- Alqifari, H.N., Coolen, F.P.A., 2019. Robustness of Nonparametric Predictive Inference for Future Order Statistics. Journal of Statistical Theory and Practice.
Bib Entry This paper considers robustness of Nonparametric Predictive Inference (NPI), in particular considering inference involving future order statistics. The concept of robust inference is usually aimed at development of inference methods which are not too sensitive to data contamination or to deviations from model assumptions. In this paper we use it in a slightly narrower sense. For our aims, robustness indicates insensitivity to small change in the data, as our predictive probabilities for order sta- tistics and statistical inferences involving future observations depend upon the given observations. We introduce some concepts for assessing the robustness of statistical procedures to the NPI framework, namely sensitivity curve and breakdown point; these classical concepts require some adoption for application in NPI. Most of our nonparametric inferences have a reasonably good robustness with regard to small changes in the data. - Coolen-Maturi, T., 2018. Three-group ROC predictive analysis for ordinal outcomes. Communications in Statistics – Theory and Methods, to appear. 46, 9476–9493.
Bib Entry Measuring the accuracy of diagnostic tests is crucial in many application areas including medicine, machine learning, and credit scoring. The receiver operating characteristic (ROC) surface is a useful tool to assess the ability of a diagnostic test to discriminate among three-ordered classes or groups. In this article, nonparametric predictive inference (NPI) for three-group ROC analysis for ordinal outcomes is presented. NPI is a frequentist statistical method that is explicitly aimed at using few modeling assumptions, enabled through the use of lower and upper probabilities to quantify uncertainty. This article also includes results on the volumes under the ROC surfaces and consideration of the choice of decision thresholds for the diagnosis. Two examples are provided to illustrate our method. - Coolen, F.P.A., Coolen-Maturi, T., Alqifari, H.N., 2018. Nonparametric predictive inference for future order statistics. Communications in Statistics - Theory and Methods, to appear. 47, 2527–2548.
Bib Entry This paper presents nonparametric predictive inference for future order statistics. Given data consisting of n real-valued observations, m future observations are considered and predictive probabilities are presented for the r-th ordered future observation. In addition, joint and conditional probabilities for events involving multiple future order statistics are presented. The paper further presents the use of such predictive probabilities for order statistics in statistical inference, in particular considering pairwise and multiple comparisons based on two or more independent groups of data. - Bakera, R.M., Coolen-Maturi, T., Coolen, F.P.A., 2017. Nonparametric Predictive Inference for Stock Returns. Journal of Applied Statistics 44, 1333–1349.
Bib Entry In finance, inferences about future asset returns are typically quantified with the use of parametric distributions and single-valued probabilities. It is attractive to use less restrictive inferential methods, including nonparametric methods which do not require distributional assumptions about variables, and imprecise probability methods which generalise the classical concept of probability to set-valued quantities. Main attractions include the flexibility of the inferences to adapt to the available data and that the level of imprecision in inferences can reflect the amount of data on which these are based. This paper introduces nonparametric predictive inference (NPI) for stock returns. NPI is a statistical approach based on few assumptions, with inferences strongly based on data and with uncertainty quantified via lower and upper probabilities. NPI is presented for inference about future stock returns, as a measure for risk and uncertainty, and for pairwise comparison of two stocks based on their future aggregate returns. The proposed NPI methods are illustrated using historical stock market data. - Coolen-Maturi, T., 2017. Predictive inference for best linear combination of biomarkers subject to limits of detection. Statistics in Medicine 36, 2844–2874.
Bib Entry Measuring the accuracy of diagnostic tests is crucial in many application areas including medicine, machine learning and credit scoring. The receiver operating characteristic (ROC) curve is a useful tool to assess the ability of a diagnostic test to discriminate between two classes or groups. In practice, multiple diagnostic tests or biomarkers are combined to improve diagnostic accuracy. Often, biomarker measurements are undetectable either below or above the so-called limits of detection (LoD). In this paper, nonparametric predictive inference (NPI) for best linear combination of two or more biomarkers subject to limits of detection is presented. NPI is a frequentist statistical method that is explicitly aimed at using few modelling assumptions, enabled through the use of lower and upper probabilities to quantify uncertainty. The NPI lower and upper bounds for the ROC curve subject to limits of detection are derived, where the objective function to maximize is the area under the ROC curve. In addition, the paper discusses the effect of restriction on the linear combination’s coefficients on the analysis. Examples are provided to illustrate the proposed method. - Yin, Y.-C., Coolen, F.P.A., Coolen-Maturi, T., 2017. An imprecise statistical method for accelerated life testing using the power-Weibull model. Reliability Engineering & System Safety 167, 158–167.
Bib Entry Accelerated life testing provides an interesting challenge for quantification of the uncertainties involved, in particular due to the required linking of the unitsâ failure times, or failure time distributions, at different stress levels. This paper provides an initial exploration of the use of statistical methods based on imprecise probabilities for accelerated life testing. We apply nonparametric predictive inference at the normal stress level, in combination with an estimated parametric power-Weibull model linking observations at different stress levels. To provide robustness with regard to this assumed link between different stress levels, we introduce imprecision by considering an interval around the parameter estimate, leading to observations at stress levels other than the normal level to be transformed to intervals at the normal level. The width of such intervals is increasing with the difference between the stress level at which a unit is tested and the normal level. The resulting inference method is predictive, so it explicitly considers the random failure time of a future unit tested at the normal level. We perform simulation studies to investigate the performance of our imprecise predictive method and to get insight into a suitable amount of imprecision for the linking between levels. We also explain how simulation studies can assist in choosing imprecision in order to provide robustness against specific biases or model misspecifications. - Coolen, F.P.A., Coolen-Maturi, T., 2016. The structure function for system reliability as predictive (imprecise) probability. Reliability Engineering & System Safety 154, 180–187.
