Bonferroni Inequality Research Articles

On approximations of data-driven chance constrained programs over Wasserstein balls

Distributionally robust chance constrained programs minimize a deterministic cost function subject to the satisfaction of one or more safety conditions with high probability, given that the probability distribution of the uncertain problem parameters affecting the safety condition(s) is only known to belong to some ambiguity set. We study three popular approximation schemes for distributionally robust chance constrained programs over Wasserstein balls, where the ambiguity set contains all probability distributions within a certain Wasserstein distance to a reference distribution. The first approximation replaces the chance constraint with a bound on the conditional value-at-risk, the second approximation decouples different safety conditions via Bonferroni's inequality, and the third approximation restricts the expected violation of the safety condition(s) so that the chance constraint is satisfied. We show that the conditional value-at-risk approximation can be characterized as a tight convex approximation, which complements earlier findings on classical (non-robust) chance constraints, and we offer a novel interpretation in terms of transportation savings. We also show that the three approximations can perform arbitrarily poorly in data-driven settings, and that they are generally incomparable with each other.

Linearization and Variance Estimation of the Bonferroni Inequality Index

Abstract The study of income inequality is important for predicting the wealth of a country. There is an increasing number of publications where the authors call for the use of several indices simultaneously to better account for the wealth distribution. Due to the fact that income data are usually collected through sample surveys, the sampling properties of income inequality measures should not be overlooked. The most widely used inequality measure is the Gini index, and its inferential aspects have been deeply investigated. An alternative inequality index could be the Bonferroni inequality index, although less attention on its inference has been paid in the literature. The aim of this paper is to address the inference of the Bonferroni index in a finite population framework. The Bonferroni index is linearized by differentiation with respect to the sample indicators which allows for conducting a valid inference. Furthermore, the linearized variables are used to evaluate the effects of the different observations on the Bonferroni and Gini indices. The result demonstrates once for all that the former is more sensitive to the lowest incomes in the distribution than the latter.

One-two dependence and probability inequalities between one- and two-sided union-intersection tests.

In a paper published in 1939 in The Annals of Mathematical Statistics, Wald and Wolfowitz discussed the possible validity of a probability inequality between one- and two-sided coverage probabilities for the empirical distribution function. Twenty-eight years later, Vandewiele and Noé proved this inequality for Kolmogorov-Smirnov type goodness of fit tests. We refer to this type of inequality as one-two inequality. In this paper, we generalize their result for one- and two-sided union-intersection tests based on positively associated random variables and processes. Thereby, we give a brief review of different notions of positive association and corresponding results. Moreover, we introduce the notion of one-two dependence and discuss relationships with other dependence concepts. While positive association implies one-two dependence, the reverse implication fails. Last but not least, the Bonferroni inequality and the one-two inequality yield lower and upper bounds for two-sided acceptance/rejection probabilities which differ only slightly for significance levels not too large. We discuss several examples where the one-two inequality applies. Finally, we briefly discuss the possible impact of the validity of a one-two inequality on directional error control in multipletesting.

New perspectives on the Gini and Bonferroni indices of inequality

This paper rigorously demonstrates that for any unequal income distribution, the well-known Gini index of inequality is bounded above by the recently revived Bonferroni inequality index. The bound is exactly attained if and only if out of n incomes in the society, n − 1 poor incomes are identical. The boundedness theorem is shown to possess a duality-type inequality implication. These two inequality metrics, two popular members of a general class of inequality indices generated by Aaberge’s (J Econ Inequal 5:305–322, 2007) ‘scaled conditional mean curve’, may lead to different directional rankings of alternative income distributions because of some important differences between them. We then explicitly examine their sensitivity to Weymark’s (Math Soc Sci 1:409–430, 1981) ‘comonotonic additivity’ postulate.

On Bounds for Probabilities of Combinations of Events, the Jordan Formula, and the Bonferroni Inequalities

This paper presents a method for deriving optimal lower and upper bounds for probabilities and conditional probabilities (given a σ-field) for various combinations of events. The optimality is understood as the possibility that inequalities become equalities for some sets of events. New generalizations of the Jordan formula and the Bonferroni inequalities are obtained. The corresponding conditional versions of these results are also considered.

