Distribution Of Unknown Parameters Research Articles

Stochastic inverse boundary element method for defect and structural reliability parameter prediction

A stochastic inverse boundary element method is developed. The stochastic boundary integral equations are combined with the improved Kalman filtering algorithm to formulate the inverse iterative scheme and minimize the difference between the initial value and the objective value. Several numerial examples which show the effectiveness of this paper, including unknown defect and unknown parameter distribution of random boundary traction, are presented and discussed.

BAYESIAN LOCAL INFLUENCE ASSESSMENTS IN A GROWTH CURVE MODEL WITH GENERAL COVARIANCE STRUCTURE

The objective of this paper is to present a Bayesian approach based on Kullback-Leibler divergence for assessing local influence in a growth curve model with general covariance structure. Under certain prior distribution assumption, the Kullback-Leibler divergence is used to measure the influence of some minor perturbation on the posterior distribution of unknown parameter. This leads to the diagnostic statistic for detecting which response is locally influential. As an application, the common covariance-weighted perturbation scheme is thoroughly considered.

Decision procedures of automated system for inspection program development

Fatigue failure probability of airframe depends on unknown distribution parameters of fatigue crack growth model. The parameters should be estimated by processing of airframe full-scale fatigue test results. A definition of p-set function as specific statistical decision function is introduced. The use of the function, which defines some p-set for random vector, provides limitation of supremum of failure probability (as a function of unknown parameter) instead of dubious compromise combination of required reliability and confidence probability. The calculations are complex enough but it can be used in frame of automated system.

Performance evaluation of closed tandem queueing networks with finite buffers

In this paper we propose a new approximation method for the analysis of closed tandem queueing networks with general service times and blocking-after-service. The principle of the method is to decompose the original network consisting of M servers into a set of M subsystems, each subsystem consisting of two servers separated by a finite buffer. In order to determine the distributions of the service times of the two servers of each subsystem, we express relationships among distributions pertaining to the different subsystems. We then propose to use a two-moment approximation. The population constraint of the closed network is taken into account by prescribing that the sum of the average buffer sizes of the subsystems is equal to the number of the customers of the network. We end up with a set of equations that characterize the unknown parameters of the service time distributions of the servers of the subsystems. An iterative procedure is then used to determine these unknown parameters. Numerical results show that the new method is quite accurate.

Empirical Stochastic Silo Load Model. II: Data Analysis

Pressure data obtained in experiments with an existing grain silo are used to estimate the parameters of a stochastic silo load model formulated by the rirst two authors in the first of three papers. The estimation procedure is based on the method of maximum likelihood as applied to point observations of a Gaussian random field with unknown distribution parameters. The justification of this method is not based on superior scientific reasoning, but rather on a pragmatic principle: The family of interpolation curves between the measured data are required to satisfy whatever given regularity conditions (continuity, differentiability, etc.) in as simple a way as possible without being in conflict with the data. If by visual inspection these interpolation curves are judged to be reasonable from an engineering point of view, then also the corresponding silo load model should be acceptable as the basis for a reliability analysis of the silo wall. Generally, this procedure leads to a conservative assessment of the reliability. This is demonstrated in the third paper.

On order statistics from non-identical right-truncated exponential random variables and some applications

By considering order statistics arising from n independent non-identically distributed right-truncated exponential random variables, we derive in this paper several recurrence relations for the single and the product moments of order statistics. These recurrence relations are simple in nature and could be used systematically in order to compute all the single and the product moments of order statistics for all sample sizes in a simple recursive manner. The results for order statistics from a multiple-outlier model (with a slippage of p observations) from a right-truncated exponential population are deduced as special cases. These results will be useful in assessing robustness properties of any linear estimator of the unknown parameter of the right-truncated exponential distribution, in the presence of one or more outliers in the sample. These results generalize those for the order statistics arising from an i.i.d. sample from a right-truncated exponential population established by Joshi (1978, 1982).

Robustness of perturbation analysis estimators for queueing systems with unknown distributions

Sample-path-based stochastic gradient estimators for performance measures of queueing systems rely on the assumption that a probability distribution of the random vector of interest (e.g., a service or interarrival time sequence) is given. In this paper, we address the issue of dealing with unknown probability distributions and investigate the robustness of such estimators with respect to possibly erroneous distribution choices. We show that infinitesimal perturbation analysis (IPA) can be robust in this sense and, in some cases, provides distribution-independent estimates. Comparisons with other gradient estimators are provided, including experimental results. We also show that finite perturbation analysis (FPA), though only providing gradient approximations, possesses some attractive robustness properties with respect to unknown distribution parameters. An application of FPA estimation is included for a queueing system performance optimization problem involving customers with real-time constraints.