Bib Entry In system reliability, the structure function models functioning of a system for given states of its components. As such, it is typically a straightforward binary function which plays an essential role in reliability assessment, yet it has received remarkably little attention in its own right. We explore the structure function in more depth, considering in particular whether its generalization as a, possibly imprecise, probability can provide useful further tools for reliability assessment in case of uncertainty. In particular, we consider the structure function as a predictive (imprecise) probability, which enables uncertainty and indeterminacy about the next task the system has to perform to be taken into account. The recently introduced concept of ‘survival signature’ provides a useful summary of the structure function to simplify reliability assessment for systems with many components of multiple types. We also consider how the (imprecise) probabilistic structure function can be linked to the survival signature. We briefly discuss some related research topics towards implementation for large practical systems and networks, and we outline further possible generalizations. - Coolen-Maturi, T., Coolen, F.P.A., Muhammad, N., 2016. Predictive inference for bivariate data: Combining nonparametric predictive inference for marginals with an estimated copula. Journal of Statistical Theory and Practice 10, 515–538.
Bib Entry This article presents a new method for prediction of an event involving a future bivariate observation. The method combines nonparametric predictive inference (NPI) applied to the marginals with a parametric copula to model and estimate the dependence structure between two random quantities, as such, the method is semiparametric. In NPI, uncertainty is quantified through imprecise probabilities. The resulting imprecision in the marginals provides robustness with regard to the assumed parametric copula. Due to the specific nature of NPI, the estimation of the copula parameter is also quite straightforward. The performance of this method is investigated via simulations, with particular attention to robustness with regard to the assumed copula in case of small data sets. The method is further illustrated via two examples, using small data sets from the literature. This article presents several novel aspects of statistical inference. First, the link between NPI and copulas is powerful and attractive with regard to computation. Second, statistical methods using imprecise probability have gained substantial attention in recent years, where typically imprecision is used on aspects for which less information is available. This article presents a different approach, namely, imprecision mainly being introduced on the marginals, for which there is typically quite sufficient information, in order to provide robustness for the harder part of the inference, namely the copula assumptions and estimation. Third, the setup of the simulations to evaluate the performance of the proposed method is novel; key to these are frequentist comparisons of the success proportion of predictions with the corresponding data-based lower and upper predictive inferences. All these novel ideas can be applied far more generally to other inferences and models, while also many alternatives can be considered. Hence, this article presents the starting point of an extensive research program towards powerful predictive inference methods for multivariate data. - Tee, K.F., Khan, L.R., Coolen-Maturi, T., 2015. Application of receiver operating characteristic curve for pipeline reliability analysis. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 229, 181–192.
Bib Entry Structural reliability analysis of buried pipeline systems is one of the fundamental issues for water and wastewater asset managers. Measuring the accuracy of a reliability analysis or a failure prediction technique is an effective approach to enhancing its applicability and provides guidance on selection of reliability or failure prediction methods. The determination of threshold value for a particular pipe failure criterion provides useful information on reliability analysis. However, this threshold value is not always known. In this article, receiver operating characteristic curve has been applied where empirical and nonparametric predictive inference techniques are used to evaluate the accuracy of pipeline reliability analysis and to predict the failure threshold value. Multi-failure conditions, namely, corrosion-induced deflection, buckling, wall thrust and bending stress have been assessed in this article. It is hoped that choosing the optimal operating point on the receiver operating characteristic curve, which involves both maintenance and financial issues, can be ideally implemented by combining the receiver operating characteristic analysis with a formal risk–cost management of underground pipelines. - Coolen, F.P.A., Coolen-Maturi, T., 2015. Predictive inference for system reliability after common-cause component failures . Reliability Engineering & System Safety 135, 27–33.
Bib Entry This paper presents nonparametric predictive inference for system reliability following common-cause failures of components. It is assumed that a single failure event may lead to simultaneous failure of multiple components. Data consist of frequencies of such events involving particular numbers of components. These data are used to predict the number of components that will fail at the next failure event. The effect of failure of one or more components on the system reliability is taken into account through the system׳s survival signature. The predictive performance of the approach, in which uncertainty is quantified using lower and upper probabilities, is analysed with the use of {ROC} curves. While this approach is presented for a basic scenario of a system consisting of only a single type of components and without consideration of failure behaviour over time, it provides many opportunities for more general modelling and inference, these are briefly discussed together with the related research challenges. - Coolen-Maturi, T., Coolen, F.P.A., 2015. Nonparametric Predictive Inference With Combined Data Under Different Right-Censoring Schemes. Journal of Statistical Theory and Practice 9, 288–304.
Bib Entry This article presents nonparametric predictive inference (NPI) for meta-analysis in which multiple independent samples of lifetime data are combined, where different censoring schemes may apply to the different samples. NPI is a frequentist statistical approach based on few assumptions and with uncertainty quantified via lower and upper probabilities. NPI has the flexibility to deal with a mixture of different types of censoring, mainly because the inferences do not depend on counterfactuals, which affect several inferences for more established frequentist approaches. We show that the combined sample, consisting of differently censored independent samples, can be represented as one sample of progressively censored data. This allows explicit formulas for the NPI lower and upper survival functions to be presented that are generally applicable. The approach is illustrated through an example using a small data set from the literature, for which several scenarios are presented. - Coolen-Maturi, T., 2014. Nonparametric predictive pairwise comparison with competing risks . Reliability Engineering & System Safety 132, 146–153.
Bib Entry In reliability, failure data often correspond to competing risks, where several failure modes can cause a unit to fail. This paper presents nonparametric predictive inference (NPI) for pairwise comparison with competing risks data, assuming that the failure modes are independent. These failure modes could be the same or different among the two groups, and these can be both observed and unobserved failure modes. {NPI} is a statistical approach based on few assumptions, with inferences strongly based on data and with uncertainty quantified via lower and upper probabilities. The focus is on the lower and upper probabilities for the event that the lifetime of a future unit from one group, say Y, is greater than the lifetime of a future unit from the second group, say X. The paper also shows how the two groups can be compared based on particular failure mode(s), and the comparison of the two groups when some of the competing risks are combined is discussed. - Coolen-Maturi, T., Coolen, F.P.A., 2014. Nonparametric predictive inference for combined competing risks data . Reliability Engineering & System Safety 126, 87–97.
