Benjamin Rohaut

MD Jan Claassen

Decision making in perceived devastating brain injury: a call to explore the impact of cognitive biases

Keywords: devastating brain injury, critical care, withdrawal of life sustaining treatments, decision making, neuroprognosis, cognitive biases

Advances in critical care medicine have led to an increase in the number of patients who survive the initial phase of critical care. For these survivors, the impact of brain injuries on the probability of regaining consciousness, further cognitive disability, and more generally speaking, their overall quality of life is tremendous. Currently, most patients with acute brain injury that die early following injury, expire in the context of limitation or withdrawal of life sustaining treatments (WLST) [START_REF] Frontera | Integrating Palliative Care Into the Care of Neurocritically Ill Patients: A Report From the Improving Palliative Care in the ICU Project Advisory Board and the Center to Advance Palliative Care[END_REF] . Decisions that shift the goals of care to aim for WLST are typically driven by discussions between caregivers and families about the likely patient's outcome. However, accurate prediction of recovery in a brain-injured patient is extremely challenging, especially during the acute phase. Timing and willingness to pursue WLST varies greatly, but if built on prognostic uncertainty, carries a high risk of promoting a selffulfilling prophecy bias 2 .

To guide physicians taking care of patients with severe acute brain injury, the Professional Standards Committee of Faculty of Intensive Care Medicine and the Intensive Care Society published in this issue of the British Journal of Anaesthesia their joint statement [START_REF] Harvey | Management of perceived devastating brain injury after hospital admission: a consensus statement from stakeholder professional organizations[END_REF] . The authors are specifically addressing the subgroup of acutely brain injured patients for which outcomes are thought to be unlikely to be compatible with a "good outcome", a scenario they gave the label of "Perceived Devastating Brain Injury" abbreviated as DBI. The authors provide a number of practical and helpful recommendations in the spirit of providing maximal medical support during the early phase following acute brain injury (within the first hours). As doubts are raised by the medical team to continue life support, the authors argue that full medical support should be continued for another 24 to 48 hours before initiating any WLST allowing for better detection of the rare patient that is able to "go on to make a good recovery despite very poor early prognosis signs".

We agree with the authors' interpretation that this prescribed "stand-by window" would ensure that judgments on prognosis that are shared with the family are grounded at least on a minimum amount of information, such as more than one clinical assessment and accumulation of recommended prognostic measures. Although providing details for the comprehensive prognostic approach were beyond the scope of this consensus statement, the authors should be commended for recommending a minimum framework allowing for more accurate prognostication such as the use of pathology specific prognostication scales and the integration of a "neurosciences team" in this decision making process.

For most common life-threatening conditions of patients with DBI, the application of the "stand-by window" will realistically result at a minimum in an admission of to an ICU. In addition to improved neurological prognostication, this extended observation window would provide more time for open discussions with families. This stronger relationship may decrease stress among relatives and also consequently increase acceptance for organ donation, when applicable. On the other hand, additional stress and traumatizing experiences may be created for families and staff by these protracted, heart-wrenching discussions. This concern needs to be acknowledged as healthcare workers as well as family members need to be provided with better access to appropriate resources to cope with the resulting stress.

The statement emphasizes fundamental ethical aspects such as respecting patient preference as a primary goal of care and the need to take into account the global social and health-system context. In the multi-cultural environments encountered world-wide particularly in the modern urban metropolis, health care professionals need to take into account the diverse cultural context of each patient when approaching these sensitive discussions. This is challenging and may require tailored training.

When compared to the recent US consensus statement [START_REF] Souter | Recommendations for the Critical Care Management of Devastating Brain Injury: Prognostication, Psychosocial, and Ethical Management[END_REF] , DBI definition is less restrictive than the one used in the American consensus statement (where DBI stands for "Neurological injury where there is an immediate threat to life from a neurologic cause") as it includes survival with poor or no functional recovery. The suggested observation time-widow (24 -48h) is also significantly smaller than in the American consensus statement (72h). It is further noteworthy that the American consensus statement is more restrictive in their DBI criteria and permissive in term of observation time, giving in that sense more time to assess potentially more severely ill patients. However, these time frames are more dictated by practical considerations of bed availability than any fundamental scientific basis. The ideal time-window is unclear and should be determined according to each individual case in the context of the available resources and the underlying pathophysiology.

What "good outcome" are we trying to predict?

An interesting issue raised in the statement is the kind of prediction we talk about. As underlined by the authors, predicting in-hospital mortality is not sufficient. Both clinicians and society are increasingly asking for more accurate prediction of long-term disability and ideally of quality of life. The definition of what constitutes a poor outcome is controversial 5 6 , but usually lumps vegetative and minimally conscious state, and in a more variable extent, severe disability together with death. This may include levels of recovery where patients are unable to experience social relations, require continued medical assistance, etc.

