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• Christensen, Dennis & Novik, Geir Petter (2026). Stop Using Limiting Stimuli as a Measure of Sensitivities of Energetic Materials. Propellants, explosives, pyrotechnics. ISSN 0721-3115. doi: 10.1002/prep.70210. Full text in Research Archive Show summary

  Accurately estimating the sensitivity of explosive materials is a potentially life‐saving task that requires standardised protocols across nations. One of the most widely applied procedures worldwide is the so‐called ‘1‐In‐6’ test from the United Nations (UN) Manual of Tests in Criteria, which estimates a ‘limiting stimulus’ for a material. In this paper we demonstrate that, despite their popularity, limiting stimuli are not a well‐defined notion of sensitivity and do not provide reliable information about a material's susceptibility to ignition. In particular, they do not permit construction of confidence intervals to quantify estimation uncertainty. We show that continued reliance on limiting stimuli through the 1‐In‐6 test has caused needless confusion in energetic materials research, both in theoretical studies and practical safety applications. To remedy this problem, we consider three well‐founded alternative approaches to sensitivity testing to replace limiting stimulus estimation. We compare their performance in an extensive simulation study and apply the best‐performing approach to real data, estimating the friction sensitivity of pentaerythritol tetranitrate (PETN).

• Simensen, Oskar Høgberg; Christensen, Dennis & Hjort, Nils Lid (2026). Random irregular histograms. Computational Statistics & Data Analysis. ISSN 0167-9473. 220. doi: 10.1016/j.csda.2026.108367. Full text in Research Archive
• Christensen, Dennis; Stoltenberg, Emil Aas & Hjort, Nils Lid (2026). Asymptotic properties of adaptive designs. Stochastic Processes and their Applications. ISSN 0304-4149. 196. doi: 10.1016/j.spa.2026.104922. Full text in Research Archive Show summary

  There exist multiple regression applications in engineering, industry and medicine where the outcomes follow an adaptive experimental design in which the next measurement depends on the previous observations, so that the observations are not conditionally independent given the covariates. In this paper we present new asymptotic theory for such processes of dependent regression data. This includes stand-alone results for adaptive covariate processes, which, in particular, enable us to develop new likelihood theory for inference on the regression model parameters. Existing results asserting asymptotic normality of the maximum likelihood estimator require regularity conditions involving the second or third derivatives of the log-likelihood. Here we instead extend the theory of differentiability in quadratic mean (DQM) to the setting of adaptive designs, which requires strictly fewer regularity assumptions than the classical theory. In doing so, we discover a new DQM assumption, which we call summable differentiability in quadratic mean (S-DQM). As applications, we first verify asymptotic normality for two classical adaptive designs, namely the Bruceton 'up-and-down' design and the Robbins–Monro design. Next, we consider a more complicated problem, namely a Markovian version of the Langlie design.

• Novik, Geir Petter; Bolsønes, Marianne; Ueland, Ragnhild & Christensen, Dennis (2025). Characteristics of triacetone triperoxide (TATP) formed in 2-propanol. Fire safety journal. ISSN 0379-7112. 153. doi: 10.1016/j.firesaf.2025.104370. Full text in Research Archive Show summary

  There have been several reports of explosions occurring as a result of handling aged 2-propanol. Several of these explosions have resulted in injuries. In some of these incidents, the formation of an explosive substance most commonly named TATP (triacetone triperoxide; 3,3,6,6,9,9-hexamethyl-1,2,4,5,7,8-hexoxonane) has been identified. Upon standing in the presence of air, it is known that 2-propanol under certain circumstances is inclined to form TATP; however, few studies have been carried out to determine its properties. Following a recent explosion in Norway involving 2-propanol, a noticeably large amount of a substance with a crystalline structure was discovered at the site of the accident. This substance was identified as TATP. The substance was analysed using nuclear magnetic resonance (NMR), Fourier transform infrared (FT-IR) and liquid chromatography-mass spectrometry (LC–MS), and its sensitivity to friction was studied using the standardised BAM friction test. The study shows that TATP can be formed naturally in 2-propanol in the presence of air, and that its characteristics are similar to that of the traditionally laboratory synthesised substance. This implies that handling ageing 2-propanol containers could be associated with severe danger, as TATP formed in 2-propanol is found to be highly sensitive to external stimuli.

