0000000000012828
AUTHOR
Maris Purgailis
Statistical validation of rival models for observable stochastic process and its identification
In this paper, for statistical validation of rival (analytical or simulation) models collected for modeling observable process in stochastic system (say, transportation or service system), a uniformly most powerful invariant (UMPI) test is developed from the generalized maximum likelihood ratio (GMLR). This test can be considered as a result of a new approach to solving the Behrens-Fisher problem when covariance matrices of multivariate normal populations (compared with respect to their means) are different and unknown. The test makes use of an invariant statistic whose distribution, under the null hypothesis, does not depend on the unknown (nuisance) parameters. The sample size and thresho…
Improved State Estimation of Stochastic Systems via a New Technique of Invariant Embedding
In this paper we construct the minimum risk estimators of state of stochastic systems. The method used is that of the invariant embedding of sample statistics in a loss function in order to form pivotal quantities, which make it possible to eliminate unknown parameters from the problem. This method is a special case of more general considerations applicable whenever the statistical problem is invariant under a group of transformations, which acts transitively on the parameter space.
Weibull Prediction Limits for a Future Number of Failures Under Parametric Uncertainty
In this paper, we present an accurate procedure, called “within-sample prediction of order statistics,” to obtain prediction limits for the number of failures that will be observed in a future inspection of a sample of units, based only on the results of the first in-service inspection of the same sample. The failure-time of such units is modeled with a two-parameter Weibull distribution indexed by scale and shape parameters β and δ, respectively. It will be noted that in the literature only the case is considered when the scale parameter β is unknown, but the shape parameter δ is known. As a rule, in practice the Weibull shape parameter δ is not known. Instead it is estimated subjectively …
ESTIMATION OF PRIVATE AND SOCIAL RATES OF RETURN TO INVESTMENTS IN EDUCATION IN LATVIA
The main aim of the paper is to evaluate the rates of return to investments in education at individual and society level as well. The task of the paper is to provide detailed analysis and estimation of the variables which impact the private and social rates of return. It is based on Professor Angel de la Fuente methodology complemented by Mincer earnings function and non-parametric DEA (Data Envelopment Analysis) method to estimate world technological frontier and the technological gap. For this purpose the authors build the matrix with respective years of schooling depending from level of schooling and birth of year taking into account the differences in schooling system since 1940ties. Th…
Stochastic Fatigue Models for Efficient Planning Inspections in Service of Aircraft Structures
For important fatigue-sensitive structures of aircraft whose breakdowns cause serious accidents, it is required to keep their reliability extremely high. In this paper, we discuss inspection strategies for such important structures against fatigue failure. The focus is on the case when there are fatiguecracks unexpectedly detected in a fleet of aircraft within a warranty period (prior to the first inspection). The paper examines this case and proposes stochastic models for prediction of fatigue-crack growth to determine appropriate inspections intervals. We also do not assume known parameters of the underlying distributions, and the estimation of that is incorporated into the analysis and d…
Finding Prediction Limits for a Future Number of Failures in the Prescribed Time Interval under Parametric Uncertainty
Computing prediction intervals is an important part of the forecasting process intended to indicate the likely uncertainty in point forecasts. Prediction intervals for future order statistics are widely used for reliability problems and other related problems. In this paper, we present an accurate procedure, called ‘within-sample prediction of order statistics', to obtain prediction limits for the number of failures that will be observed in a future inspection of a sample of units, based only on the results of the first in-service inspection of the same sample. The failure-time of such units is modeled with a two-parameter Weibull distribution indexed by scale and shape parameters β and δ, …
Optimal Adaptive Inspection Planning Process in Service of Fatigued Aircraft Structures
In this paper, a control theory is used for planning inspections in service of fatigue-sensitive aircraft structure components under crack propagation. One of the most important features of control theory is its great generality, enabling one to analyze diverse systems within one unified framework. A key idea, which has emerged from this study, is the necessity of viewing the process of planning in-service inspections as an adaptive control process. Adaptation means the ability of self-modification and self-adjustment in accordance with varying conditions of environment. The adaptive control of inspection planning process in service of fatigued aircraft structures differs from ordinary stoc…