Predicting long-term recovery of consciousness and cognitive recovery at an early point in time, which may occur several months or even years after brain injury, is enormously difficult and inaccurate at this time [START_REF] Estraneo | Late recovery after traumatic, anoxic, or hemorrhagic long-lasting vegetative state[END_REF] . Moreover, over such a long time-scale, patient and relatives' conception of what is a "good outcome" often change dramatically. This value change may occur in either direction. Consequently, the most relevant "perceived" outcome may not be the "here-and-now" judgment, but the "there-and-later" estimation by patients and relatives. Despite much recent progress and several on-going studies, for now and probably many years to come, caregivers would have no choice but to accept an inherent amount of uncertainty while making decisions in these scenarios.

The statement provides a potential framework for future research suggesting to not study mortality rate as an outcome whenever possible and to rather focus on disability or quality of life. This is not novel but needs to be stated for these extremely sick patients. The confounding effect of a self-fulfilling prophecy is minimized by this approach, even if, as mentioned in the statement, one could argue that outcome can even be worsened by inappropriate care based on incorrect prognostication among survivors.

Potential other solutions will involve creative integration of clinical and scientific communities, providing generalized data sharing enabling long term pooled analysis, and the use of more integrated study designs to real-world clinical practice and the design of "pragmatic trials" 8 9 . It is encouraging to see that an international group of experts has joined forces with support from the J.M. McDonald Foundation under the leadership of Nicholas D.

Schiff from Weill Cornell Medical Center to collect a large sample of brain injured patients with acute and chronic disorders of consciousness with the goal of developing more accurate prognostication tools and gain insights into the recovery process.

Psychological issues in neuro-prognostication

We would like to initiate a discussion about the fundamentals of the decision-making process as it relates to decisions for WLST in acute brain injury, which is notably not discussed in this statement (and only briefly mentioned in the American consensus statement).

The chosen acronym "DBI" (standing for "perceived Devastating Brain Injury") assigns this condition a label, which corresponds less to a patient's condition and more to the supposed future neurological recovery of the patient. This future state is based on the prediction of the medical team which is illustrated by the adjective "perceived" as well as the provided definition: "Any neurological condition that is assessed at the time of hospital admission as an immediate threat to life or incompatible with good functional recovery and where early limitation or withdrawal of therapy is being considered". Although this particular definition can be misleading (especially when the acronym DBI looses its "perceived" adjective in the text) it is noteworthy that this consensus statement mainly concerns a "real life" clinical situation (integrating the on-going cognitive processes into the caregiver's minds) than a classically nosological entity. We think this semantic distinction if of prime importance since we know how far off our current prognostication tools are from being perfect, especially when applied within the first 72 hours of acute brain injury. Despite this, caregivers leading WLST discussions are typically overly pessimistic and frequently inaccurate in predicting outcomes 2 10 11 .

Prognostication of the outcome of perceived DBI can be a very complex process holding contradictory elements of variable uncertainties. In such complex situations, we have known for decades that many ''decisional short-cuts" can lead toward systematic and predictable errors known as 'cognitive biases' [START_REF] Tversky | Judgment under Uncertainty: Heuristics and Biases[END_REF] . Cognitive biases can be viewed as the cognitive equivalent of visual illusions at the perceptive level (reveling how information can be processed erroneously). The field of medicine is no exception [START_REF] Elstein | Clinical judgment: psychological research and medical practice[END_REF] . The availability bias for instance (i.e., the tendency to make judgments of likelihood based on the ease of recall rather than on actual probabilities), tends to make us think first about a rare diagnosis we recently experienced (being more heavily influenced by a recent patient, or data seen at a conference that we just returned from). This may still cloud our judgment even if we consciously know that the case or data pertained to a very rare diagnosis.

There is a growing interest in studying the impact of these biases on diagnostic and therapeutic decision making 14 15 . Unfortunately, studies on how biases affect prognostication are scarce but the existing data is alarming. Neurologists with expertise in stroke care estimating the probability of key clinical outcomes such as mortality or disability at discharge have been shown to fall within the 95% CI of the actually observed outcomes in less than 20% of cases [START_REF] Saposnik | JURaSSiC: accuracy of clinician vs risk score prediction of ischemic stroke outcomes[END_REF] .

