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 …

A New Technique of Invariant Statistical Embedding and Averaging in Terms of Pivots for Improvement of Statistical Decisions Under Parametric Uncertainty

In this chapter, a new technique of invariant embedding of sample statistics in a decision criterion (performance index) and averaging this criterion via pivotal quantities (pivots) is proposed for intelligent constructing efficient (optimal, uniformly non-dominated, unbiased, improved) statistical decisions under parametric uncertainty. 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, the technique of invariant statistical embedding and averaging in terms of pivotal quantities (ISE&APQ) is independent of the choice of priors and represents …

Optimal airline seat inventory control for multi‐leg flights

Airline seat inventory control is about “selling the right seats to the right people at the right time”. In this paper, the problem of determining optimal booking policy for multiple fare classes in a pool of identical seats for multi‐leg flights is considered. During the time prior to departure of a multi‐leg flight, decisions must be made concerning the allocation of reserved seats to passengers requesting space on the full or partial spans of the flight. It will be noted that in the case of multi‐leg flights the long‐haul passengers are often unable to obtain seats because the shorter‐haul passengers block them. For large commercial airlines, efficiently setting and updating seat allocat…

A New Simple Computational Method of Simultaneous Constructing and Comparing Confidence Intervals of Shortest Length and Equal Tails for Making Efficient Decisions Under Parametric Uncertainty

A confidence interval is a range of values that provides the user with useful information about how accurately a statistic estimates a parameter. In the present paper, a new simple computational method is proposed for simultaneous constructing and comparing confidence intervals of shortest length and equal tails in order to make efficient decisions under parametric uncertainty. This unified computational method provides intervals in several situations that previously required separate analysis using more advanced methods and tables for numerical solutions. In contrast to the Bayesian approach, the proposed approach does not depend on the choice of priors and is a novelty in the theory of st…

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…

A Novel Intelligent Technique for Product Acceptance Process Optimization on the Basis of Misclassification Probability in the Case of Log-Location-Scale Distributions

In this paper, to determine the optimal parameters of the product acceptance process under parametric uncertainty of underlying models, a new intelligent technique for optimization of product acceptance process on the basis of misclassification probability is proposed. It allows one to take into account all possible situations that may occur when it is necessary to optimize the product acceptance process. The technique is based on the pivotal quantity averaging approach (PQAA) which allows one to eliminate the unknown parameters from the problem and to use available statistical information as completely as possible. It is conceptually simple and easy to use. One of the most important featur…

A New Technique for Vibration-Based Diagnostics of Fatigued Structures Based on Damage Pattern Recognition via Minimization of Misclassification Probability

Vibration-based diagnostics provide various methods to detect, locate, and characterize damage in structural and mechanical systems by examining changes in measured vibration response. Research in vibration-based damage recognition has been rapidly expanding over the last few years. The basic idea behind this technology is that modal parameters (notably frequencies, mode shapes, and modal damping) are functions of the physical properties of the structure (mass, damping, and stiffness). Therefore, changes in the physical properties will cause detectable changes in the modal properties. In investigations, many techniques were applied to recognize damage in structural and mechanical systems, b…

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…

Decision Making and Optimization for 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 inspections will lead to higher cost. When there is a fatigue reliability requirement, the problem is usually to develop …

A New Technique for Education Process Optimization via the Dual Control Approach

Adaptive dual control in one biomedical problem

In this paper, the following biomedical problem is considered. People are subjected to a certain chemotherapeutic treatment. The optimal dosage is the maximal dose for which an individual patient will have toxicity level that does not cross the allowable limit. We discuss sequential procedures for searching the optimal dosage, which are based on the concept of dual control and the principle of optimality. According to the dual control theory, the control has two purposes that might be conflicting: one is to help learning about unknown parameters and/or the state of the system (estimation); the other is to achieve the control objective. Thus the resulting control sequence exhibits the closed…

A New Intelligent Technique of Constructing Optimal Airline Seat Protection Levels for Multiple Nested Fare Classes of Single-Leg Flights

A new, rigorous formulation of the optimization problem of airline seat protection levels for multiple nested fare classes is presented. A number of results useful for practical application are obtained. A numerical example is given.

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…

A New Dynamic Model for Anticipatory Adaptive Control of Airline Seat Reservation via Order Statistics of Cumulative Customer Demand

This paper deals with dynamic anticipatory adaptive control of airline seat reservation for the stochastic customer demand that occurs over time T before the flight is scheduled to depart. It is assumed that time T is divided into m periods, namely a full fare period and m−1 discounted fare periods. The fare structure is given. An airplane has a seat capacity of U. For the sake of simplicity, but without loss of generality, we consider (for illustration) the case of nonstop flight with two fare classes (business and economy). The proposed policies of the airline seat inventory control are based on the use of order statistics of cumulative customer demand, which have such properties as bivar…

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…

Adaptive Planning in-Service Inspections of Fatigued Structures in Damage Tolerance Situations via Observations of Crack Growth Process