Upper limits in experiments with background or measurement errors

It is shown that upper limits can be calculated in classical statistics for measurements contaminated by background or distorted owing to the finite resolution of the apparatus. The a posteriori probability distributions, as fixed by the experimental results, are used for the background and the measurement errors, respectively. Contrary to the Bayesian approach, assumptions on prior distributions of unknown parameters are avoided.

Construction of control plans using a quantitative index with two-sided bounds

The article constructs control plans using a normally distributed quantitative index with unknown distribution parameters and two-sided bounds on the index.

Maximum Likelihood Stacking in White Gaussian Noise with Unknown Variances

In seismic exploration, the process of estimating the source produced signal common to a collection of sampled data recordings contaminated with additive noise is known as stacking. Stacking is used when a low energy acoustic source can be repeated in the same physical location (or nearly so) and a collection of data recordings are gathered at the same electromechanical transducer from the multiple source initiations. This stacking process is primarily aimed at reducing environmental noise which frequently dominates each individual recording when low energy sources are used. This is in contrast to other techniques used to reduce or eliminate so called source generated noise which behaves as signal in the stacking process. Specifically, the problem we address is the simultaneous maximum likelihood (ML) estimation of signal values and unknown noise distribution parameters from a collection of sampled data recordings, each containing the same signal sequence. In this paper we restrict our consideration to independent, zero mean, Gaussian noise variates with unknown variance which may differ between data records. This model is justified when a broadband analog noise source is subsequently low pass filtered along with the signal and the result sampled at the Nyquist rate. The linear filtering tends to produce a Gaussian noise process and the Nyquist rate sampling produces approximately uncorrelated samples. The ML estimates of signal values and noise variances are obtained numerically by steepest ascent.

Solution of underdetermined electromagnetic and seismic problems by the maximum entropy method

Many inversion problems require solution of a Fredholm integral equation of the form T(\bar{r}) = \int D(\bar{r},\bar{r}')\sigma (\bar{r}') dV' , where T is the observable, D is an operator, and \sigma is the unknown parameter distribution. Examples occur in the areas of radiation and scattering, tomography, and geotomography. We reduce the equation to matrix form and apply a maximum entropy technique based on the first principle of data reduction to obtain a most probable \sigma distribution. We illustrate the technique by synthetic data examples of geotomography assuming straight rays, with and without noise. The examples show how sharp anomalies may be identified in grossly underdetermined situations. We outline the algorithm used and describe some computational properties. Our method suggests a way of overcoming the ill-conditioned nature of Fredholm integral equation inversion.

A minimum Bayes risk approach to optimal portfolio choice

This paper presents a Bayes decision approach to optimal portfolio choice under uncertainty. The approach ties together the estimation and decision problems. It duly recognizes the uncertainty of estimation of unknown parameters of asset return distributions by postulating a posterior PDF. It differentiates between the true-optimal solution and a quasi-optimal solution and defines the difference between the realized utilities under these two different solutions as a loss. Then it treats the expected loss with respect to the posterior PDF as the Bayes risk. The optimal portfolio choice is formulated as a constrained optimization (nonlinear programming) problem. The Bayes solution is derived in general and is solved for diffuse and conjugate priors in particular. An illustrative example is provided. The numerical results are reported and compared to those obtained according to the traditional method that ignores estimation risk. To observe how estimation risk affects optimal portfolio decision, the Bayes ex...

Inference concerning the population correlation coefficient from bivariate normal samples based on minimal observations

Let (X, Y) be bivariate normally distributed with means (μ 1, μ 2), variances (σ 1 2, σ 2 2 ) and correlation between X and Y equal to ρ. Let (X i, Y i ) be independent observations on (X, Y) for i=1,2,..., n. Because of practical considerations only Z i =min (X i, Y i) is observed. In this paper, as in certain routine applications, assuming the means and the variances to be known in advance, an unbiased consistent estimator of the unknown distribution parameter ρ is proposed. A comparison between the traditional maximum likelihood estimator and the unbiased estimator is made. Finally, the problem is extended to multivariate normal populations with common mean, common variance and common non-negative correlation coefficient.