Many biases may cloud the judgment of a physician or a healthcare team member providing a prognostic assessment including availability bias, confirmation bias, anchoring or framing effect (for details please refer to Table 1). For instance, prognosticating recovery of a DBI patient, we can error in two fundamentally different directions: (1) Predicting a good prognosis for a patient that would actually have a bad outcome (e.g. permanent vegetative or minimally conscious state for example). ( 2) Predicting a poor prognosis, that will lead to a WLST, in a patient that could have actually recover with a good outcome. Both errors may have dramatic consequences that we would like to avoid. However, there is a fundamental difference between these two scenarios: in the second case the patient generally dies after a WLST decision and the ground truth will remain unknown. In other words, it will be impossible to know if our prognosis was actually correct or wrong. This is not true for the first type of error, where distraught family members may confront us with the inaccuracy of our prediction should the patient not recover.

The loss aversion effect (our tendency to view losses as larger than corresponding gains) may bias towards decisions that prevent the risk of discovering that we were wrong.

We do not run this risk when giving a poor prognosis. Loss aversion can lead to contradictory cognitive contents if evidence does not fit perfectly a poor prognosis generating a state of "mental discomfort" (called cognitive dissonance) [START_REF] Festinger | A theory of cognitive dissonance[END_REF] . To minimize this discomfort, we may be tempted to unconsciously readjust the weight of each line of evidence in order to better fit the poor prognosis. Framing effect and confirmation bias may lead us to overestimate pejorative elements and minimize the good ones, while substitution bias results in prognostic overconfidence (Table 1). Scrutinizing the decision making process with insights gained from cognitive psychology allows us to better understand and minimize the impact of systematic biases that may lead us to more proactively argue for WLST. The first step in the right direction is being aware of the neuropsychological driving forces of decision making at the Table 1. Common cognitive biases that may underlie systematic errors in neuroprognostication of patients with devastating brain injury.

individual but also the group level [START_REF] Bang | Making better decisions in groups[END_REF] . This kind of approach was elegantly illustrated in the recent article by Braxton and colleagues outlining cognitive biases leading to an overinvestment in futile therapeutics at the end of life in the ICU setting [START_REF] Braxton | Escalation of Commitment in the Surgical ICU[END_REF] .

Consensus statements are crucial even in the absence of high quality data as they provide a generally agreed upon framework to guide management and decision making

Cognitive bias Description Example

Confirmation bias to look for or to interpret evidence to support prior hypothesis rather than look for disconfirming evidence.

looking evidence to support the presumed prognosis rather than contradictory elements.

Availability bias

judgments of likelihood or percentages based on ease of recall (greater "availability" in memory) rather than on actual probabilities.

overestimate the likelihood of a prognosis based on a recent experience with a similar case.

Anchoring effect

to rely heavily on one piece of information when making decisions (usually the first piece of information acquired: the "anchor").

focusing on salient features in the patient's presentation too early in the prognosis process and failing to adjust this initial impression in the light of new information.

Framing effect

to draw different conclusions from the same information, depending on how that information is presented.

allowing the way evidence is framed or whom the information came from to influence prognosis making.

Loss aversion

to view losses as looming larger than corresponding gains.

continue with a given prognosis, even though it may not fit the new evidence (avoiding the loss of "being right"). Attribute substitution answering a complex, difficult question by substituting it by a related but simpler one. translate a legitimate high confidence in diagnosis elements into an overconfidence on prognosis issue.

Sunk-cost effect

to allow previously spent time, money, or effort to influence present or future decisions.

overestimation of a good prognosis if a lot of resources (typically surgery or organ supply) have been successful (in term of short outcome).

Dunning Krúger effect

tendency for unskilled individuals to overestimate their own ability ("illusory superiority") and the tendency for experts to underestimate their own ability. being over confident in a prognosis in case of a lake of knowledge in this specific field (in comparison to expert).

Bandwagon effect

to do (or believe) things because many other people do (or believe) the same rely too much on apparent consensus and/or common practices.

Commission bias

to favour action rather than inaction. jumping to a withdrawal of care procedure (+/-organ donation) rather than just giving more time to get more information.

Blind obedience

to show undue deference to authority or technology.

relying too much on a unique expert opinion or test result.

processes in these extremely stressful life-and-death decisions. However, we think that both the present and former American statements only go halfway. The stand-by window should depend on the underlying pathophysiology but likely should be enlarged. Beyond the scope of these consensus statements, experts need to demand development of reliable and accurate prognostic tools to guide decision-making processes for these fragile patients. Lastly research into and training of health care professionals in recognizing and coping with the impact of biases on decision-making needs to be supported. This heightened awareness of the dangers of cognitive biases for decision-making will hopefully encourage physicians to better use available prognostic tools and to develop new computerized clinical decision support systems [START_REF] Garg | Effects of computerized clinical decision support systems on practitioner performance and patient outcomes: a systematic review[END_REF] .

