Point and Interval Estimation, From One-Order Statistic, of the Location Parameter of an Extreme-Value Distribution with Known Scale Parameter and of the Scale Parameter of a Weibull Distribution with Known Shape Parameter

This paper derives a one-order statistic estimator ?mn b for the location parameter of the (first) extreme-value distribution of smallest values with cumulative distribution function F(x;u,b) = 1 - exp {-exp[(x-u)/b]} using the minimum-variance unbiased one-order statistic estimator for the scale parameter of an exponential distribution, as was done in an earlier paper for the scale parameter of a Weibull distribution. It is shown that exact confidence bounds, based on one-order statistic, can be easily derived for the location parameter of the extreme-value distribution and for the scale parameter of the Weibull distribution, using exact confidence bounds for the scale parameter of the exponential distribution. The estimator for u is shown to be b ln cmn + xmn, where xmn is the mth order statistic from an ordered sample of size n from the extreme-value distribution with scale parameter b and Cmn is the coefficient for a one-order statistic estimator of the scale parameter of an exponential distribution. Values of the factor cmn, which have previously viously been tabulated for n = 1(1)20, are given for n = 21(1)40. The ratios of the mean-square-errors of the maximum-likelihood estimators based on m order statistics to those of the one-order statistic estimators for the location parameter of the extreme-value distribution and the scale parameter of the Weibull distribution are investigated by Monte Carlo methods. The use of the table and related tables is discussed and illustrated by numerical examples.     

One-Order-Statistic Conditional Estimators of Shape Parameters of Limited and Pareto Distributions and Scale Parameters of Type II Asymptotic Distributions of Smallest and Largest Values

One-order-statistic estimators are derived for the shape parameter K of the limited distribution function F1(x, ?, K) = 1 - (? - x)K and the Pareto distribution function F2(y, ?, K) = 1 - (y - ?)-K, given the location parameters ? and ?, respectively. Similar estimators are derived for the scale parameters v1 and Vn, of the Type II asymptotic distributions of smallest and largest values, F3(w, v1, K) = 1 - exp[-(w/v1)-K] and F4(z, vn K) = exp [-(z/vn)-K], given the shape parameter K and assuming the location parameter is zero. The one-order-statistic estimators are K?|? = -1/Cmn 1n(? - xmn) for the limited distribution, K?|? = 1/Cmn 1n(ymn - ?) for the Pareto distribution, ?1|K = Cmn-1/K Wmn and ?n|K = Cmn-1/K Zn-m+1,n for the Type II distributions of smallest and largest values, where Xmn, Ymn, Wmn, Zmn are the mth order statistics of samples of size n from the respective distributions and Cmn is the coefficient for a one-order-statistic estimator of the scale parameter of an exponential distribution, which has been tabled in an earlier paper. It is shown that exact confidence bounds can be easily derived for these parameters using exact confidence bounds for the scale parameter of the exponential distribution. Use of the estimators is illustrated by numerical examples.     

Empirical Bayes Estimators of Reliability for Lognormal Failure Model

Empirical Bayes (EB) procedures are considered for estimating the reliability R(t;?,?) = gaufc[(ln t -?)/?] for the lognormal failure model. EB estimators are obtained for the 2 cases: i)? is unknown and ? is known, and both ? and ? are unknown. The empirical Cdf of the maximum likelihood estimators of the parameters is used to obtain the EB estimators. ii) A smooth EB estimator of R(t;?,?) is developed when ? is unknown and ? is known. A modification of this estimator is proposed for both ? and ? unknown. In both cases, EB estimators are obtained for complete samples at each testing stage. Monte Carlo simulations are presented to compare the EB estimators and the maximum likelihood (ML) estimators of R(t;?,?). The simulations indicate that the smooth EB estimators have smaller mean squared errors than the other EB estimators or the ML estimators.     