Life Tests Under Dependent Competing Causes of Failure

In a life testing situation the failure of an individual, either a living organism or an inanimate object, may be classified into one of k(< 1) mutually exclusive classes, usually causes of failure. One often has dependent causes of failure in actual physical situations, i.e., the theoretical lifetime of an individual failing from one cause may be correlated with the theoretical lifetime of the same individual failing from a different cause. This paper i) discusses some properties of a bivariate Weibull distribution and ii) is concerned with estimating, by the method of maximum likelihood, the unknown parameters of life distributions belonging to two particular parametric families, viz., bivariate normal and bivariate Weibull, when the causes of failure are dependent. An example involving the failure of small electrical appliances is analyzed and compared with an analysis which assumes the causes of failure to be independent.

The equivalence condition for two important adaptation algorithms and its relation to efficient estimates of probability distribution parameters

A comparison is made between the convergence rates of two important algorithms in adaptation theory; one of them minimizes the empirical risk, while the other minimizes the Bayes a posteriori risk and is calculated according to the stipulated system of probability distributions which depend on a set of unknown parameters. The class of distributions is established for which the convergence rates of the indicated algorithms coincide. The relation between the conditions governing the indicated coincidence and the conditions governing the existence of efficient estimates of the unknown probability distribution parameters is investigated.

Two techniques for assessment of subjective probability distributions — An experimental study

Subjects were asked to assess their subjective probability distributions for unknown parameters of Bernoulli processes. The processes were generated by random devices like, for instance, irregular dice. The assessments were based on two assessment techniques. One asked for the median and quartiles of the distributions, the other asked for the impact of four hypothetical samples. The main purpose of the study was to study the resulting two sets of distributions. The results show substantial differences between the distributions using the two techniques but these differences decreased as the experiment progressed.

The effect of learning on the assessment of subjective probability distributions

Subjects were asked to assess subjective probability distributions for unknown parameters of a Bernoulli process. The process was generated by random devices such as oddly shaped dice. One group of subjects received feedback between the sessions in the form of the true values of the parameters. The members of this group improved their performance more than the second group. They also acquired more confidence in their assessments.

Frequency Functions Fitted by Moments

THE method of moments has been much used to fit frequency functions whether in the Pearson or the Gram-Charlier or the Edgeworth system. In the monograph in which R. A. Fisher laid the foundation for so much of the modern development of mathematical statistics and, in particular, for his theory of the estimation of values of unknown parameters of frequency distributions of known type, he indicated that type of frequency function which could be fitted with perfect efficiency, as he termed it, by the use of the first four moments.' It is an important matter in statistical theory to follow up this suggestion of Fisher's even though one may not often fit the type of function proposed. The frequency function in question, viz.,

An apparent inconsistency arising from tests of significance based on fiducial distributions of unknown parameters

It has been observed that the Behrens and Fisher test of the difference of the means of two samples gives a smaller percentage of significant results than might be expected on the analogy of the ordinary t test with a pooled estimate of variance. The cause of this apparent anomaly is explained, and it is shown that the criticisms of the test to which the anomaly has given rise have their origin in (a) neglect of the relevant information provided by the estimated values of the variances, and (b) an insufficient appreciation of the fiducial basis of all tests of significance (including the ordinary t test) on small samples.It is pointed out that Sukhatme's table (constructed for the Behrens and Fisher test) also provides a test for the weighted mean of the means of two sets of observations, concerning whose relative accuracy no prior knowledge is available.

Quantile regression for censored mixed-effects models with applications to HIV studies.

HIV RNA viral load measures are often subjected to some upper and lower detection limits depending on the quantification assays. Hence, the responses are either left or right censored. Linear/nonlinear mixed-effects models, with slight modifications to accommodate censoring, are routinely used to analyze this type of data. Usually, the inference procedures are based on normality (or elliptical distribution) assumptions for the random terms. However, those analyses might not provide robust inference when the distribution assumptions are questionable. In this paper, we discuss a fully Bayesian quantile regression inference using Markov Chain Monte Carlo (MCMC) methods for longitudinal data models with random effects and censored responses. Compared to the conventional mean regression approach, quantile regression can characterize the entire conditional distribution of the outcome variable, and is more robust to outliers and misspecification of the error distribution. Under the assumption that the error term follows an asymmetric Laplace distribution, we develop a hierarchical Bayesian model and obtain the posterior distribution of unknown parameters at the pth level, with the median regression (p = 0.5) as a special case. The proposed procedures are illustrated with two HIV AIDS studies on viral loads that were initially analyzed using the typical normal (censored) mean regression mixed-effects models, as well as a simulation study.