Bib Entry The nonparametric predictive inference (NPI) approach for competing risks data has recently been presented, in particular addressing the question due to which of the competing risks the next unit will fail, and also considering the effects of unobserved, re-defined, unknown or removed competing risks. In this paper, we introduce how the {NPI} approach can be used to deal with situations where units are not all at risk from all competing risks. This may typically occur if one combines information from multiple samples, which can, e.g. be related to further aspects of units that define the samples or groups to which the units belong or to different applications where the circumstances under which the units operate can vary. We study the effect of combining the additional information from these multiple samples, so effectively borrowing information on specific competing risks from other units, on the inferences. Such combination of information can be relevant to competing risks scenarios in a variety of application areas, including engineering and medical studies. - Aboalkhair, A.M., Coolen, F.P.A., MacPhee, I.M., 2014. Nonparametric predictive inference for reliability of a k-out-of-m:G system with multiple component types . Reliability Engineering & System Safety 131, 298–304.
Bib Entry Nonparametric predictive inference for system reliability has recently been presented, with specific focus on k-out-of-m:G systems. The reliability of systems is quantified by lower and upper probabilities of system functioning, given binary test results on components, taking uncertainty about component functioning and indeterminacy due to limited test information explicitly into account. Thus far, systems considered were series configurations of subsystems, with each subsystem i a ki-out-of- m i : G system which consisted of only one type of components. Key results are briefly summarized in this paper, and as an important generalization new results are presented for a single k-out-of-m:G system consisting of components of multiple types. The important aspects of redundancy and diversity for such systems are discussed. - Coolen, F.P.A., Himd, S.B., 2014. Nonparametric Predictive Inference for Reproducibility of Basic Nonparametric Tests. Journal of Statistical Theory and Practice 8, 591–618.
Bib Entry Reproducibility of tests is an important characteristic of the practical relevance of test outcomes. Recently, there has been substantial interest in the reproducibility probability (RP), where not only its estimation but also its actual definition and interpretation are not uniquely determined in the classical frequentist statistics framework. Nonparametric predictive inference (NPI) is a frequentist statistics approach that makes few assumptions, enabled by the use of lower and upper probabilities to quantify uncertainty, and that explicitly focuses on future observations. The explicitly predictive nature of NPI provides a natural formulation for inferences on RP. In this article, we introduce the NPI approach to RP for some basic nonparametric tests. - Coolen, F.P.A., Coolen-Maturi, T., Al-nefaiee, A.H., 2014. Nonparametric predictive inference for system reliability using the survival signature. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 228, 437–448.
Bib Entry The survival signature has recently been presented as an attractive concept to aid quantification of system reliability. It has similar characteristics as the system signature, which is well established, but contrary to the latter it is easily applicable to systems with multiple types of components. We present an introductory overview of the survival signature together with new results to aid computation. We develop nonparametric predictive inference for system reliability using the survival signature. The focus is on the failure time of a system, given failure times of tested components of the same types as used in the system. - Al-nefaiee, A.H., Coolen, F.P.A., 2013. Nonparametric predictive inference for system failure time based on bounds for the signature. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 227, 513–522.
Bib Entry System signatures provide a powerful framework for reliability assessment for systems consisting of exchangeable components. The use of signatures in nonparametric predictive inference has been presented and leads to lower and upper survival functions for the system failure time, given failure times of tested components. However, deriving the system signature is computationally complex. This article presents how limited information about the signature can be used to derive bounds on such lower and upper survival functions and related inferences. If such bounds are sufficiently decisive they also indicate that more detailed computation of the system signature is not required. - Aboalkhair, A.M., Coolen, F.P.A., MacPhee, I.M., 2013. Nonparametric predictive reliability of series of voting systems . European Journal of Operational Research 226, 77–84.
Bib Entry Nonparametric Predictive Inference (NPI) for system reliability reflects the dependence of reliabilities of similar components due to limited knowledge from testing. {NPI} has recently been presented for reliability of a single voting system consisting of multiple types of components. The components are all assumed to play the same role within the system, but with regard to their reliability components of different types are assumed to be independent. The information from tests is available per type of component. This paper presents {NPI} for systems with subsystems in a series structure, where all subsystems are voting systems and components of the same type can be in different subsystems. As {NPI} uses only few modelling assumptions, system reliability is quantified by lower and upper probabilities, reflecting the limited information in the test data. The results are illustrated by examples, which also illustrate important aspects of redundancy and diversity for system reliability. - Coolen, F.P.A., Coolen-Schrijner, P., Coolen-Maturi, T., Elkhafifi, F.F., 2013. Nonparametric predictive inference for ordinal data. Communications in Statistics - Theory and Methods 42, 3478–3496.
Bib Entry Nonparametric predictive inference (NPI) is a powerful frequentist statistical framework based only on an exchangeability assumption for future and past observations, made possible by the use of lower and upper probabilities. In this article, NPI is presented for ordinal data, which are categorical data with an ordering of the categories. The method uses a latent variable representation of the observations and categories on the real line. Lower and upper probabilities for events involving the next observation are presented, and briefly compared to NPI for non ordered categorical data. As application, the comparison of multiple groups of ordinal data is presented. - Houlding, B., Coolen, F.P.A., 2012. Nonparametric predictive utility inference . European Journal of Operational Research 221, 222–230.
Bib Entry We consider the natural combination of two strands of recent statistical research, i.e., that of decision making with uncertain utility and that of Nonparametric Predictive Inference (NPI). In doing so we present the idea of Nonparametric Predictive Utility Inference (NPUI), which is suggested as a possible strategy for the problem of utility induction in cases of extremely vague prior information. An example of the use of {NPUI} within a motivating sequential decision problem is also considered for two extreme selection criteria, i.e., a rule that is based on an attitude of extreme pessimism and a rule that is based on an attitude of extreme optimism. - Elkhafifi, F.F., Coolen, F.P.A., 2012. Nonparametric predictive inference for accuracy of ordinal diagnostic tests. Journal of Statistical Theory and Practice 6, 681–697.