Empirical Bayes Estimation in the Weibull Distribution

In part I empirical Bayes estimation procedures are introduced and employed to obtain an estimator for the unknown random scale parameter of a two-parameter Weibull distribution with known shape parameter. In part II, procedures are developed for estimating both the random scale and shape parameters. These estimators use a sequence of maximum likelihood estimates from related reliability experiments to form an empirical estimate of the appropriate unknown prior probability density function. Monte Carlo simulation is used to compare the performance of these estimators with the appropriate maximum likelihood estimator. Algorithms are presented for sequentially obtaining the reduced sample sizes required by the estimators while still providing mean squared error accuracy compatible with the use of the maximum likelihood estimators. In some cases whenever the prior pdf is a member of the Pearson family of distributions, as much as a 60% reduction in total test units is obtained. A numerical example is presented to illustrate the procedures.     

Bayesian Estimation of Parameters of Life Distributions and Reliability from Type II Censored Samples

The Bayesian approach to reliability estimation from Type II censored samples is discussed here with emphasis on obtaining natural conjugate prior distributions. The underlying sampling distribution from which the censored samples are drawn follows a generalized life model (GLM) which includes a model proposed by Epstein and Sobel, Weibull, exponential, and Rayleigh distributions as special cases. Results are given for the Type II asymptotic distribution of largest values, Pareto, and Limited distribution. The natural conjugate prior, Bayes estimate for the generalized scale parameter, posterior risk, Bayes risk and Bayes estimate of the reliability function were derived for the distributions studied. In every case the natural conjugate prior is a 2-parameter family which provides a wide range of possible prior knowledge. Conjugate diffuse priors were derived. A diffuse prior, also called a quasi-pdf, is not a pdf because its integral is not unity. It represents roughly an informationless prior state of knowledge. The proper choice of the parameter for the diffuse prior leads to maximum likelihood, classical uniform minimum-variance unbiased estimator, and an admissible biased estimator with minimum mean square error as the generalized Bayes estimate. A feature of the GLM is the increasing function g(·) with possible applications in accelerated testing. KG(·) is a s-complete s-sufficient statistic for ?, and KG(·)/m is a maximum likelihood estimate for ?. Similar results were obtained for the Pareto, Type II asymptotic distribution of extremes, Pareto (associated with Pearl-Reed growth distribution) and others.     

Time-censored ramp tests with stress bound for Weibull lifedistribution

This paper considers ramp tests for Weibull life distribution when there are limitations on test stress and test time. The inverse power law and a cumulative exposure model are assumed. Maximum likelihood estimators of model parameters and their asymptotic covariance matrix are shown. The optimum ramp test plans are given which minimize the asymptotic variance of the ML estimator of a specified quantile of log(life) at design constant stress. The effects of the pre-estimates of design parameters are studied     

Empirical Bayes Approach to Reliability Estimation for the Exponential Distribution

In the empirical Bayes (EB) approach to a reliability estimation for the exponential distribution, a prior distribution is placed on the family of prior gamma distributions to produce an EB estimator. The EB estimator is asymptotically optimal and admissible. The results of a Monte Carlo study are presented to demonstrate the favorable risk behavior of the EB estimator as a Bayes estimator if the sample size is large.     

Parameter estimator based on a minimum discrepancy criterion: aBayesian approach

A new estimation criterion based on the discrepancy between the estimator's error covariance and its information lower bound is proposed. This discrepancy measure criterion tries to take the information content of the observed data into account. A minimum discrepancy estimator (MDE) is then obtained under a linearity assumption. This estimator is shown to be equivalent to the maximum likelihood estimator (MLE), if one assumes that a linear efficient estimator exists and the prior distribution of parameters is uniform. Moreover, it is equivalent to the minimum variance unbiased estimator (MVUE) if the MDE is required to be unbiased. Illustrative examples of MDE and its comparisons with other estimators are given     

Comparing between estimation approaches: admissible and dominating linear estimators

We treat the problem of evaluating the performance of linear estimators for estimating a deterministic parameter vector x in a linear regression model, with the mean-squared error (MSE) as the performance measure. Since the MSE depends on the unknown vector x, a direct comparison between estimators is a difficult problem. Here, we consider a framework for examining the MSE of different linear estimation approaches based on the concepts of admissible and dominating estimators. We develop a general procedure for determining whether or not a linear estimator is MSE admissible, and for constructing an estimator strictly dominating a given inadmissible method so that its MSE is smaller for all x. In particular, we show that both problems can be addressed in a unified manner for arbitrary constraint sets on x by considering a certain convex optimization problem. We then demonstrate the details of our method for the case in which x is constrained to an ellipsoidal set and for unrestricted choices of x. As a by-product of our results, we derive a closed-form solution for the minimax MSE estimator on an ellipsoid, which is valid for arbitrary model parameters, as long as the signal-to-noise-ratio exceeds a certain threshold.     