On estimates for probabilities of combinations of events, the Jordan formula and the Bonferroni inequalities

On estimates for probabilities of combinations of events, the Jordan formula and the Bonferroni inequalities

Cross-Screening in Observational Studies That Test Many Hypotheses

ABSTRACTWe discuss observational studies that test many causal hypotheses, either hypotheses about many outcomes or many treatments. To be credible an observational study that tests many causal hypotheses must demonstrate that its conclusions are neither artifacts of multiple testing nor of small biases from nonrandom treatment assignment. In a sense that needs to be defined carefully, hidden within a sensitivity analysis for nonrandom assignment is an enormous correction for multiple testing: In the absence of bias, it is extremely improbable that multiple testing alone would create an association insensitive to moderate biases. We propose a new strategy called “cross-screening,” different from but motivated by recent work of Bogomolov and Heller on replicability. Cross-screening splits the data in half at random, uses the first half to plan a study carried out on the second half, then uses the second half to plan a study carried out on the first half, and reports the more favorable conclusions of the two studies correcting using the Bonferroni inequality for having done two studies. If the two studies happen to concur, then they achieve Bogomolov–Heller replicability; however, importantly, replicability is not required for strong control of the family-wise error rate, and either study alone suffices for firm conclusions. In randomized studies with just a few null hypotheses, cross-screening is not an attractive method when compared with conventional methods of multiplicity control. However, cross-screening has substantially higher power when hundreds or thousands of hypotheses are subjected to sensitivity analyses in an observational study of moderate size. We illustrate the technique by comparing 46 biomarkers in individuals who consume large quantities of fish versus little or no fish. The R package CrossScreening on CRAN implements the cross-screening method. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

Decomposing the Bonferroni Inequality Index by Subgroups: Shapley Value and Balance of Inequality

Additive decomposability is an interesting feature of inequality indices which, however, is not always fulfilled; solutions to overcome such an issue have been given by Deutsch and Silber (2007) and by Di Maio and Landoni (2017). In this paper, we apply these methods, based on the “Shapley value” and the “balance of inequality” respectively, to the Bonferroni inequality index. We also discuss a comparison with the Gini concentration index and highlight interesting properties of the Bonferroni index.

An adaptive Mantel-Haenszel test for sensitivity analysis in observational studies.

In a sensitivity analysis in an observational study with a binary outcome, is it better to use all of the data or to focus on subgroups that are expected to experience the largest treatment effects? The answer depends on features of the data that may be difficult to anticipate, a trade-off between unknown effect-sizes and known sample sizes. We propose a sensitivity analysis for an adaptive test similar to the Mantel-Haenszel test. The adaptive test performs two highly correlated analyses, one focused analysis using a subgroup, one combined analysis using all of the data, correcting for multiple testing using the joint distribution of the two test statistics. Because the two component tests are highly correlated, this correction for multiple testing is small compared with, for instance, the Bonferroni inequality. The test has the maximum design sensitivity of two component tests. A simulation evaluates the power of a sensitivity analysis using the adaptive test. Two examples are presented. An R package, sensitivity2x2xk, implements the procedure.

High Confidence Intervals Applied to Aircraft Altitude Prediction

This paper describes the application of high-confidence-interval prediction methods to the aircraft trajectory prediction problem, more specifically to the altitude prediction during climb. We are interested in methods for finding two-sided intervals that contain, with a specified confidence, at least a desired proportion of the conditional distribution of the response variable. This paper introduces two-sided Bonferroni-quantile confidence intervals, which is a new method for obtaining high-confidence two-sided intervals in quantile regression. This paper also uses the Bonferroni inequality to propose a new method for obtaining tolerance intervals in least squares regression. The latter has the advantages of being reliable, fast, and easy to calculate. We compare physical point-mass models to the introduced models on an air traffic management data set composed of traffic at major French airports. Experimental results show that the proposed interval prediction models perform significantly better than the conventional point-mass model currently used in most trajectory predictors. When comparing with a recent state-of-the-art point-mass model with adaptive mass estimation, the proposed methods give altitude intervals that are slightly wider but more reliable.

Undulatory Theory with Paraconsistent Logic (Part I): Quantum Logical Model with Two Wave Functions

Paraconsistent logic (PL) is a non-classical logic that accepts contradiction in its foundations. It can be represented in the form of paraconsistent annotated logic with annotation of two values (PAL2v). When used to model quantum phenomena, PAL2v is called paraquantum logic (PQL). In this work, the concept of PQL is applied to create a logical model presenting the fundamental principles of quantum mechanics that support particle-wave theory. This study uses the well-known Young’s double-slit experiment, wherein quantum phenomena appear when a monochromatic light beam passes through the two slits. We focused on a reference point located between the slits, where we observed the effects of two types of wave interferences in a region defined as a two-wave region (2W region). Considering that the effect in this 2W region is very similar to that studied by Huygens, we adopt a paraquantum logical model in which a particle (or quantum) is represented by two wave functions. The two wave functions result in four State Vectors (Ket, Bra,,) in the PQL Lattice that express the symmetry and the entanglement of Quantum Mechanics. The constructed model adapts well to the quantum phenomena, is strongly consistent, and can be considered as an innovative form of analysis in the field of quantum mechanics. Based on this model, we present in two parts (Part I and Part II) the comparative analysis of values found in SchrÖdinger’s equation and probabilistic models of wave-particle theory using Bonferroni inequality.