Bib Entry We introduce nonparametric predictive inference (NPI) for accuracy of diagnostic tests with ordinal outcomes, with the inferences based on data for a disease group and a non-disease group. We introduce empirical and NPI lower and upper receiver operating characteristic (ROC) curves and the corresponding areas under the curves, and we prove that these are nested, with the latter equal to the NPI lower and upper probabilities for correctly ordered future observations from the non-disease and disease groups. We discuss the use of the Youden index related to the NPI lower and upper ROC curves in order to determine the optimal cutoff point for the test. - Coolen, F.P.A., Al-nefaiee, A.H., 2012. Nonparametric predictive inference for failure times of systems with exchangeable components. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 226, 262–273.
Bib Entry The theory of system signatures (Samaniego, 2007) provides a powerful framework for reliability assessment for systems consisting of exchangeable components. For a system with m components, the signature is a vector containing the probabilities for the events that the system fails at the moment of the jth ordered component failure time, for all j = 1,…, m. As such, the signature represents the structure of the system. This paper presents how signatures can be used within nonparametric predictive inference, a statistical framework which uses few modelling assumptions enabled by the use of lower and upper probabilities to quantify uncertainty. The main result is the use of signatures to derive lower and upper survival functions for the failure time of systems with exchangeable components, given failure times of tested components that are exchangeable with those in the system. In addition, it is shown how the failure times of two such systems can be compared. This paper is the first in which signatures are combined with theory of lower and upper probabilities; related research challenges are briefly discussed. - Coolen-Maturi, T., Coolen-Schrijner, P., Coolen, F.P.A., 2012. Nonparametric Predictive Inference for Binary Diagnostic Tests. Journal of Statistical Theory and Practice 6, 665–680.
Bib Entry Measuring the accuracy of diagnostic tests is crucial in many application areas, including medicine, health care, and data mining. Good methods for determining diagnostic accuracy provide useful guidance on selection of patient treatment, and the ability to compare different diagnostic tests has a direct impact on quality of care. In this paper nonparametric predictive inference (NPI) for accuracy of diagnostic tests with binary test results is presented and discussed, together with methods for comparison of two such tests. NPI does not aim at inference for an entire population but instead explicitly considers future observations, which is particularly suitable for inference to support decisions on medical diagnosis for one future patient, or for a predetermined number of future patients, so the NPI approach provides an attractive alternative to standard methods. - Coolen-Maturi, T., Coolen-Schrijner, P., Coolen, F.P.A., 2012. Nonparametric predictive multiple comparisons of lifetime data. Communications in Statistics - Theory and Methods 41, 4164–4181.
Bib Entry We consider lifetime experiments to compare units from different groups, where the units’ lifetimes may be right censored. Nonparametric predictive inference for comparison of multiple groups is presented, in particular lower and upper probabilities for the event that a specific group will provide the largest next lifetime. We include the practically relevant consideration that the overall lifetime experiment may be terminated at an early stage, leading to simultaneous right-censoring of all units still in the experiment. - Elsaeiti, M.A., Coolen, F.P.A., 2012. A Nonparametric predictive approach to sequential acceptance problems. Journal of Statistical Theory and Practice 6, 383–401.
Bib Entry Sequential acceptance problems are considered with the aim to select candidates from a group, with the candidates observed sequentially, either per individual or in subgroups, and with the ordering of an individual compared to previous candidates and those in the same subgroup available. For given total group size, this problem can in principle be solved by dynamic programming, but the computational effort required makes this not feasible for problems once the number of candidates to be selected and the total group size are not small. We present a new heuristic approach to such problems, based on the principles of nonparametric predictive inference, and we study its performance via simulations, which are also used to compare the method with some alternatives. The approach is easy to implement and computationally straightforward. - Coolen-Maturi, T., Coolen, F.P.A., 2011. Unobserved, re-defined, unknown or removed failure modes in competing risks. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 225, 461–474.
Bib Entry Recently the nonparametric predictive approach to inference for competing risks was introduced by Maturi et al. (2010, J. Risk Reliab. 224, 11–26). In this paper further results for such inferences are presented, with focus on four important and closely related aspects. First, the effect of defined failure modes which thus far have not yet caused failures is studied. Second, the effect of re-defining (groups of) failure modes is considered, followed by a discussion of possible unknown, so undefined, failure modes. Finally, the effect of removal of failure modes is illustrated. - Abellán, J., Baker, R.M., Coolen, F.P.A., 2011. Maximising entropy on the nonparametric predictive inference model for multinomial data . European Journal of Operational Research 212, 112–122.
Bib Entry The combination of mathematical models and uncertainty measures can be applied in the area of data mining for diverse objectives with as final aim to support decision making. The maximum entropy function is an excellent measure of uncertainty when the information is represented by a mathematical model based on imprecise probabilities. In this paper, we present algorithms to obtain the maximum entropy value when the information available is represented by a new model based on imprecise probabilities: the nonparametric predictive inference model for multinomial data (NPI-M), which represents a type of entropy-linear program. To reduce the complexity of the model, we prove that the NPI-M lower and upper probabilities for any general event can be expressed as a combination of the lower and upper probabilities for the singleton events, and that this model can not be associated with a closed polyhedral set of probabilities. An algorithm to obtain the maximum entropy probability distribution on the set associated with NPI-M is presented. We also consider a model which uses the closed and convex set of probability distributions generated by the NPI-M singleton probabilities, a closed polyhedral set. We call this model A-NPI-M. A-NPI-M can be seen as an approximation of NPI-M, this approximation being simpler to use because it is not necessary to consider the set of constraints associated with the exact model. - Roelofs, V.J., Coolen, F.P.A., Hart, A.D.M., 2011. Nonparametric Predictive Inference for Exposure Assessment. Risk Analysis 31, 218–227.