• Christensen, Dennis & Moen, Per August Jarval (2025). perms: Likelihood-free estimation of marginal likelihoods for binary response data in Python and R. Journal of Computational Science. ISSN 1877-7503. 84. doi: 10.1016/j.jocs.2024.102467. Full text in Research Archive Show summary

  In Bayesian statistics, the marginal likelihood (ML) is the key ingredient needed for model comparison and model averaging. Unfortunately, estimating MLs accurately is notoriously difficult, especially for models where posterior simulation is not possible. Recently, Christensen (2023) introduced the concept of permutation counting, which can accurately estimate MLs of models for exchangeable binary responses. Such data arise in a multitude of statistical problems, including binary classification, bioassay and sensitivity testing. Permutation counting is entirely likelihood-free and works for any model from which a random sample can be generated, including nonparametric models. Here we present perms, a package implementing permutation counting. As a result of extensive optimisation efforts, perms is computationally efficient and able to handle large data problems. It is available as both an R package and a Python library. A broad gallery of examples illustrating its usage is provided, which includes both standard parametric binary classification and novel applications of nonparametric models, such as changepoint analysis. We also cover the details of the implementation of perms and illustrate its computational speed via a simple simulation study.

• Novik, Geir Petter & Christensen, Dennis (2024). Increased impact sensitivity in ageing high explosives; analysis of Amatol extracted from explosive remnants of war. Royal Society Open Science. 11(3). doi: 10.1098/rsos.231344. Full text in Research Archive Show summary

  Millions of tonnes of explosive remnants of war remain in nature, and the volume is continuously growing. The explosive legacy of wars represents an increasing threat to the environment and to societal safety and security. As munitions continue to deteriorate, harmful constituents will eventually leak into the environment, poisoning ecological receptors and contaminating the surrounding soil and groundwater. Moreover, deterioration due to exposure to various environmental factors may ultimately cause the munitions to become increasingly sensitive to external stimuli and susceptible to accidental detonation. To thoroughly assess how to address these ageing munitions, we must first establish certain threshold values for the safe and secure handling and final disposal of the explosive ordnance. One key factor is to establish how the impact sensitivity of the explosives evolves over time. In the present work, we investigated the high explosive substance Amatol extracted from ageing explosive remnants of war. The results obtained in the analysis indicate that the high explosives in the examined specimens generally were much more sensitive to impact than previously assumed. Furthermore, the analysis revealed that the standardised methodology of impact sensitivity testing was insufficient for estimating the sensitivities in question, and a more careful statistical analysis was required.

• Christensen, Dennis (2024). Inference for Bayesian nonparametric models with binary response data via permutation counting. Bayesian Analysis. ISSN 1936-0975. doi: 10.1214/22-BA1353. Full text in Research Archive Show summary

  Since the beginning of Bayesian nonparametrics in the early 1970s, there has been a wide interest in constructing models for binary response data. Such data arise naturally in problems dealing with bioassay, current status data and sensitivity testing, and are equivalent to left and right censored observations if the inputs are one-dimensional. For models based on the Dirichlet process, inference is possible via Markov chain Monte Carlo (MCMC) simulations. However, there exist multiple processes based on different principles, for which such MCMC-based methods fail. Examples include logistic Gaussian processes and quantile pyramids. These require MCMC for posterior inference given exact observations, and thus become intractable when the data comprise both left and right censored observations. Here we present a new importance sampling algorithm for nonparametric models given exchangeable binary response data. It can be applied to any model from which samples can be generated, or even only approximately generated. The main idea behind the algorithm is to exploit the symmetries introduced by exchangeability. Calculating the importance weights turns out to be equivalent to evaluating the permanent of a certain class of (0,1)-matrix, which we prove can be done in polynomial time by deriving an explicit algorithm.