Rohaut & Claassen. BJA 2018

Acknowledgements

We thank Mariam Chammat, Albert Moukheiber and Lionel Naccache for their very constructive remarks.

Declaration of interest

None declared.


Rohaut Benjamin

Lionel Naccache

Dr Benjamin Rohaut

What are the boundaries of unconscious semantic cognition?

What are the scope and the limits, if any, of unconscious cognition? Can the human mind access to the meaning of a printed word without any awareness of the stimulus? During the last 20 past years, evidence supporting the existence of unconscious semantic processing of words accumulated (for a short review see [START_REF] Naccache | Conscious influences on subliminal cognition exist and are asymmetrical: validation of a double prediction[END_REF]. But still, many questions remain unanswered. How deep can these processes occur? Can multiple words, sets of words or complex syntactic structures such as sentences be processed unconsciously? Can multiple meanings of polysemous words be represented unconsciously?

In this issue of the EJN, Nakamura and colleagues report a study assessing precisely how many words can be integrated unconsciously [START_REF] Nakamura | Neural capacity limits during unconscious semantic processing[END_REF].

They presented subjects with series of masked flashed words followed by a consciously visible target, and probed priming effects through a combination of behavioural and eventrelated potentials (ERP) measures. While unmasked visible prime words influenced the processing of target words irrespective of the number of words separating the prime word from the target, masked priming effects were much more limited: semantic priming effects of masked words vanished when two or more masked words were inserted between the masked prime and the visible target. This result could suggest that unconscious semantic processing is limited to only two items (the masked prime and the inserted word), as proposed by the authors. Alternatively, given that the number of inserted masked word was confounded with the temporal distance between prime and target words, it may also simply reflect the shortlived limitation of unconscious representations. Future experiments manipulating the number of inserted words and the temporal gap as orthogonal factors will allow a better understanding this interesting result. One may also wonder if the masked words inserted between the prime and the target were really semantically processed and integrated unconsciously together.

While a recent ERP study demonstrated that two masked words presented simultaneously could be semantically integrated [START_REF] Van Gaal | Can the meaning of multiple words be integrated unconsciously[END_REF], the Nakamura et al. study shows that this is also possible when the two words are presented sequentially.

The question of the depth of unconscious semantic cognition has been the subject of many phantasms among the general public, not to mention advertisers and politics. These limits have been fluctuating gently over the time thought. For instance, for more than 35 years, scientists used to consider that the multiple meanings of a polysemous words (e.g. the word "bank", that can be the bank of the river or the bank where you save your money) could be processed in a parallel manner at the unconscious levels irrespectively to the contextual content [START_REF] Marcel | Conscious and preconscious recognition of polysemous words: Locating the selective effects of prior verbal context[END_REF]. This view was particularly popular among the general public, corresponding to a less "limited" or a "more open" unconscious mental life. Recently, trying to replicate this seminal work and, using a very similar ERP approach that the one reported by Nakamura in this issue, we demonstrated that unconscious semantic processing of polysemous word was actually very similar to what we observe in the conscious condition [START_REF] Rohaut | Unconscious semantic processing of polysemous words is not automatic[END_REF]. Unconscious semantic processing of polysemous words was strongly influenced by the current conscious semantic context. These strong influences of the current conscious posture on unconscious processing seem extremely general.

Finally, in addition to these fundamental scientific questions, it is important to note that understanding unconscious cognition is also of great interest in medical science. Probing residual cognition in unresponsive patients (in a comatose, vegetative or minimally conscious state) is of prime importance to better diagnose their level of consciousness and to elaborate a more reliable prognosis of their consciousness recovery. Capitalizing on fundamental research, several groups have attempt to develop reliable EEG semantic paradigms allowing to detect EEG signatures of unconscious semantic processing. To date, results have been very mitigated though, mainly because of the great inter-individual variability that lead to a very poor sensitivity at the individual level, even in healthy subjects [START_REF] Steppacher | N400 predicts recovery from disorders of consciousness[END_REF][START_REF] Balconi | Disorders of Consciousness and N400 ERP Measures in Response to a Semantic Task[END_REF][START_REF] Cruse | The reliability of the N400 in single subjects: Implications for patients with disorders of consciousness[END_REF][START_REF] Erlbeck | Task instructions modulate the attentional mode affecting the auditory MMN and the semantic N400[END_REF][START_REF] Rohaut | Probing ERP correlates of verbal semantic processing in patients with impaired consciousness[END_REF][START_REF] Beukema | A hierarchy of event-related potential markers of auditory processing in disorders of consciousness[END_REF]. Currently, one of the most promising approach could be the detection of brain modulations in response to the narrative content of natural speech [START_REF] Ivan | Divergent neural responses to narrative speech in disorders of consciousness[END_REF].