Bib Entry Exposure assessment for food and drink consumption requires the combining of information about people’s consumption of products with concentration data sets to provide predictions for chemical intake by humans. In this article, we present a method called nonparametric predictive inference (NPI) for exposure assessment. NPI is a distribution-free method relying only on Hill’s assumption . Effectively, is a postdata exchangeability assumption, which is a natural starting point for nonparametric statistics. For further discussion we refer to works by Hill and Coolen. We illustrate how NPI can be implemented to produce predictions for an individual’s exposure based on consumption, body weight, and concentration data. NPI has the advantage that we do not have to assume a distribution to implement it. There may, however, be information available to suggest a distribution for a random quantity. Therefore, we present an NPI-Bayes hybrid method where this information can be taken into account by using Bayesian methods while using NPI for the other random quantities in the model. - Baker, R.M., Coolen, F.P.A., 2010. Nonparametric predictive category selection for multinomial data. Journal of Statistical Theory and Practice 4, 509–526.
Bib Entry A new method is presented for selecting a single category or the smallest subst of categories, based on observations from a multinomial data set, where the selection criterion is a minimally required lower probability that (at least) a specific number of future observations will belong to that category or subset of categories. The inferences about the future observations are made using an extension of Coolen and Augustin’s nonparametric predictive inference (NPI) model to a situation with multiple future observations. - Maturi, T.A., Coolen-Schrijner, P., Coolen, F.P.A., 2010. Nonparametric predictive inference for competing risks. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 224, 11–26.
Bib Entry In reliability, failure data often correspond to competing risks, where several failure modes can cause a unit to fail. This paper presents nonparametric predictive inference (NPI) for competing risks data, assuming that the different failure modes are independent. NPI is a statistical approach based on few assumptions, with inferences strongly based on data and with uncertainty quantified via lower and upper probabilities. The focus is on the lower and upper probabilities for the event that a future unit will fail due to a specific failure mode. The paper illustrates the effect of grouping different failure modes together, and some special cases and features are discussed. It is also shown that NPI can easily deal with competing risks data resulting from experiments with progressive censoring. Furthermore, new formulae are presented for the NPI lower and upper survival functions. - Coolen, F.P.A., Augustin, T., 2009. A nonparametric predictive alternative to the Imprecise Dirichlet Model: The case of a known number of categories . International Journal of Approximate Reasoning 50, 217–230.
Bib Entry Nonparametric predictive inference (NPI) is a general methodology to learn from data in the absence of prior knowledge and without adding unjustified assumptions. This paper develops {NPI} for multinomial data when the total number of possible categories for the data is known. We present the upper and lower probabilities for events involving the next observation and several of their properties. We also comment on differences between this {NPI} approach and corresponding inferences based on Walley’s Imprecise Dirichlet Model. - Coolen, F.P.A., Elsaeiti, M.A., 2009. Nonparametric Predictive Methods for Acceptance Sampling. Journal of Statistical Theory and Practice 3, 907–921.
Bib Entry We present methods for basic acceptance sampling for attributes, based on the nonparametric predictive inferential approach for Bernoulli data presented by Coolen (1998), which is extended for this application. We consider both acceptance sampling based on destructive tests and on non-destructive tests. Attention is mostly restricted to single-stage sampling, but extension to two-stage sampling is also considered and discussed. - Coolen-Schrijner, P., Maturi, T.A., Coolen, F.P.A., 2009. Nonparametric Predictive Precedence Testing for Two Groups. Journal of Statistical Theory and Practice 3, 273–287.
Bib Entry This paper presents a statistical method for comparison of two groups based on nonparametric predictive inference (NPI). NPI is a statistical approach based on few modelling assumptions, with inferences strongly based on data and uncertainty quantified via lower and upper probabilities. Lifetimes of units from groups X and Y are compared, based on observed lifetimes from an experiment that may have ended before all units failed. We present upper and lower probabilities for the event that the lifetime of a future unit from X is less than the lifetime of a future unit from Y, and we compare this approach with traditional precedence testing. - MacPhee, I.M., Coolen, F.P.A., Aboalkhair, A.M., 2009. Nonparametric predictive system reliability with redundancy allocation following component testing. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 223, 181–188.
- Maturi, T.A., Coolen-Schrijner, P., Coolen, F.P.A., 2009. Nonparametric predictive pairwise comparison for real-valued data with terminated tails . International Journal of Approximate Reasoning 51, 141–150.
Bib Entry This paper presents a statistical method for comparison of two groups of real-valued data, based on nonparametric predictive inference (NPI), with the tails of the data possibly terminated, leading to small values being left-censored and large values being right-censored. Such tails termination can occur due to several reasons, including limits of detection, consideration of outliers, and specific designs of experiments. {NPI} is a statistical approach based on few assumptions, with inferences strongly based on data and with uncertainty quantified via lower and upper probabilities. We present {NPI} lower and upper probabilities for the event that the value of a future observation from one group is less than the value of a future observation from the other group, and we discuss several special cases that relate to well-known statistical problems. - Coolen-Schrijner, P., Coolen, F.P.A., MacPhee, I.M., 2008. Nonparametric predictive inference for system reliability with redundancy allocation. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 222, 463–476.
Bib Entry This paper presents lower and upper probabilities for the reliability of k-out-of-m systems, which include series and parallel systems, and of series systems with independent ki-out-of-mi subsystems, for which optimal redundancy allocation is also presented in case of zero-failure testing. First, attention is restricted to k-out-of-m systems with exchangeable components. The lower and upper probabilities for successful functioning of the system are based on the nonparametric predictive inferential (NPI) approach for Bernoulli data. In this approach, it is assumed that test data are available on the components, and that the future components to be used in the system are exchangeable with these. Thereafter, systems are considered that consist of a series of independent subsystems, with subsystem i a ki-out-of-mi system consisting of exchangeable components. For such systems, an algorithm for optimal redundancy allocation after zero-failure testing is presented. A particularly attractive feature of NPI in reliability, with lower and upper probabilities, is that data containing zero failures can be dealt with in an attractive manner. - Arts, G.R.J., Coolen, F.P.A., 2008. Two Nonparametric Predictive Control Charts. Journal of Statistical Theory and Practice 2, 499–512.