• Christensen, Dennis; Novik, Geir Petter & Unneberg, Erik (2024). Estimating sensitivity with the Bruceton method: Setting the record straight. Propellants, explosives, pyrotechnics. ISSN 0721-3115. 49(7). doi: 10.1002/prep.202400022. Full text in Research Archive Show summary

  Accurate estimates of sensitivities of energetic materials are crucial for ensuring safe production, transport, usage and destruction of explosives. When estimating sensitivities, researchers most commonly follow the NATO standard guidelines (STANAGs), in which the Bruceton method is imposed. Introduced in 1948, this method contains (i) an experimental design for choosing which stimulus levels to measure at and (ii) a recipe for computing sensitivity estimates. Although the former experimental design is supported by both theory and simulations, few modern researchers are aware that the latter recipe was only intended as a pen-and-paper approximation of the maximum likelihood estimates, which are easy to compute today. The persistent use of this outdated approximation has led to many unfortunate misconceptions amongst users of the Bruceton method, including the rejection of many perfectly valid data sets and neglect of uncertainty assessments via confidence intervals. This is both dangerous and unnecessarily wasteful. This paper sets the record straight and explains how researchers should estimate sensitivity via maximum likelihood estimation and how to construct confidence intervals. It also shows explicitly how wasteful said approximation is via both simulations and with real data.

• Jensen, Tomas Lunde; Moxnes, John Fredrik; Unneberg, Erik & Christensen, Dennis (2020). Models for predicting impact sensitivity of energetic materials based on the trigger linkage hypothesis and Arrhenius kinetics. Journal of Molecular Modeling. ISSN 1610-2940. doi: 10.1007/s00894-019-4269-z. Full text in Research Archive Show summary

  In order to predict the impact sensitivity of high explosives, we designed and evaluated several models based on the trigger linkage hypothesis and the Arrhenius equation. To this effect, we calculated the heat of detonation, temperature of detonation and bond dissociation energy for 70 energetic molecules. The bond dissociation energy divided by the temperature of detonation proved to be a good predictor of the impact sensitivity of nitroaromatics, with a coefficient of determination (R2) of 0.81. A separate Bayesian analysis gave similar results, taking model complexity into account. For nitramines, there was no relationship between the impact sensitivity and the bond dissociation energy. None of the models studied gave good predictions for the impact sensitivity of liquid nitrate esters. For solid nitrate esters, the bond dissociation energy divided by the temperature of detonation showed promising results (R2 = 0.85), but since this regression was based on only a few data points, it was discredited when model complexity was accounted for by our Bayesian analysis. Since the temperature of detonation correlated with the impact sensitivity for nitroaromatics, nitramines and nitrate esters, we consider it to be one of the leading predictive factors of impact sensitivity for energetic materials.

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• Christensen, Dennis (2025). Adaptive designs in sensitivity testing of explosives. Full text in Research Archive
• Christensen, Dennis (2025). Asymptotic theory of up-and-down designs. Full text in Research Archive
• Christensen, Dennis; Heijden, Antoine van der & Novik, Geir Petter (2025). Replacing the Bruceton approximation by maximum likelihood estimation. Full text in Research Archive Show summary

  Many researchers consider the statistical estimation of sensitivities of energetic materials to be a confusing topic. This is partly because the NATO Standardisation Agreements (STANAGs) on sensitivity testing advocate for an outdated pen-and-paper approximation rather than the direct computation of maximum likelihood estimates. The present paper aims to assist researchers in transitioning from this approximation to maximum likelihood estimation. In doing so, we clarify some prevalent misconceptions related to sensitivity testing.

