal parameter estimator derived for the noncoherent case (or its large-sample realizations) asymptotically achieves the lowest possible estimation error variance (corresponding to the coherent Cramer-Rao bound)     

Maximum-likelihood versus maximum a posteriori parameter estimation of physiological system models: the C-peptide impulse response case study

Maximum-likelihood (ML), also given its connection to least-squares (LS), is widely adopted in parameter estimation of physiological system models, i.e., assigning numerical values to the unknown model parameters from the experimental data. A more sophisticated but less used approach is maximum a posteriori (MAP) estimation. Conceptually, while ML adopts a Fisherian approach, i.e., only experimental measurements are supplied to the estimator, MAP estimation is a Bayesian approach, i.e., a priori available statistical information on the unknown parameters is also exploited for their estimation. In this paper, after a brief review of the theory behind ML and MAP estimators, we compare their performance in the solution of a case study concerning the determination of the parameters of a sum of exponential model which describes the impulse response of C-peptide (CP), a key substance for reconstructing insulin secretion. The results show that MAP estimation always leads to parameter estimates with a precision (sometimes significantly) higher than that obtained through ML, at the cost of only a slightly worse fit. Thus, a three exponential model can be adopted to describe the CP impulse response model in place of the two exponential model usually identified in the literature by the ML/LS approach. Simulated case studies are also reported to evidence the importance of taking into account a priori information in a data poor situation, e.g., when a few or too noisy measurements are available. In conclusion, our results show that, when a priori information on the unknown model parameters is available, Bayes estimation can be of relevant interest, since it can significantly improve the precision of parameter estimates with respect to Fisher estimation. This may also allow the adoption of more complex models than those determinable by a Fisherian approach.     

Reliability analysis of accelerated life-test data from arepairable system

This paper analyzes accelerating testing of a repairable item modeled by a nonhomogeneous Poisson process with covariates. We extensively analyze a single accelerating variable with two stress levels, and derive closed-form maximum likelihood (ML) solutions. These closed-form solutions provide: (1) an easier way to obtain point estimates of the unknown parameters under usual operating conditions, and (2) a way to obtain confidence intervals on the process parameters and function thereof which are more accurate than those based on asymptotic normality of ML estimates. We analyze the circumstances in which a drawback to closed-form estimation arises, and guides the extent that our procedures may equally apply. An example application drawn from a real situation of accelerated testing is presented, and numerical estimates based on our procedures are derived and discussed. Theoretical and simulation results show that estimation procedures based on the power-law process and regression methods can be a flexible, useful tool for reliability analysis of a repairable item     

Estimation of parameters of life from progressively censored data using Burr-XII model

Based on progressively Type-II censored samples, the maximum likelihood, and Bayes estimators for some lifetime parameters (reliability, and hazard functions), as well as the parameters of the Burr-XII model, are derived. The Bayes estimators are obtained using both the symmetric (Squared Error, SE) loss function, and asymmetric (LINEX, and General Entropy, GE) loss functions. This was done with respect to the conjugate prior for the one shape parameter, and discrete prior for the other parameter of this model. Also the existence, uniqueness, and finiteness of the ML parameter estimates for this type of censoring are discussed. A practical example consisting of data from an accelerated test on insulating fluid reported by Nelson (1982) was used for illustration, and comparison. Finally, some numerical results using simulation study concerning different sample sizes, and progressive censoring schemes were reported.     

一种新的K分布形状参数估计器

该文提出了一种将已有的U估计器和X估计器结合起来估计K分布形状参数的新估计器。仿真结果表明,与以前提出的方法相比,新估计器在我们所关心的小v值范围能够提供更加准确的形状参数估计,几乎相当于通过数值方法计算的最大似然估计器的性能。     

Modeling the distribution of DCT coefficients for JPEG reconstruction

In the paper, the one moment (OM) method for the estimation of the shape parameter of generalized Gaussian distribution (GGD) is derived from the two moments method in the case when the moments converge in the limits to the same value. The one moment method reduces to the maximum likelihood (ML) method in the special case when the moment equals the shape parameter. The proposed method exhibits smaller complexity of calculations over ML keeping the same error.Assuming Laplacian distribution, there exists a method for optimally biasing the reconstruction levels for the quantized AC discrete cosine transform (DCT) coefficients using only the quantized ones available at the JPEG decoder [J.R. Price, M. Rabbani, Biased reconstruction for JPEG decoding, IEEE Signal Process. Lett. 6 (12) (1999) 297–299; R. Krupiński, J. Purczyński, First absolute moment and variance estimators used in JPEG reconstruction, IEEE Signal Process. Lett. 11 (8) (2004) 674–677].Many researchers stated that the subset of images can be modeled with GGD with the shape parameter lower than 1. By assuming a source signal with GGD with the exponent 0.5, equations in a closed form for the centroid reconstruction can be obtained as it cannot be done for a GGD model. The ML method of discrete GGD 0.5 is derived, which requires the estimation of only one parameter. For selected images, the values of PSNR coefficients are compared for both distributions.     