Bib Entry Control charts based on distributional assumptions may be unsatisfactory if the assumptions are violated, as the average run length (ARL) might deviate from its anticipated value. Control limits for such charts are based on estimates of parameters, which might cause biased ARL. We introduced extrema charts based on nonparametric predictive inference (NPI) (Arts, Coolen and van der Laan, 2004), which do not suffer from these problems. Such charts use a reference set, and resulting inferences are exactly calibrated (Lawless and Fredette, 2005). It was shown that extrema charts perform well when compared to X charts, and they can quickly detect medium to large shifts. In this paper, we explore two alternative charts based on NPI, focussing on their data requirement and performance in detecting smaller shifts. We consider charts that use other order statistics than the extrema, and charts that use subsequent samples rather than single samples. The main conclusions are that the use of order statistics other than extrema might require large reference sets, hence seems to be less attractive, but the NPI subsequent sampling chart is an attractive alternative to our extrema chart. - Coolen, F.P.A., 2007. Non-parametric prediction of unobserved failure modes. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 221, 207–216.
Bib Entry This paper presents the application of a recently developed non-parametric predictive inferential approach for multinomial data to the problem of prediction of occurrence of new failure modes. These inferences are in terms of lower and upper probabilities for the next observation. The lower probability of occurrence of a new failure mode is zero in all cases, as the data never suggest strongly that there have to be further failure modes. The main interest is in the upper probability that the next observed failure is caused by a new failure mode. - Coolen-Schrijner, P., Coolen, F.P.A., 2007. Nonparametric predictive comparison of success-failure data in reliability. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 221, 319–327.
Bib Entry Suppose a technical unit is required to perform a particular task in the future, and that several different types of the unit are available. The unit could be a system or a component, the different types might be different designs or units made by different producers. Units of each type have been tested, and the result of each test of a unit is success or failure to perform the task required. It is assumed that the available test data consist of the number of tests of units per type together with the numbers of successes in these tests. In this paper comparison of different types of units on the basis of such success-failure data is considered, where it is explicitly assumed that interest is in the future performance of m ≥1 units per type. Both pairwise and multiple comparisons of such different types of units are presented, with the overall idea of selecting the particular type of unit that is likely to lead to fewest failures in the m future tasks. The use of upper and lower probabilities makes it possible to work in a non-parametric statistical framework which requires only few modelling assumptions. The influence of the value of m on the inferences is studied. In addition, special attention is given to cases where for some types of units few, or even zero, failures were observed during testing. - Coolen, F.P.A., 2004. On the Use of Imprecise Probabilities in Reliability. Quality and Reliability Engineering International 20, 193–202.
Bib Entry Theory of imprecise probability generalizes classical probability theory, by assigning to each event an interval instead of a single number. In this paper, we briefly discuss this generalization and some recently suggested applications of imprecise probabilities in reliability. We also comment on challenges for research and applications. Copyright © 2004 John Wiley & Sons, Ltd. - Coolen-Schrijner, P., Coolen, F.P.A., 2004. Non-parametric Predictive Inference for Age Replacement with a Renewal Argument. Quality and Reliability Engineering International 20, 203–215.
Bib Entry We consider an age replacement problem with cost function based on the renewal reward theorem. However, instead of assuming a known probability distribution for the lifetimes, we apply Hill’s assumption A(n) for predicting probabilities for the lifetime of a future item. Lower and upper bounds for the survival function of a future item are used, resulting in upper and lower cost functions. Minimizing these upper and lower cost functions to obtain the optimal age replacement times is simplified due to the special form of these functions.To discuss some features of our approach, we first study the consequences of using n equally spaced percentiles from a known distribution instead of n observed data. Secondly, we report on a simulation study where the lifetimes are simulated from known distributions, so that the optimal replacement times corresponding to our approach can be compared with the theoretical optimal replacement times. - Arts, G.R.J., Coolen, F.P.A., van der Laan, P., 2004. Nonparametric Predictive Inference in Statistical Process Control. Quality Technology & Quantitative Management 1, 201–216.
Bib Entry Statistical process control (SPC) is used to decide when to stop a process as confidence in the quality of the next item(s) is low. Information to specify a parametric model is not always available, and as SPC is of a predictive nature, we present a control chart developed using nonparametric predictive inference. The proposed ‘extrema chart’, based on the extrema of a sample of observations from the process, is a generalisation of an existing nonparametric method, which controls a process using single observations. We examine the average run length (ARL) of both the one-sided and two-sided extrema chart, and a simulation study is presented to compare the extrema chart with the well known X¯ chart and CUSUM chart. The disadvantage of these charts is that when the process mean and variation of the in-control process have to be estimated, the ARL is biased. This is not an issue for the extrema chart, as no knowledge about the underlying distribution is required. - Coolen, F.P.A., Coolen-Schrijner, P., 2003. A nonparametric predictive method for queues . European Journal of Operational Research 145, 425–442.
Bib Entry This paper presents a novel statistical approach to queues. Instead of studying characteristics of an assumed parametric stochastic model, the method uses information in the form of observed service times per queue and, while adding a minimum of additional assumptions, develops predictive probability results for the waiting time for customers in a queue. We show how these results can be used in a multi-queue problem to assign arriving customers to queues, aiming at minimisation of waiting times. - Coolen, F.P.A., Yan, K.J., 2003. Nonparametric predictive inference for grouped lifetime data . Reliability Engineering & System Safety 80, 243–252.