This original approach that capitalize on fundamental research [START_REF] Hasson | Intersubject synchronization of cortical activity during natural vision[END_REF] emphasizes how fruitful can be the bidirectional approach of exploring cognition in both unconscious patients and conscious subjects.


MD Bertrand Hermann

Gwen Goudard

Karine Courcoux

Mélanie Valente

Sebastien Labat

Lucienne Despois

MA, RN Julie Bourmaleau

Louise Richard-Gilis

MD, PhD Frédéric Faugeras

MD Sophie Demeret

( R Eegt

MD, PhD Jacobo D Sitt

MD, PhD Lionel Naccache

MD, PhD Benjamin Rohaut

"DoC-feeling": a new behavioural tool to help diagnose the Minimally Conscious State

come

Introduction

Accurate diagnosis of the level of consciousness in a brain damaged patient is of great importance to better predict recovery. Disorders of Consciousness (DoC) taxonomy has been recently challenged [START_REF] Naccache | Minimally conscious state or cortically mediated state?[END_REF][START_REF] Tim | Reforming the taxonomy in disorders of consciousness[END_REF][START_REF] Bernat | Nosologic considerations in disorders of consciousness[END_REF] but schematically includes the unresponsive wakefulness syndrome (UWS, also termed vegetative state) and the minimally conscious state (MCS). The detection of MCS has a huge prognostic impact since functional outcome is dramatically better for MCS patients [START_REF] Luauté | Long-term outcomes of chronic minimally conscious and vegetative states[END_REF][START_REF] Noé | Behavioral recovery in disorders of consciousness: a prospective study with the Spanish version of the Coma Recovery Scale-Revised[END_REF][START_REF] Bruno | Prognosis of patients with altered state of consciousness[END_REF][START_REF] Klein | Rehabilitation outcome of unconscious traumatic brain injury patients[END_REF][START_REF] Faugeras | Survival and consciousness recovery are better in the minimally conscious state than in the vegetative state[END_REF]. However, assessing consciousness in DoC patients can be challenging and in such cases, clinicians may need dedicated clinical tools and brain imaging techniques specifically designed to probe consciousness [START_REF] Giacino | Disorders of consciousness after acquired brain injury: the state of the science[END_REF]. However, even when using dedicated clinical tools such as the Coma Recovery Scale Revised (CRS-R; [START_REF] Giacino | The JFK Coma Recovery Scale-Revised: measurement characteristics and diagnostic utility[END_REF]), a unique assessment remains associated with a high frequency of diagnostic error [START_REF] Schnakers | Diagnostic accuracy of the vegetative and minimally conscious state: Clinical consensus versus standardized neurobehavioral assessment[END_REF][START_REF] Rohaut | Neurology of consciousness impairments[END_REF]. To circumvent this limitation, repeated clinical assessments have been proposed but this can be limited by the availability of trained clinicians [START_REF] Wannez | The repetition of behavioral assessments in diagnosis of disorders of consciousness[END_REF].

In this study, we aimed at evaluating the diagnostic accuracy of pooled nursing staff estimations of the level of consciousness in DoC patients. Through their clinical practice, nursing staff (i.e. nurses and nursing assistants) accumulates extended observation time of patients that they care for. Interacting with patients through standardized procedures (such as nursing care, medication administration, blood sample, etc...), they spontaneously generate an estimation of the level of consciousness of the patient. Pooling opinions of several individuals have been shown to outperform the individual judgements [START_REF] Kurvers | Boosting medical diagnostics by pooling independent judgments[END_REF][START_REF] Bang | Making better decisions in groups[END_REF]. In this study, we hypothesized that pooling individual nursing staff estimations of the level of consciousness can improve the detection of MCS.

Methods

Evaluation of consciousness

Reference standard

Patients were admitted to the NeuroIntensive Care Unit and were observed for at least one week during which they encompassed multiple neurological assessments and brain imagery such as high-density EEG, event-related potentials, MRI and PET-scan.