• Christensen, Dennis; Heijden, Antoine van der & Novik, Geir Petter (2025). An improved statistical analysis of 72 sensitivity datasets. Full text in Research Archive Show summary

  This paper addresses the statistical estimation of sensitivities of energetic materials to friction and impact. The NATO Standardisation Agreements (STANAGs) for sensitivity testing still rely on a pen-and-paper approximation of maximum likelihood estimation from 1948. Although useful for its time, this approximation is today unnecessary as maximum likelihood estimation is easily implemented in virtually any general-purpose programming language and spreadsheet software. In this paper we show how the use of the approximation manifests itself in practice by analysing 72 datasets for impact and friction sensitivity of multiple energetic materials. For a substantial minority of the datasets, the use of the approximation is not valid, as a certain simplification of the likelihood function cannot be applied. Despite the STANAGs’ requirement to discard such datasets, we find that the maximum likelihood estimates exist and provide perfectly valid results for all of them. For the remaining datasets, where the use of the approximation is valid, we find that it performs poorly when constructing confidence intervals. For extreme quantiles, we obtain a relative error as high as 20%. In conclusion, our paper serves as further evidence that the 1948 approximation should be abolished and replaced with direct maximum likelihood estimation.

• Løland, Anders; Christensen, Dennis & Novik, Geir Petter (2024). Hva er dumpet ammunisjon? [Internet]. Sannsynligvis VIKTIG (podkast). Full text in Research Archive
• Christensen, Dennis (2024). Marginal likelihood estimation of nonparametric models via permutation counting. Full text in Research Archive
• Christensen, Dennis; Stoltenberg, Emil Aas & Hjort, Nils Lid (2024). Theory for adaptive designs in regression. arXiv.org. ISSN 2331-8422. doi: 10.48550/arXiv.2312.13387. Full text in Research Archive
• Løland, Anders; Christensen, Dennis & Novik, Geir Petter (2024). Hvordan kan vi måle følsomhet i eksplosiver? [Internet]. Sannsynligvis VIKTIG (podkast). Full text in Research Archive
• Christensen, Dennis & Nygård, Sebastian Teigen (2023). Ethnic differences in children’s sports participation; causal evidence of discrimination in Norway. Full text in Research Archive
• Christensen, Dennis (2023). A symmetry-based simulation method for Bayesian nonparametric models with binary response data. Full text in Research Archive
• Christensen, Dennis (2023). A symmetry-based simulation method for Bayesian nonparametric models with binary response data. Full text in Research Archive
• Christensen, Dennis (2023). A symmetry-based simulation method for Bayesian nonparametric models with binary response data. Full text in Research Archive
• Christensen, Dennis & Moen, Per August Jarval (2023). Fast implementation of a general importance sampling algorithm for Bayesian nonparametric models with binary responses. Full text in Research Archive
• Christensen, Dennis; Unneberg, Erik; Høyheim, Eirik; Jensen, Tomas Lunde & Hjort, Nils Lid (2023). Improved measurements of impact sensitivities of energetic materials. Full text in Research Archive Show summary

  Accurate estimation of impact sensitivity is crucial for safe production, handling, storage and transport of energetic materials. Indeed, molecular characteristics will affect sensitivity, and for solid materials, factors like particle size, lattice defects and morphology also play a role and make reproducibility difficult. As various synthesis and recrystallisation methods may lead to differences in crystal properties, it is important to determine the impact sensitivity whenever an energetic material is prepared. Of particular interest is the median ℎ50, namely the impact energy level at which there is a probability of 50% of an explosion occurring. This value has been shown to correlate with quantum chemical properties of the energetic material in question, providing insight into the underlying causes which govern sensitivity. However, in practical applications, it may be more important to estimate extreme values like the 99% quantile ℎ99. In addition to providing point estimates, we would like to derive confidence intervals to address their uncertainty. In this work, we cover the most common methods for constructing such confidence intervals (the delta method, Fieller’s theorem and the likelihood-ratio test) and compare their performance on sensitivity data via simulations. Our experiments indicate that Fieller’s theorem is the superior method, and we therefore use it to construct confidence intervals for ℎ50 and ℎ99 for cyclotetramethylene- tetranitramine (HMX), using new data. Based on our results, we formulate recommendations for researchers measuring sensitivities of synthesised molecules.

• Christensen, Dennis (2022). Nonparametric Bayesian sensitivity testing with optimal design. Full text in Research Archive
• Christensen, Dennis; Haug, Ola; Kunimitsu, Taro; Kolstø, Johannes Voll & Lenkoski, Alex (2024). Climate Hazards and Collateral Value: A Survey of Recent Literature. Norsk Regnesentral. Full text in Research Archive

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