Bib Entry This paper presents the application of a recently introduced nonparametric predictive inferential method to grouped lifetime data. Such data consist of numbers of events and numbers of right-censorings per interval, for a finite partition of the time-axis, without further information on the exact event and censoring times. Optimal bounds are derived for probabilities of events in terms of a future observation, where the bounds correspond to specific configurations of the event and censoring times per interval. The method is illustrated, and briefly compared to some alternative nonparametric approaches, via an example. - Coolen, F.P.A., Coolen-Schrijner, P., Yan, K.J., 2002. Nonparametric predictive inference in reliability . Reliability Engineering & System Safety 78, 185–193.
Bib Entry We introduce a recently developed statistical approach, called nonparametric predictive inference (NPI), to reliability. Bounds for the survival function for a future observation are presented. We illustrate how NPI can deal with right-censored data, and discuss aspects of competing risks. We present possible applications of NPI for Bernoulli data, and we briefly outline applications of NPI for replacement decisions. The emphasis is on introduction and illustration of NPI in reliability contexts, detailed mathematical justifications are presented elsewhere. - Coolen, F.P.A., 1998. Low structure imprecise predictive inference for Bayes’ problem . Statistics & Probability Letters 36, 349–357.
Bib Entry This paper presents direct conditional imprecise probabilities for the number of successes in a finite number of future trials, given information about a finite number of past trials. A simple underlying process determining failures or successes is assumed, related to Bayes’ postulate, and Hill’s A(n) assumption is used. The results are related to the type of predictive inference known as low structure or black-box inference.

- Aboalkhair, A.M., Coolen, F.P.A., MacPhee, I.M., 2012. Nonparametric predictive inference for reliability of a series of subsystems with multiple component types, in: Reliability, Risk Management (Proceedings Esrel 2011 Risk Management (Proceedings Esrel 2011 Troyes). C. Bérenguer, A.G., Soares, C.G. (Eds.), Advances in Safety. Taylor & Francis Group, London, . (Abstract page 168 of Book of Abstracts, pp. 1069–1077.
Bib Entry - Coolen, F.P.A., 2011. Nonparametric predictive inference in reliability and risk: recent developments, in: Proceedings SSARSth Summer Safety & Reliability Seminars. Krzysztof Kolowrocki and Joanna Soszynska-Budny, Volume 1 (1/2), pp 39-50, Journal of Polish Safety and Reliability Association. Eds, pp. 2011–5.
Bib Entry - Coolen, F.P.A., 2011. Nonparametric predictive inference, in: Lovric, M. (Ed.), International Encyclopedia of Statistical Science. Springer, pp. 968–970.
- Al-Nefaiee, A., Coolen, F.P.A., 2011. Nonparametric predictive inference for failure times of systems consisting of exchangeable components, in: Prescott, D., Remenyte-Prescott, R. (Eds.), Proceedings Advances in Risk and Reliability Technology Symposium 2011. pp. 226–236.
Bib Entry - Coolen-Maturi, T., Coolen, F.P.A., 2011. On unobserved failure modes in competing risks, in: Prescott, D., Remenyte-Prescott, R. (Eds.), Proceedings Advances in Risk and Reliability Technology Symposium 2011. pp. 340–352.
Bib Entry - Baker, R.M., Coolen-Schrijner, P., Coolen, F.P.A., Augustin, T., 2011. Nonparametric predictive inference with subcategory data, in: ISIPTA’11. pp. 41–50.
- Aboalkhair, A.M., Coolen, F.P.A., MacPhee, I.M., 2011. Nonparametric predictive inference for system reliability with subsystems consisting of multiple component types, in: Prescott, D., Remenyte-Prescott, R. (Eds.), Proceedings Advances in Risk and Reliability Technology Symposium 2011. pp. 322–339.
Bib Entry - Crossman, R.J., Abellan, J., Augustin, T., Coolen, F.P.A., 2011. Building imprecise classification trees with entropy ranges, in: ISIPTA’11. pp. 129–138.
- Aboalkhair, A.M., Coolen, F.P.A., MacPhee, I.M., 2010. Nonparametric predictive inference for reliability of voting systems with multiple component types, in: Chateauneuf, M.N., Guérin, F. (Eds.), A. May 2010, Clermont-Ferrand, France, Proceedings Third International Conference on Accelerated Life Testing, Reliability-based Analysis and Design, pp. 175–182.
Bib Entry - Coolen, F.P.A., Maturi, T.A., 2010. Nonparametric predictive inference for order statistics of future observations, in: Borgelt, C., Gonzalez-Rodriguez, G., Trutschnig, W., Lubiano, M.A., Gil, M.A., Grzegorzewski, P., Hryniewicz, O. (Eds.), Combining Soft Computing and Statistical Methods in Data Analysis. Springer, Berlin (Advances in Intelligent and Soft Computing 77), . (Proceedings of the 5th International Conference on Soft Methods in Probability and Statistics, Oviedo, Spain, 28 Sept - 1 Oct 2010, pp. 97–104.
Bib Entry - Coolen, F.P.A., Coolen-Schrijner, P., Maturi, T.A., 2010. On nonparametric predictive inference for ordinal data, in: Kruse, R., Hoffmann, F. (Eds.), Eyke Hüllermeier. Computational Intelligence for Knowledge-Based Systems Design, Proceedings of the 13th International Conference on Information Processing and Management of Uncertainty, IPMU 2010, Dortmund, Germany, June 28 - July 2, 2010. Springer, Berlin, pp. 188–197.
Bib Entry - Maturi, T.A., Coolen-Schrijner, P., Coolen, F.P.A., 2009. On nonparametric predictive inference for competing risks, in: Proceedings 18th ARTS. pp. 196–211.
Bib Entry - Baker, R.M., Coolen, F.P.A., 2009. Category selection for multinomial data, in: Proceedings ISIPTA’09. pp. 21–30.
- Coolen, F.P.A., Aboalkhair, A.M., MacPhee, I.M., 2009. Nonparametric predictive system reliability with all subsystems consisting of one type of component, in: Kallen, M.J., Kuniewski, S.P. (Eds.), Risk and Decision Analysis in Maintenance Optimization and Flood Management. IOS Press (Delft University Press), Amsterdam, pp. 85–98.