Clinical assessments consisted of at least 3 neurological exams which included the Coma Recovery Scale-Revised (CRS-R; [START_REF] Giacino | The JFK Coma Recovery Scale-Revised: measurement characteristics and diagnostic utility[END_REF]), performed by expert clinicians (BH, BR, FF, LN) belonging to an external expert team in DoC patients. CRS-R scoring ranges from 0 to 23 and is based on the presence or absence of responses on a set of hierarchically ordered items testing auditory, visual, motor, oromotor, communication and arousal function. State of consciousness (i.g., UWS, MCS and conscious state) depends on specific key behaviours probed during the CRS-R assessment. For instance, visual pursuit, reproducible movements to command and/or complex motor behaviour scores for MCS [START_REF] Giacino | The JFK Coma Recovery Scale-Revised: measurement characteristics and diagnostic utility[END_REF]. We used the highest level of consciousness among all the CRS-R performed on a dedicated patient as the reference standard. Each patient was thus labelled as being in a UWS or MCS. MCS was the target condition.

Index test

Nursing staff (nurses and nursing assistants) taking care of a DoC patient was asked to fill in a form at the end of their shift containing a scale called "DoC-feeling". DoCfeeling was designed as a 100 mm visual analog scale aiming at quantifying the best patient's consciousness level observed during the shift. We specifically asked caregivers to rate their "gut feeling" about the best level of consciousness observed during the shift or the "présence" (presence), using the French idiom "le patient estil là?" which is very close to the English one "Is there anybody home?" (Figure 1 andS1). This wording reproduced the commonly used language to communicate observations relative to consciousness level of a patient among caregivers. Individual DoC-feeling ratings were collected prospectively. Caregivers were blinded to the previous caregivers' ratings and to the reference standard (the CRS-R) and expert physicians were blinded to the index test.

In order to obtain a final global metric, for each patient, all ratings were pooled using Each patient was evaluated around 3 times by disorders of consciousness (DoC) experts using the Coma Recovery Scale -Revised (CRS-R). In parallel, nursing staff members reported their daily observations using the DoC-feeling visual analog scale. The reference standard was defined as the best state of consciousness observed during one of the CRS-R and the patient was coded as being in an Unresponsive Wakefulness Syndrome (UWS) or a Minimally Conscious State (MCS) accordingly (reference standard). All individual DoCfeeling obtained during the whole hospital stay were pooled and the median value (represented by the vertical dashed line) of the polled results was defined as the DoC-feeling score (index test).

the median to obtain the DoC-feeling score that constituted the index test of this study.

Clinical data

Demographics, aetiology and time from the acute brain injury were collected. In addition to CRS-R and DoC-feeling ratings, we also collected complementary metrics (such as the classical distinction between wakefulness and awareness, interaction during nursing and/or painful care) using the same VAS approach (see supplementary material) as well as the best FOUR-score observed during each shift [START_REF] Wijdicks | Validation of a new coma scale: The FOUR score[END_REF].

Statistics

Our primary objective was to evaluate the diagnostic accuracy of the index test called "DoC-feeling score" to detect the target condition (MCS) as defined by the standard reference (best CRS-R).

First, to evaluate the association of individual DoC-feeling ratings with the standard reference, we computed a linear mixed model (LMM) using DoC-feeling individual ratings as the dependent variable, the state of consciousness as the fixed effect explanatory variable and patients as well as raters as random effects. LMM provides the optimal approach in order to consider the non-independence between DoC-feeling ratings due to the repeated measurements over time at both the patient level (same patient rated by several raters) and the rater level (several ratings by rater).

We next pooled the individual ratings obtained for each patient using the median to obtain the DoC-feeling score (index test). We thus obtained a DoC-feeling score as well as a reference standard label (UWS or MCS) for each patient. We did a direct comparison of the scores between the two populations using a Wilcoxon-Mann-Whitney test, and in order to assess diagnostic accuracy of DoC-feeling scores to detect MCS (target condition) we computed the area under the ROC curve (AUC) and report sensitivities and specificities for several cut-offs of DoC-feeling scores. All statistical tests were two-sided. Categorical variables were expressed as numbers (percentage), quantitative variables as median [interquartile range]. Analyses were performed using the R statistical software version 3.4.1 [START_REF] Development | R: A language and environment for statistical computing[END_REF]. LMM were performed using the lme4 and nlme packages [START_REF] Pinheiro | nlme: Linear and Nonlinear Mixed Effects Models[END_REF]. AUC, sensitivity and specificity with their 95% CI were computed using 2000 stratified bootstrap replicates (AUC) and binomial test (sensitivity and specificity) respectively using the pROC package [START_REF] Robin | pROC: an open-source package for R and S+ to analyze and compare ROC curves[END_REF].

The Standards for Reporting Diagnostic Accuracy (STARD) were followed thoroughly [START_REF] Cohen | STARD 2015 guidelines for reporting diagnostic accuracy studies: explanation and elaboration[END_REF].