Maximum likelihood estimation of Burr XII distribution parameters under Type II censoring

The mathematical theory for the point estimation of the parameters of the Burr Type XII distribution by maximum likelihood (ML) is developed for Type II censored samples. Also derived are necessary and sufficient conditions on the sample data that guarantee the existence, uniqueness and finiteness of the ML parameter estimates for all possible permissible parameter combinations. The asymptotic theory of ML is invoked to obtain approximate confidence intervals for the ML parameter estimates. An application to reliability data arising in a life test experiment is discussed.     

A new unwindowed lattice filter for RLS estimation

Adaptive lattice algorithms are derived for the solution of unwindowed least squares estimation problems for AR and FIR models. The basic approach is to embed the unwindowed problem in a larger prewindowed problem and then eliminate superfluous terms in the lattice. Initializations are given to allow the lattice to use no initial parameter estimates or to include initial parameter estimates with desired weightings in the quadratic criterion for parameter estimation. A numerical example is given     

A model-based approach for estimation of two-dimensional maximum entropy power spectra

A stochastic model-based approach is presented for estimation of the two-dimensional maximum entropy power spectrum (MEPS) from given finite uniform array data. The method consists of fitting an appropriate two-dimensional noncausal Gaussian-Markov random field (GMRF) model to the given data using the maximum likelihood (ML) technique for parameter estimation. The nonlinear criterion function used for ML estimation is similar in structure to the function arising in the deterministic approach of Lang and McClellan. The model-based approach provides new insights into the two-dimensional MEPS estimation problem. For example, using the asymptotic normality of ML estimates, we derive simultaneous confidence bands for the estimated MEPS. It turns out that when the true correlations are generated by a noncausal GMRF model, the two-dimensional MEPS can be obtained by solving linear equations. This approach also suggests techniques for realizing two-dimensional GMRF models from the given correlation data. Several numerical examples are given to illustrate the usefulness of the approach.     

Maximum likelihood estimation of exponential distribution parameters using interval data

This paper addresses the problem of estimating, by the method of maximum likelihood (ML), the location parameter (when present) and scale parameter of the exponential distribution (ED) from interval data. Interval data are defined as two data values that surround an unknown failure observation. Such observations occur naturally, during periodic inspections, for example, when only the time interval during which the failure occurred is known. The appropriate (conditional) log-likelihood functions are derived, as are expressions for the asymptotic variances and covariances of the ML parameter estimates. To illustrate the calculations involved, two numerical examples are discussed.     

Noncausal 2-D spectrum estimation for direction finding

A two-dimensional noncausal autoregressive (NCAR) plus additive noise model-based spectrum estimation method is presented for planar array data typical of signals encountered in array processing applications. Since the likelihood function for NCAR plus noise data is nonlinear in the model parameters and is further complicated by the unknown variance of the additive noise, computationally intensive gradient search algorithms are required for computing the estimates. If a doubly periodic lattice is assumed, the complexity of the approximate maximum likelihood (ML) equation is significantly reduced without destroying the theoretical asymptotic properties of the estimates and degrading the observed accuracy of the estimated spectra. Initial conditions for starting the approximate ML computation are suggested. Experimental results that can be used to evaluate the signal-plus-noise approach and compare its performance to those of signal-only methods are presented for Gaussian and simulated planar array data. Statistics of estimated spectrum parameters are given, and estimated spectra for signals with close spatial frequencies are shown. The approximate ML parameter estimate's asymptotic properties, such as consistency and normality, are established, and lower bounds for the estimate's errors are derived, assuming that the data are Gaussian     

Solving ML equations for 2-parameter Poisson-process models forungrouped software-failure data

Existence conditions are given for maximum likelihood (ML) parameter estimates for several families of 2-parameter software-reliability Poisson-process models. For each such model, the ML equations can be expressed in terms of one equation in one unknown. Bounds are given on solutions to these one equation problems to serve as initial intervals for search algorithms like bisection. Uniqueness of the solutions is established in some cases. Solutions are also tabulated for certain simple cases. Results are given for ungrouped failure data (exact times are available for all failures). ML estimation problems for such a situation are treated as limiting cases of problems based on failure times grouped into intervals of decreasing mesh     

Unbiased Maximum-Likelihood Estimation of a Weibull Percentile when the Shape Parameter is Known

The maximum-likelihood (ML) estimator for a percentile of a Weibull distribution with a known shape parameter is considered. Multiplicative correction factors are listed for rendering the ML estimator mean or median unbiased in the cases where the samples are type II censored with or without replacement. The correction factors depend upon the number of failures and the shape parameter but are independent of the sample size and the percentile being estimated.     