Bib Entry - Elsaeiti, M., Coolen, F.P.A., 2009. Nonparametric predictive inference for acceptance sampling with destructive tests, in: Proceedings 18th ARTS. pp. 212–220.
Bib Entry - Maturi, T.A., Coolen-Schrijner, P., Coolen, F.P.A., 2009. Nonparametric predictive multiple comparisons with censored data and competing risks, in: Proceedings ISIPTA’09. pp. 307–316.
- Aboalkhair, A.M., Coolen, F.P.A., MacPhee, I.M., 2009. Nonparametric predictive system reliability with redundancy allocation following component testing, in: Proceedings 18th ARTS. pp. 396–407.
Bib Entry - Coolen, F.P.A., Coolen-Schrijner, P., 2008. Nonparametric predictive inference for k-out-of-m systems, in: Advances in Mathematical Modeling for Reliability. T. Bedford, J. Quigley, L. Walls, B. Alkali, A. Daneshkhah, G. Hardman. IOS Press, Amsterdam, pp. 185–192.
Bib Entry - Maturi, T.A., Coolen-Schrijner, P., Coolen, F.P.A., 2008. Lifetime comparisons with early termination of experiments, in: Proceedings ISNI’08. pp. 127–130.
Bib Entry - Coolen, F.P.A., 2008. On nonparametric predictive inference for Bernoulli quantities with set-valued data, in: Soft Methods for Handling Variability and Imprecision. D. Dubois, M. Asuncion Lubiana, H. Prade, M. Angeles Gil, P. Grzegorzewski, O. Hryniewicz. Springer, pp. 85–91.
Bib Entry - Coolen, F.P.A., Maturi, T.A., 2008. On nonparametric predictive inference with incomplete data, in: Proceedings ISNI’08. pp. 80–84.
Bib Entry - Maturi, T.A., Coolen-Schrijner, P., Coolen, F.P.A., 2008. Early termination of experiments in nonparametric predictive comparisons, in: Proceedings IWAP’08 (CD-Rom.
- Coolen, F.P.A., Augustin, T., 2007. Multinomial nonparametric predictive inference with sub-categories, in: Proceedings ISIPTA’07. pp. 77–86.
- Coolen, F.P.A., Houlding, B., Parkinson, S.G., 2007. Jury size and composition - a predictive approach, in: Proceedings ISIPTA’07. pp. 87–96.
- Coolen-Schrijner, P., Maturi, T., Coolen, F.P.A., 2007. Nonparametric predictive precedence testing for two groups, in: Proceedings MMR’07 (CD-ROM).
- Coolen, F.P.A., Coolen-Schrijner, P., 2007. Nonparametric predictive inference for Bernoulli quantities: two examples, in: Proceedings 56th ISI Conference (CD-ROM.
- Coolen, F.P.A., Coolen-Schrijner, P., 2007. Nonparametric predictive inference for voting systems, in: Proceedings MMR’07 (CD-ROM.
- Coolen-Schrijner, P., Coolen, F.P.A., 2007. Nonparametric predictive comparison of success-failure data in reliability, in: Proceedings 17th ARTS. pp. 255–273.
Bib Entry - Coolen, F.P.A., 2006. Coolen-Schrijner, in: P. Springer, Comparing proportions data with few successes. In: Soft Methods for Integrated Uncertainty Modelling, eds: J. Lawry, E. Miranda, A. Bugarin, S. Li, M. Angeles Gil, P. Grzegorzewski, O. Hryniewicz, pp. 241–248.
Bib Entry - Coolen, F.P.A., 2006. The occurrence of not yet observed failure modes, in: Safety and Reliability for Managing Risk. C. Guedes Soares, E. Zio. Taylor & Francis, pp. 881–888.
Bib Entry - Coolen, F.P.A., Augustin, T., 2005. Learning from multinomial data: a nonparametric predictive alternative to the Imprecise Dirichlet Model, in: Proceedings ISIPTA’05. pp. 125–134.
- Coolen-Schrijner, P., Coolen, F.P.A., Shaw, S.C., 2004. Adaptive (opportunity-based) age replacement strategies, in: Proceedings 5th MIMAR. pp. 75–80.
Bib Entry - Coolen, F.P.A., Yan, K.J., 2003. Nonparametric predictive comparison of two groups of lifetime data, in: Proceedings ISIPTA’03. pp. 148–161.
Bib Entry - Coolen-Schrijner, P., Coolen, F.P.A., 2003. Nonparametric predictive inference for age replacement with a renewal argument, in: Proceedings 15th ARTS. pp. 39–54.
Bib Entry - Coolen, F.P.A., Yan, K.J., 2002. The use of right-censored data in nonparametric predictive inference, in: Proceedings MMR’02. pp. 155–158.
Bib Entry

- Coolen-Maturi, T., Elkhafifi, F.F., Coolen, F.P.A., 2013. Nonparametric predictive inference for three-group ROC analysis.
Bib Entry Measuring the accuracy of diagnostic tests is crucial in many application areas including medicine, health care and credit scoring. Good methods for determining diagnostic accuracy in medicine provide useful guidance on selection of patient treatment according to the severity of their health status. The receiver operating characteristic (ROC) surface is a useful tool to assess the ability of a diagnostic test to discriminate among three ordered classes or groups. In this paper, nonparametric predictive inference (NPI) for three-group ROC analysis for continuous data is presented. NPI is a frequentist statistical method that is explicitly aimed at using few modelling assumptions in addition to data, enabled through the use of lower and upper probabilities to quantify uncertainty. Furthermore, it focuses exclusively on future observations, which provides an alternative perspective to the usual approaches which typically aim at estimation of characteristics of assumed underlying populations. This focus on prediction may be particularly rele- vant if one considers decisions about a diagnostic test that must be applied to a future patient. This paper presents the NPI approach to three-group ROC analysis, including results on the volumes under the ROC surfaces and consideration of the choice of decision thresholds for the diagnosis.

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