Required traffic micro‐simulation runs for reliable multivariate performance estimates

SummaryPrevious methods to calculate the minimum number of traffic micro‐simulation runs do not consider multiple measures of performance simultaneously at an overall confidence level, which can lead to unreliable simulation outputs. This paper describes new methodologies for calculating the minimum number of traffic micro‐simulation runs for multivariate estimates at an overall confidence level. Simultaneous confidence intervals obtained from multiple comparisons in statistical theory such as the Bonferroni inequality and simultaneous confidence interval method are used to estimate multiple measures of performance with allowable errors at an overall confidence level. Measures of performance can be means and standard deviations. Results of numerical analysis based on an example corridor suggest that the proposed methods provide improved means of assessing statistical accuracy of multiple measures of performance. Results also indicate that the minimum number of runs is influenced by not only the sample size issue but also the complexity of the traffic system. Copyright © 2015 John Wiley & Sons, Ltd.

Implementing Intersection Bounds in Stata

We present the clrbound, clr2bound, clr3bound, and clrtest commands for estimation and inference on intersection bounds as developed by Chernozhukov, Lee, and Rosen (2013, Econometrica 81: 667–737). The intersection bounds framework encompasses situations where a population parameter of interest is partially identified by a collection of consistently estimable upper and lower bounds. The identified set for the parameter is the intersection of regions defined by this collection of bounds. More generally, the methodology can be applied to settings where an estimable function of a vector-valued parameter is bounded from above and below, as is the case when the identified set is characterized by conditional moment inequalities. The commands clrbound, clr2bound, and clr3bound provide bound estimates that can be used directly for estimation or to construct asymptotically valid confidence sets. clrtest performs an intersection bound test of the hypothesis that a collection of lower intersection bounds is no greater than zero. The command clrbound provides bound estimates for one-sided lower or upper intersection bounds on a parameter, while clr2bound and clr3bound provide two-sided bound estimates using both lower and upper intersection bounds. clr2bound uses Bonferroni's inequality to construct two-sided bounds that can be used to perform asymptotically valid inference on the identified set or the parameter of interest, whereas clr3bound provides a generally tighter confidence interval for the parameter by inverting the hypothesis test performed by clrtest. More broadly, inversion of this test can also be used to construct confidence sets based on conditional moment inequalities as described in Chernozhukov, Lee, and Rosen (2013). The commands include parametric, series, and local linear estimation procedures.

Successive Elimination Algorithm for Constrained One-bit Transform Based Motion Estimation Using the Bonferroni Inequality

Successive Elimination Algorithm for Constrained One-bit Transform Based Motion Estimation Using the Bonferroni Inequality

Pitfalls in Clinical Research. The Case of the New Antineoplastic Agents

Pitfalls are often present in clinical research. In this paper, we highlight some shortcomings affecting the comparative studies on the efficacy of the new drugs in Oncology. More precisely, we will focus on the frequent choice of an intermediate endpoint - instead of the final endpoint of effectiveness - and the also frequent practice to allow the patient to crossover to the alternative treatment, when the proof of a greater effectiveness has not yet been reached. The effects of the Bonferroni's inequality on the results of a clinical study, and the choice of the comparator, as well as the use of the placebo, are also discussed. Finally, the importance of an appropriate procedure to determine the appropriate sample size is pointed out. An example from published studies is presented after the discussion of each issue.