Bounds on maximum likelihood ratio-part I: application to antenna array detection-estimation with perfect wavefront coherence

The multiple hypothesis testing problem of the detection-estimation of an unknown number of independent Gaussian point sources is adequately addressed by likelihood ratio (LR) maximization over the set of admissible covariance matrix models. We introduce nonasymptotic lower and upper bounds for the maximum LR. Since LR optimization is generally a nonconvex multiextremal problem, any practical solution could now be tested against these bounds, enabling a high probability of recognizing nonoptimal solutions. We demonstrate that in many applications, the lower bound is quite tight, with approximate maximum likelihood (ML) techniques often unable to approach this bound. The introduced lower bound analysis is shown to be very efficient in determining whether or not performance breakdown has occurred for subspace-based direction-of-arrival (DOA) estimation techniques. We also demonstrate that by proper LR maximization, we can extend the range of signal-to-noise ratio (SNR) values and/or number of data samples wherein accurate parameter estimates are produced. Yet, when the SNR and/or sample size falls below a certain limit for a given scenario, we show that ML estimation suffers from a discontinuity in the parameter estimates: a phenomenon that cannot be eliminated within the ML paradigm.     

Parameter estimation for the 3-parameter Gamma distribution usingthe continuation method

This paper shows a maximum-likelihood (ML) parameter estimation algorithm for the 3-parameter Gamma distribution. The algorithm, a combination of the continuation method and the extended Gamma distribution model, can find the local ML estimates of the parameters without a careful selection of the starting point in the iterative process. This algorithm is more efficient than previous algorithms, and can find the multiple local ML estimates     

Weibull accelerated life testing with unreported early failures

Situations arise in life testing where early failures go unreported, e.g. a technician believes an early failure is “his fault” or “premature” and must not be recorded. Consequently, the reported data come from a truncated distribution and the number of unreported early failures is unknown. Inferences are developed for a Weibull accelerated life-testing model in which transformed scale and shape parameters are expressed as linear combinations of functions of the environment (stress). Coefficients of these combinations are estimated by maximum likelihood methods which allow point, interval, and confidence bound estimates to be computed for such quantities of interest for a given stress level as the shape parameter, the scale parameter, a selected quantile, the reliability at a particular time, and the number of unreported early failures. The methodology allows lifetimes to be reported as exact, right censored, or interval-valued, and to be subject optionally to testing protocols which establish thresholds below which lifetimes go unreported. A broad spectrum of applicability is anticipated by virtue of the substantial generality accommodated in both stress modeling and data type     

A Spatiotemporal Framework for Estimating Trial-to-Trial Amplitude Variation in Event-Related MEG/EEG

A spatiotemporal framework for estimating trial-to-trial variability in evoked response (ER) data is presented. Spatial and temporal bases capture the aspects of the response that are consistent across trials, while the basis expansion coefficients represent the variable components of the response. We focus on the simplest case of constant spatiotemporal response shape and varying amplitude across trials. Two different constraints on the amplitude evolution are employed to effectively integrate the individual responses and improve robustness at low SNR. The linear dynamical system response constraint estimates the current trial amplitude as an unknown constant scaling of the estimate in the previous trial plus zero-mean Gaussian noise with unknown variance. The independent response constraint estimates response amplitudes across trials as independent Gaussian random variables having unknown mean and variance. We develop a generalized expectation-maximization algorithm to obtain the maximum-likelihood (ML) estimates of the signal waveform, noise covariance matrix, and unknown constraint parameters. ML source localization is achieved by scanning the likelihood over different sets of spatial bases. We demonstrate the variability estimation and source localization effectiveness of the proposed algorithms using both real and simulated ER data.     

Fitting graphs and numerical tables for a single parameterrepresentation of the shape of programme sound signal distributions

In order to represent the statistical character of a sound signal simply and with a single parameter, numerical tables and graphs of theoretical distributions for the estimation of a parameter called the shape parameter of the distribution are presented. The theoretical distributions are based on a statistical model of broadcast signals which was derived from the analysis of variations in measured distribution. The statistical functions dealt with are PDFs (probability density functions) and CDFs (cumulative distribution functions) for instantaneous amplitude and power, for RMS-valued intensity fluctuations, and for one-minute mean powers and peak powers. A list of the values of the shape parameter estimated for CCIR data is given, from which almost all the CCIR data can be reproduced by using the theoretical distribution curves and the numerical tables presented