Estimation of Constrained Parameters With Guaranteed MSE Improvement

We address the problem of estimating an unknown parameter vector x in a linear model y=Cx+v subject to the a priori information that the true parameter vector x belongs to a known convex polytope X. The proposed estimator has the parametrized structure of the maximum a posteriori probability (MAP) estimator with prior Gaussian distribution, whose mean and covariance parameters are suitably designed via a linear matrix inequality approach so as to guarantee, for any xisinX, an improvement of the mean-squared error (MSE) matrix over the least-squares (LS) estimator. It is shown that this approach outperforms existing "superefficient" estimators for constrained parameters based on different parametrized structures and/or shapes of the parameter membership region X     

Bayes estimation of the linear hazard-rate model

In life testing, the failure-time distributions are often specified by choosing an appropriate hazard-rate function. The class of life-time distribution characterized by a linear hazard-rate includes the one-parameter exponential and Rayleigh distributions. Usually the parameters of the linear hazard-rate model are estimated by the method of least squares. This work is concerned with Bayes estimation of the two-parameters from a type-2 censored sample. Monte Carlo simulation is used to compare the Bayes risk of the regression estimator with the minimum Bayes risk. Discrete mixtures of decreasing failure rate distributions are known to have decreasing failure rates. The authors prove that the result holds for continuous mixtures as well     

On estimating the shape parameter of the Weibull distribution by shrinkage towards an interval

This paper proposes some shrunken estimators for the shape parameter of the Weibull distribution under censored sampling when some apriori or guessed interval containing the parameter β is available. The extensions of the work done in Pandey and Singh (1984) have been considered. Comparisons of the proposed estimators with the usual unbiased estimator, in terms of mean squared error are made. It is found that the proposed estimators are preferable to the usual estimator in some guessed interval of the parameter space of β.     

Estimators for type-II censored (log)normal samples

Three common estimators for the parameters of the log-normal distribution are evaluated for censored samples. Correction factors which eliminate essentially all the bias, and formulas for the standard deviations of the estimators, are presented. It is reported that the Persson-Rootzen estimators are about as good as the maximum-likelihood estimators, without the penalty of requiring iterative (computer) optimization. Also, the estimators resulting from (least squares) fitting a line to the plot of log lifetimes on normal (Gaussian) probability paper are reasonably good. Formulas are given for obtaining these latter estimators without actually plotting the points. The author simulated 5 k to 30 k samples (more samples for smaller N for each case) and calculated the following: the means, standard deviations, and third moments of each estimator; correlations between the two members of each pair; comparisons between the estimators; and simple corrections to improve the performance of the estimators     

Bayes estimation of reliability in stress-strength model of Weibull distribution with equal scale parameters

The paper provides a Bayesian approach to inference about the reliability in a multicomponent stress-strength system. We consider Bayes' estimator of the system reliability from data consisting of a random sample from the stress distribution and one from the strength distribution when the two distributions are Weibull with equal and known scale parameters. The estimator of λ, ratio of two shape parameters, is also considered. The proposed estimators can be compared with the maximum likelihood estimators (mles). However, the comparison is carried out for single component stress-strength system and the Monte Carlo efficiencies are obtained. It is found that the proposed estimators are better than the corresponding mles.     