Comparison of Power for Multiple Comparison Procedures

The number of methods for evaluating, and possibly making statistical decisions about, null contrasts - or their small sub-set, multiple comparisons - has grown extensively since the early 1950s. That demonstrates how important the subject is, but most of the growth consists of modest variations of the early methods. This paper examines nine fairly basic procedures, six of which are methods designed to evaluate contrasts chosen post hoc, i.e., after an examination of the test data. Three of these use experimentwise or familywise type 1 error rates (Scheffé 1953, Tukey 1953, Newman-Keuls, 1939 and 1952), two use decision-based type 1 error rates (Duncan 1951 and Rodger 1975a) and one (Fisher's LSD 1935) uses a mixture of the two type 1 error rate definitions. The other three methods examined are for evaluating, and possibly deciding about, a limited number of null contrasts that have been chosen independently of the sample data - preferably before the data are collected. One of these (planned t-tests) uses decision-based type 1 error rates and the other two (one based on Bonferroni's Inequality 1936, and the other Dunnett's 1964 Many-One procedure) use a familywise type 1 error rate. The use of these different type 1 error rate definitionsA creates quite large discrepancies in the capacities of the methods to detect true non-zero effects in the contrasts being evaluated. This article describes those discrepancies in power and, especially, how they are exacerbated by increases in the size of an investigation (i.e., an increase in J, the number of samples being examined). It is also true that the capacity of a multiple contrast procedure to 'unpick' 'true' differences from the sample data is influenced by the type of contrast the procedure permits. For example, multiple range procedures (such as that of Newman-Keuls and that of Duncan) permit only comparisons (i.e., two-group differences) and that greatly limits their discriminating capacity (which is not, technically speaking, their power). Many methods (those of Scheffé, Tukey's HSD, Newman-Keuls, Fisher's LSD, Bonferroni and Dunnett) place their emphasis on one particular question, "Are there any differences at all among the groups?" Some other procedures concentrate on individual contrasts (i.e., those of Duncan, Rodger and Planned Contrasts); so are more concerned with how many false null contrasts the method can detect. This results in two basically different definitions of detection capacity. Finally, there is a categorical difference between what post hoc methods and those evaluating pre-planned contrasts can find. The success of the latter depends on how wisely (or honestly well informed) the user has been in planning the limited number of statistically revealing contrasts to test. That can greatly affect the method's discriminating success, but it is often not included in power evaluations. These matters are elaborated upon as they arise in the exposition below. DOI:10.2458/azu_jmmss_v4i1_rodger

Comparison of Power for Multiple Comparison Procedures

The number of methods for evaluating, and possibly making statistical decisions about, null contrasts - or their small sub-set, multiple comparisons - has grown extensively since the early 1950s. That demonstrates how important the subject is, but most of the growth consists of modest variations of the early methods. This paper examines nine fairly basic procedures, six of which are methods designed to evaluate contrasts chosen post hoc, i.e., after an examination of the test data. Three of these use experimentwise or familywise type 1 error rates (Scheffé 1953, Tukey 1953, Newman-Keuls, 1939 and 1952), two use decision-based type 1 error rates (Duncan 1951 and Rodger 1975a) and one (Fisher's LSD 1935) uses a mixture of the two type 1 error rate definitions. The other three methods examined are for evaluating, and possibly deciding about, a limited number of null contrasts that have been chosen independently of the sample data - preferably before the data are collected. One of these (planned t-tests) uses decision-based type 1 error rates and the other two (one based on Bonferroni's Inequality 1936, and the other Dunnett's 1964 Many-One procedure) use a familywise type 1 error rate. The use of these different type 1 error rate definitionsA creates quite large discrepancies in the capacities of the methods to detect true non-zero effects in the contrasts being evaluated. This article describes those discrepancies in power and, especially, how they are exacerbated by increases in the size of an investigation (i.e., an increase in J, the number of samples being examined). It is also true that the capacity of a multiple contrast procedure to 'unpick' 'true' differences from the sample data is influenced by the type of contrast the procedure permits. For example, multiple range procedures (such as that of Newman-Keuls and that of Duncan) permit only comparisons (i.e., two-group differences) and that greatly limits their discriminating capacity (which is not, technically speaking, their power). Many methods (those of Scheffé, Tukey's HSD, Newman-Keuls, Fisher's LSD, Bonferroni and Dunnett) place their emphasis on one particular question, "Are there any differences at all among the groups?" Some other procedures concentrate on individual contrasts (i.e., those of Duncan, Rodger and Planned Contrasts); so are more concerned with how many false null contrasts the method can detect. This results in two basically different definitions of detection capacity. Finally, there is a categorical difference between what post hoc methods and those evaluating pre-planned contrasts can find. The success of the latter depends on how wisely (or honestly well informed) the user has been in planning the limited number of statistically revealing contrasts to test. That can greatly affect the method's discriminating success, but it is often not included in power evaluations. These matters are elaborated upon as they arise in the exposition below. DOI:10.2458/azu_jmmss_v4i1_rodger