Technique of Statistical Validation of Rival Models for Fatigue Crack Growth Process and Its Identification
The development of suitable models of stochastic crack growth process is important for the reliability analysis of fatigued structures as well as the scheduling of inspection and repair/replacement maintenance. Based on modifications of the solution of the deterministic differential equation for the crack growth rate, where a stochastic nature of this rate is expressed by a random disturbance embedded in the solution of the differential equation, the simple stochastic models are presented for practical applications. Each of these models represents a stochastic version of the solution of the Paris-Erdogan law equation. The models take into account the random disturbance parameters while main…
Improvement of Inventory Control under Parametric Uncertainty and Constraints
The aim of the present paper is to show how the statistical inference equivalence principle (SIEP), the idea of which belongs to the authors, may be employed in the particular case of finding the effective statistical decisions for the multi-product inventory problems with constraints. To our knowledge, no analytical or efficient numerical method for finding the optimal policies under parametric uncertainty for the multi-product inventory problems with constraints has been reported in the literature. Using the (equivalent) predictive distributions, this paper represents an extension of analytical results obtained for unconstrained optimization under parametric uncertainty to the case of con…
Unbiased Simultaneous Prediction Limits on Observations in Future Samples
This paper provides procedures for constructing unbiased simultaneous prediction limits on the observations or functions of observations of all of k future samples using the results of a previous sample from the same underlying distribution belonging to invariant family. The results have direct application in reliability theory, where the time until the first failure in a group of several items in service provides a measure of assurance regarding the operation of the items. The simultaneous prediction limits are required as specifications on future life for components, as warranty limits for the future performance of a specified number of systems with standby units, and in various other app…
Inspection Policies in Service of Fatigued Aircraft Structures
Fatigue is one of the most important problems of aircraft arising from their nature as multiple-component structures, subjected to random dynamic loads. For guaranteeing safety, the structural life ceiling limits of the fleet aircraft are defined from three distinct approaches: Safe-Life, Fail-Safe, and Damage Tolerance approaches. The common objectives to define fleet aircraft lives by the three approaches are to ensure safety while at the same time reducing total ownership costs. In this paper, the damage tolerance approach is considered and the focus is on the inspection scheme with decreasing intervals between inspections. The paper proposes an analysis methodology to determine appropri…
Inventory Control Under Parametric Uncertainty of Underlying Models
A large number of problems in inventory control, production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty of underlying models. In the present paper we consider the case, where it is known that the underlying distribution belongs to a parametric family of distributions. The problem of determining an optimal decision rule in the absence of complete information about the underlying distribution, i.e., when we specify only the functional form of the distribution and leave some or all of its parameters unspecified, is seen to be a standard problem of statistical estimation. Unfortunately, the clas…
Prediction Model Selection and Spare Parts Ordering Policy for Efficient Support of Maintenance and Repair of Equipment
The prediction model selection problem via variable subset selection is one of the most pervasive model selection problems in statistical applications. Often referred to as the problem of subset selection, it arises when one wants to model the relationship between a variable of interest and a subset of potential explanatory variables or predictors, but there is uncertainty about which subset to use. Several papers have dealt with various aspects of the problem but it appears that the typical regression user has not benefited appreciably. One reason for the lack of resolution of the problem is the fact that it has not been well defined. Indeed, it is apparent that there is not a single probl…
Stochastic Decision Support Models and Optimal Stopping Rules in a New Product Lifetime Testing
Determining when to stop a statistical test is an important management decision. Several stopping criteria have been proposed, including criteria based on statistical similarity, the probability that the system has a desired reliability, and the expected cost of remaining faults. This paper presents a new stopping rule in fixed-sample testing based on the statistical estimation of total costs involved in the decision to continue beyond an early failure as well as a stopping rule in sequential-sample testing to determine when testing should be stopped. The paper considers the problem that can be stated as follows. A new product is submitted for lifetime testing. The product will be accepted …