Results

Patients characteristics

Seventy-two patients were eligible during the inclusion period, 23 were not included because of a lack of informed consent from a legal representative. Two patients were excluded because they had been diagnosed as conscious ("Exit-MCS"). Forty-seven patients were included in the analysis (see Figure 2). Median age was 49 [32-62] years and 50% (n=36) were women. Main aetiologies of brain injury included anoxia (53%) and traumatic brain injury (17%). Delay between acute brain injury (ABI) and the evaluation was 134 [40-762] days (see Table 1).

Discussion

In the present study we developed and assessed a new diagnostic tool called DoCfeeling to help diagnose MCS. This score which pools repeated measures obtained among several caregivers over several days of evaluation showed a very good accuracy to diagnose MCS.

While this tool is not intended to replace the clinical examination nor the current CRS-R gold standard, we propose it as a complementary one. Indeed, we believe that the implementation of DoC-feeling could improve the overall diagnostic accuracy, taking advantage of valuable information collected by all caregivers involved in the care of a DoC patient. Caregivers are trained to evaluate pain and suffering in patients during all delivered procedures. These procedures constitute standardized interactions that can allow the generation of very reliable heuristics to assess one's percept in term of pain suffering and also consciousness.

Quantifying expertise that is not restricted to physicians is of prime interest.

Capitalizing on assessments of consciousness gathered at any hour of the day and through multiple observers may also potentially increases our ability to detect signs of consciousness in these patients who usually show large fluctuations of cognitive state and of arousal. This tool may also help to better describe and quantify these fluctuations. Additionally it also enables to acknowledge the caregiver group expertise and to increase care team attention through a coherent and cumulative set of observational data.

The good accuracy of DoC-feeling obtained in our setting is likely to be generalizable elsewhere. Firsts, as the distribution of CRS-R scores obtained in this cohort spanned most of the possible CRS-R scores, it is unlikely that the accuracy of DoC-feeling was simply a result of two easily discernible patients' clusters. Second, as all the patients included in this study, either in an acute or a chronic stage, were specifically referred to our institution to assess their level of consciousness, it is most likely that our cohort was actually representative of patient for whom the diagnosis was not trivial.

Our study presents some limitations inherent to the aim of developing a pragmatic and easily implementable tool in daily clinical practice. First, in addition to their clinical observations, caregivers might have been influenced by other factors that would have been very difficult to control. For instance, caregivers might have been influenced by insights from other caregivers. However, the variability of individual ratings for a given patient (that tended to increase over time, see supplementary material) suggests that caregivers did report their own perception independently from each other.

Moreover, interaction in small groups could have actually had a positive effect since pools of small groups insights' have been shown to outperform the overall judgment of the group (effect known as wisdom of the crowds [START_REF] Navajas | Aggregated knowledge from a small number of debates outperforms the wisdom of large crowds[END_REF]). This kind of tool might be less prone to individual subjective bias that can be observed during decision making under high degree of uncertainty such as assessment of DoC patients [START_REF] Rohaut | Decision making in perceived devastating brain injury: a call to explore the impact of cognitive biases[END_REF]. Caregivers could have also been biased by the perception of patients' relatives, although it is commonly acknowledged that relatives frequently lack objectivity (in both direction) in such dramatic situations [START_REF] Rohaut | Neurology of consciousness impairments[END_REF]. Caregivers' judgments could have been biased by classical predictors of consciousness recovery such as aetiology or delay from ABI.

Finally, although the number of involved float staff members and the result of a preliminary survey assessing prior knowledge of regular nursing staff on DoC (supplementary material) suggest together that DoC-feeling should be accurate in other settings, the monocentric design of this study requests external validation.

Despite these limitations, we think that the implementation of DoC-feeling score can significantly improve diagnostic accuracy and confidence in the diagnosis when supporting other metrics (i.e., CRS-R and brain imagery). Moreover, when incongruent with other metrics, DoC-feeling score could also be useful. Indeed, this could either suggest that clinical elements have been missed by physician while performing punctual CRS-R assessments but it could also reveal, in case of discrepancy with all the other markers (clinical and brain imagery), a possible misperception of patient's consciousness level that need to be acknowledged and, considered in medical decisions. This last point could be crucial in bridging the gap between the caregiver's team and the patient's relatives in situations of conflict.

In conclusion we propose a new diagnostic tool called DoC-feeling that can help in improving the diagnostic accuracy of MCS and thus, promote better prognostic decisions for DoC patients alongside other clinical and brain imagery tools.

Faugeras; Bertrand Hermann; Pauline Pérez; Federico Raimondo; Benjamin Rohaut;

Johan Stender; Mélanie Valente and Jacobo Sitt.