Comparison of Power for Multiple Comparison Procedures

The number of methods for evaluating, and possibly making statistical decisions about, null contrasts - or their small sub-set, multiple comparisons - has grown extensively since the early 1950s. That demonstrates how important the subject is, but most of the growth consists of modest variations of the early methods. This paper examines nine fairly basic procedures, six of which are methods designed to evaluate contrasts chosen post hoc, i.e., after an examination of the test data. Three of these use experimentwise or familywise type 1 error rates (Scheffé 1953, Tukey 1953, Newman-Keuls, 1939 and 1952), two use decision-based type 1 error rates (Duncan 1951 and Rodger 1975a) and one (Fisher's LSD 1935) uses a mixture of the two type 1 error rate definitions. The other three methods examined are for evaluating, and possibly deciding about, a limited number of null contrasts that have been chosen independently of the sample data - preferably before the data are collected. One of these (planned t-tests) uses decision-based type 1 error rates and the other two (one based on Bonferroni's Inequality 1936, and the other Dunnett's 1964 Many-One procedure) use a familywise type 1 error rate. The use of these different type 1 error rate definitionsA creates quite large discrepancies in the capacities of the methods to detect true non-zero effects in the contrasts being evaluated. This article describes those discrepancies in power and, especially, how they are exacerbated by increases in the size of an investigation (i.e., an increase in J, the number of samples being examined). It is also true that the capacity of a multiple contrast procedure to 'unpick' 'true' differences from the sample data is influenced by the type of contrast the procedure permits. For example, multiple range procedures (such as that of Newman-Keuls and that of Duncan) permit only comparisons (i.e., two-group differences) and that greatly limits their discriminating capacity (which is not, technically speaking, their power). Many methods (those of Scheffé, Tukey's HSD, Newman-Keuls, Fisher's LSD, Bonferroni and Dunnett) place their emphasis on one particular question, "Are there any differences at all among the groups?" Some other procedures concentrate on individual contrasts (i.e., those of Duncan, Rodger and Planned Contrasts); so are more concerned with how many false null contrasts the method can detect. This results in two basically different definitions of detection capacity. Finally, there is a categorical difference between what post hoc methods and those evaluating pre-planned contrasts can find. The success of the latter depends on how wisely (or honestly well informed) the user has been in planning the limited number of statistically revealing contrasts to test. That can greatly affect the method's discriminating success, but it is often not included in power evaluations. These matters are elaborated upon as they arise in the exposition below.

Multiple comparisons with a control for exponential location parameters under heteroscedasticity

In this paper, a new design-oriented two-stage two-sided simultaneous confidence intervals, for comparing several exponential populations with control population in terms of location parameters under heteroscedasticity, are proposed. If there is a prior information that the location parameter of k exponential populations are not less than the location parameter of control population, one-sided simultaneous confidence intervals provide more inferential sensitivity than two-sided simultaneous confidence intervals. But the two-sided simultaneous confidence intervals have advantages over the one-sided simultaneous confidence intervals as they provide both lower and upper bounds for the parameters of interest. The proposed design-oriented two-stage two-sided simultaneous confidence intervals provide the benefits of both the two-stage one-sided and two-sided simultaneous confidence intervals. When the additional sample at the second stage may not be available due to the experimental budget shortage or other factors in an experiment, one-stage two-sided confidence intervals are proposed, which combine the advantages of one-stage one-sided and two-sided simultaneous confidence intervals. The critical constants are obtained using the techniques given in Lam [9,10]. These critical constant are compared with the critical constants obtained by Bonferroni inequality techniques and found that critical constant obtained by Lam [9,10] are less conservative than critical constants computed from the Bonferroni inequality technique. Implementation of the proposed simultaneous confidence intervals is demonstrated by a numerical example.

Inference From Lumen Degradation Data Under Wiener Diffusion Process

The lumen degradation of light emitting diodes subject to increasing stress loading is investigated by using a cumulative damage model. The cumulative damage process is taken as a Wiener diffusion process with a drift which depends on two stress loadings. General statistical inferences on the parameters and percentiles of the light emitting diode lifetime distribution are presented based on the cumulative damage measurements, collected from a two-variable constant-stress loading accelerated degradation test. Approximate lower <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</i> -confidence bounds of the light emitting diode lighting lifetime percentiles are given using the Fisher information of the maximum-likelihood estimators, and Bonferroni's inequality. The application of the proposed method is illustrated by a lumen cumulative damage data set of high power light emitting diodes.