Second-order parameter estimation

This work provides a general framework for the design of second-order blind estimators without adopting any approximation about the observation statistics or the a priori distribution of the parameters. The proposed solution is obtained minimizing the estimator variance subject to some constraints on the estimator bias. The resulting optimal estimator is found to depend on the observation fourth-order moments that can be calculated analytically from the known signal model. Unfortunately, in most cases, the performance of this estimator is severely limited by the residual bias inherent to nonlinear estimation problems. To overcome this limitation, the second-order minimum variance unbiased estimator is deduced from the general solution by assuming accurate prior information on the vector of parameters. This small-error approximation is adopted to design iterative estimators or trackers. It is shown that the associated variance constitutes the lower bound for the variance of any unbiased estimator based on the sample covariance matrix. The paper formulation is then applied to track the angle-of-arrival (AoA) of multiple digitally-modulated sources by means of a uniform linear array. The optimal second-order tracker is compared with the classical maximum likelihood (ML) blind methods that are shown to be quadratic in the observed data as well. Simulations have confirmed that the discrete nature of the transmitted symbols can be exploited to improve considerably the discrimination of near sources in medium-to-high SNR scenarios.     

Properties of a consistent estimation procedure in ultrastructural model when reliability ratio is known

To overcome the problem of inconsistency of the ordinary least squares (OLS) estimators for regression coefficients in a linear measurement error model, we assume the reliability ratio to be known a priori and utilize this information to form consistent estimators in an ultrastructural models. Assuming the distributions of measurement errors and the random error component to be not necessarily normal, the efficiency properties of the estimator for slope parameter have been analysed and the effect of departure from normality is examined.     

Asymptotic properties of 2-D MUSIC estimator and comparison to 2-D MP estimator

The MUSIC estimator of two-dimensional frequencies (2-D MUSIC) is studied assuming a one-measurement data model with deterministic phases and additive complex white Gaussian noise. The large sample estimation covariance of the 2-D MUSIC is derived and compared to that of the 2-D matrix pencil (MP) estimator. The theoretical estimation variances for both the MP and MUSIC estimators are compared with the simulated MP and MUSIC estimation variances and the Cramer-Rao Bound (CRB). In the single 2-D sinusoid case, the most revealing form of the estimation covariance for both estimators are provided. The results shown in this paper are valid for a median range of SNR.     

Wavelet-based estimators of scaling behavior

Various wavelet-based estimators of self-similarity or long-range dependence scaling exponent are studied extensively. These estimators mainly include the (bi)orthogonal wavelet estimators and the wavelet transform modulus maxima (WTMM) estimator. This study focuses both on short and long time-series. In the framework of fractional autoregressive integrated moving average (FARIMA) processes, we advocate the use of approximately adapted wavelet estimators. For these "ideal" processes, the scaling behavior actually extends down to the smallest scale, i.e., the sampling period of the time series, if an adapted decomposition is used. But in practical situations, there generally exists a cutoff scale below which the scaling behavior no longer holds. We test the robustness of the set of wavelet-based estimators with respect to that cutoff scale as well as to the specific density of the underlying law of the process. In all situations, the WTMM estimator is shown to be the best or among the best estimators in terms of the mean-squared error (MSE). We also compare the wavelet estimators with the detrended fluctuation analysis (DFA) estimator which was previously proved to be among the best estimators which are not wavelet-based estimators. The WTMM estimator turns out to be a very competitive estimator which can be further generalized to characterize multiscaling behavior     

Performance of quadratic time-frequency distributions as instantaneous frequency estimators

General performance analysis of the shift covariant class of quadratic time-frequency distributions (TFDs) as instantaneous frequency (IF) estimators, for an arbitrary frequency-modulated (FM) signal, is presented. Expressions for the estimation bias and variance are derived. This class of distributions behaves as an unbiased estimator in the case of monocomponent signals with a linear IF. However, when the IF is not a linear function of time, then the estimate is biased. Cases of white stationary and white nonstationary additive noises are considered. The well-known results for the Wigner distribution (WD) and linear FM signal, and the spectrogram of signals whose IF may be considered as a constant within the lag window, are presented as special cases. In addition, we have derived the variance expression for the spectrogram of a linear FM signal that is quite simple but highly signal dependent. This signal is considered in the cases of other commonly used distributions, such as the Born-Jordan and the Choi-Williams distributions. It has been shown that the reduced interference distributions outperform the WD but only in the case when the IF is constant or its variations are small. Analysis is extended to the IF estimation of signal components in the case of multicomponent signals. All theoretical results are statistically confirmed.