Detection and Recognition of Target Signals in Radar Clutter via Adaptive CFAR Tests
In this paper, adaptive CFAR tests are described which allow one to classify radar clutter into one of several major categories, including bird, weather, and target classes. These tests do not require the arbitrary selection of priors as in the Bayesian classifier. The decision rule of the recognition techniques is in the form of associating the p-dimensional vector of observations on the object with one of the m specific classes. When there is the possibility that the object does not belong to any of the m classes, then this object is to be classified as belonging to one of the m classes or to class m+1 whose distribution is unspecified. The tests are invariant to intensity changes in the …
Weibull Model for Dynamic Pricing in e-Business
As is the case with traditional markets, the sellers on the Internet do not usually know the demand functions of their customers. However, in such a digital environment, a seller can experiment different prices in order to maximize his profits. In this paper, we develop a dynamic pricing model to solve the pricing problem of a Web-store, where seller sets a fixed price and buyer either accepts or doesn’t buy. Frequent price changes occur due to current market conditions. The model is based on the two-parameter Weibull distribution (indexed by scale and shape parameters), which is used as the underlying distribution of a random variable X representing the amount of revenue received in the sp…
Solving the Problems of Inspection Planning under Parametric Uncertainty of Underlying Models
Certain fatigued structures must be inspected in order to detect fatigue damages that would otherwise not be apparent. A technique for obtaining optimal inspection strategies is proposed for situations where it is difficult to quantify the costs associated with inspections and undetected failure. For fatigued structures for which failures (fatigue damages) are only detected at the time of inspection, it is important to be able to determine the optimal times of inspection. Fewer inspections will lead to lower fatigue reliability of the structure upon demand, and frequent inspection will lead to higher cost. When there is a fatigue reliability requirement, the problem is usually to develop an…
Invariant Embedding Technique and Its Applications for Improvement or Optimization of Statistical Decisions
In the present paper, for improvement or optimization of statistical decisions under parametric uncertainty, a new technique of invariant embedding of sample statistics in a performance index is proposed. This technique represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics. Unlike the Bayesian approach, an invariant embedding technique is independent of the choice of priors. It allows one to eliminate unknown parameters from the problem and to find the best invariant decision rule, which has smaller risk than any of the well-known decision rules. To illustrate the proposed technique, applica…
Adaptive Stochastic Airline Seat Inventory Control under Parametric Uncertainty
Airline seat inventory control is a very profitable tool in the airline industry. The problem of adaptive stochastic airline seat inventory control lies at the heart of airline revenue management. This problem concerns the allocation of the finite seat inventory to the stochastic customer demand that occurs over time before the flight is scheduled to depart. The objective is to find the right combination of customers of various fare classes on the flight such that revenue is maximized. In this paper, the static and dynamic policies of stochastic airline seat inventory control (airline booking) are developed under parametric uncertainty of underlying models, which are not necessarily alterna…
Improvement of Statistical Decisions under Parametric Uncertainty
A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Decision‐making under uncertainty is a central problem in statistical inference, and has been formally studied in virtually all approaches to inference. The aim of the present paper is to show how the invariant embedding technique, the idea of which belongs to the authors, may be employed in the particular case of finding the improved statistical decisions under parametric uncertainty. This technique represents a simple and computationally attractive statistical method based on the constructive use of the i…
Selection of the Best Subset of Variables in Regression and Time Series Models
The problem of variable selection is one of the most pervasive model selection problems in statistical applications. Often referred to as the problem of subset selection, it arises when one wants to model the relationship between a variable of interest and a subset of potential explanatory variables or predictors, but there is uncertainty about which subset to use. Several papers have dealt with various aspects of the problem but it appears that the typical regression user has not benefited appreciably. One reason for the lack of resolution of the problem is the fact that it is has not been well defined. Indeed, it is apparent that there is not a single problem, but rather several problems …