From an engineering standpoint the fatigue life of a fatigued structure consists of two periods: (i) crack initiation period, which starts with the first load cycle and ends when a technically detectable crack is presented, and (ii) crack propagation period, which starts with a technically detectable crack and ends when the remaining cross section can no longer withstand the loads applied and fails statically. The main aim of this paper is to present more accurate innovative stochastic fatigue model for adaptive planning inspections of fatigued structures in damage tolerance situations via observations of crack growth process during a crack propagation period. A new crack growth equation is…

New Procedures of Pattern Classification for Vibration-Based Diagnostics via Neural Network

In this paper, the new distance-based embedding procedures of pattern classification for vibration-based diagnostics of gas turbine engines via neural network are proposed. Diagnostics of gas turbine engines is important because of the high cost of engine failure and the possible loss of human life. Engine monitoring is performed using either ‘on-line’ systems, mounted within the aircraft, that perform analysis of engine data during flight, or ‘off-line’ ground-based systems, to which engine data is downloaded from the aircraft at the end of a flight. Typically, the health of a rotating system such as a gas turbine is manifested by its vibration level. Efficiency of gas turbine monitoring s…

Statistical Techniques for Validation of Simulation and Analytic Stochastic Models

In this paper, we consider the problem of statistical validation of multivariate stationary response simulation and analytic stochastic models of observed systems (say, transportation or service systems), which have p response variables. The problem is reduced to testing the equality of the mean vectors for two multivariate normal populations. Without assuming equality of the covariance matrices, it is referred to as the Behrens–Fisher problem. The main purpose of this paper is to bring to the attention of applied researchers the satisfactory tests that can be used for testing the equality of two normal mean vectors when the population covariance matrices are unknown and arbitrary. To illus…

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 …

A New Technique for Optimization of Product Acceptance Process in Terms of Misclassification Probability

A product acceptance process is an inspecting one in statistical quality control or reliability tests, which are used to make decisions about accepting or rejecting lots of products to be submitted. This process is important for industrial and business purposes of quality management. To determine the optimal parameters of the product acceptance process under parametric uncertainty of underlying lifetime models (in terms of misclassification probability), a new optimization technique is proposed. The most popular lifetime distribution used in the field of product acceptance is a two-parameter Weibull distribution, with the assumption that the shape parameter is known. Such oversimplified ass…

Improved Planning In-Service Inspections of Fatigued Aircraft Structures Under Parametric Uncertainty of Underlying Lifetime 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 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 inspection…

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…

Statistical validation of simulation models of observable systems

In this paper, for validating computer simulation models of real, observable systems, an 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 two multivariate normal populations (compared with respect to their means) are different and unknown. The test is based on invariant statistic whose distribution, under the null hypothesis, does not depend on the unknown (nuisance) parameters. The sample size and threshold of the UMPI test are determined from minimization of the weighted sum of the model builder's risk a…

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…

Effective state estimation of stochastic systems

In the present paper, for constructing the minimum risk estimators of state of stochastic systems, a new technique of invariant embedding of sample statistics in a loss function 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 estimator, which has smaller risk than any of the well‐known estimators. There exists a class of control systems where observations are not …

Novel Approaches to Prediction of a Future Number of Failures Based on Previous In-Service Inspections

In this chapter, we present novel approaches to predictions of the number of failures that will be observed in a future inspection of a sample of units, based only on the results of the previous in-service inspections of the same sample. The failure-time of such units is modeled with a distribution from a two-parameter Weibull distribution. The different cases of parametric uncertainty are considered. The pivotal quantity averaging approach proposed here for constructing point prediction and simple prediction limits emphasizes pivotal quantities relevant for eliminating unknown parameters from the problems and represents a special case of the method of invariant embedding of sample statisti…

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 …

Intelligent Constructing Exact Tolerance Limits for Prediction of Future Outcomes Under Parametric Uncertainty

The problem of constructing one-sided exact statistical tolerance limits on the kth order statistic in a future sample of m observations from a distribution of log-location-scale family on the basis of an observed sample from the same distribution is considered. The new technique proposed here emphasizes pivotal quantities relevant for obtaining tolerance factors and is applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. The exact tolerance limits on order statistics associated with sampling from underlying distributions can be found easily and quickly making tables, simulation, Monte Carlo estimated percen…

OPTIMAL AIRLINE SEAT INVENTORY CONTROL FOR MULTI-LEG FLIGHTS

Abstract For large commercial airlines, efficiently setting and updating seat allocation targets for each passenger category on each multi-leg flight is an extremely difficult problem. This paper presents static and dynamic models of airline seat inventory control for multi-leg flights with multiple fare classes, which allow one to maximize an expected contribution to profit. The dynamic model uses the most recent demand and capacity information and allows one to allocate seats dynamically and anticipatory over time.

A Novel Intelligent Technique of Invariant Statistical Embedding and Averaging via Pivotal Quantities for Optimization or Improvement of Statistical Decision Rules under Parametric Uncertainty

In the present paper, for intelligent constructing efficient (optimal, uniformly non-dominated, unbiased, improved) statistical decisions under parametric uncertainty, a new technique of invariant embedding of sample statistics in a decision criterion and averaging this criterion over pivots’ probability distributions 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, the technique of invariant statistical embedding and averaging via pivotal quantities (ISE&APQ) is independent of the choice of priors and represents a novelty i…