We thank Raphael Porcher for his help on statistical issues and Jan Claassen for his review of the final manuscript.

Patients

  All patients referred for evaluation of consciousness at the Department of Neurology of La Pitié-Salpêtrière Hospital, Paris, between February 2016 and October 2017 were screened prospectively. On hospital admission patients' relatives were approached to give consent for participation to the study. All patients with a UWS or MCS condition and consent were eligible. The protocol conformed the Declaration of Helsinki, the French regulations, and was approved by the local ethic committee (Comité de Protection des Personnes; CPP n° 2013-A01385-40; Ile de France 1; Paris, France).

Figure 1 .

 1 Figure 1. DoC-feeling score.Each patient was evaluated around 3 times by disorders of consciousness (DoC) experts using the Coma Recovery Scale -Revised (CRS-R). In parallel, nursing staff members reported their daily observations using the DoC-feeling visual analog scale. The reference standard was defined as the best state of consciousness observed during one of the CRS-R and the patient was coded as being in an Unresponsive Wakefulness Syndrome (UWS) or a Minimally Conscious State (MCS) accordingly (reference standard). All individual DoCfeeling obtained during the whole hospital stay were pooled and the median value (represented by the vertical dashed line) of the polled results was defined as the DoC-feeling score (index test).

Figure 2 .

 2 Figure 2. Flow chart. Flow chart representing the repartition of patients while using a DoC-feeling score (index test) cut-off value of 16.7 mm. CRS-R: Coma Recovery Scale -Revised; MCS: Minimally Conscious State; Exit-MCS: Patient able to communicate reliably or to use objects functionally; UWS: Unresponsive Wakefulness Syndrome.

Figure 3 .

 3 Figure 3. Individual DoC-feeling ratings. DoC-feeling ratings tended to be smaller in UWS patients when compared to MCS patients. All measures are presented (n=692) alongside boxplots helping to visualize the median and the interquartile ranges for both UWS (on the left in red) and MCS (on the right in turquoise) patients. Patients are ordered according to the median values of DoC-feeling ratings (DoC-feeling scores).

Figure 4 .

 4 Figure 4. DoC-feeling scores. Median of the individual ratings obtained for each patient were pooled to obtain one DoC-feeling score per patient. DoC-feeling scores tended to be smaller for UWS than for MCS patients (a, b) and to correlate with the total of the CRS (a). Area under the ROC curve (c), sensitivity (Se) and specificity (Sp) for several cutoffs (d) revealed performances at identifying the MCS. ***: p<0.001.

Table 1 . Patient characteristics.

 1

	Results are expressed in n(%) or median[InterQuartile Range] as appropriate.
	ABI: Acute brain injury; CRS-R: Coma Recovery Scale revised; IQR: Inter-Quartile
	Range; MCS: Minimally Conscious State; NA: Nursing Assistant; Nb: Number; TBI:
	Traumatic Brain Injury; UWS: Unresponsive Wakefulness Syndrome. * PET-scan
	was performed only in patients free of mechanical ventilation.

Acknowledgments:

We thank all the members of the Pitié-Salpétrière hospital Neuro-ICU lead by Sophie Demeret (Medical Director), Julie Bourmaleau and Louise Richard-Gilis, (Head Nurses) who have participated in this study (alphabetic order): Jérémie Abitbol , Fatiha Ait Yata Azzi, Binta Bah Fatoumata, Francis Bolgert, Sandrine Briand, Sandra Coelho, Alexia Camuzat, Marie-Chantal Colmar, Flora Cherruault, Cecile Chordi, Véronique Cottin, Bintou Coulibaly, Karine Courcoux, Mélanie Dalibard, Lucienne Despois, Estelle Dumarey, Atef El Ouarghi, Helene Espiand, Cécilia Eltebert, Fabrice Fanhan, Agnès Flament, Suzelle Fontano-Marie, Pascale Fournier, Céline Frammezelle, Gwen Goudard, Alexandra Grinéa, Nouara Harchaoui, Marie Harmancij, Claire Jacqueminet, Charlotte Janvier, Jamila Kebli, Bouchra Khedaoui, Sébastien Labat, Aurélie Lemoal, Kim Louis-Joseph, Brice Lucas, Valérie Maes, Sophie Maillard, Romain Maurel, Madely Petit, Floriane Pépin, Isabelle Picot, Eva Proneur, Manuela Roselmac, Sylviane Saintini, Mélody Seidel, Yolène Sully, Kelly Tcha, Laura Verbaux, Nicolas Weiss, Kelly Yanganju.

We thank all the members of the PICNIC-Lab "DoC-Team", dedicated to the improvement of care of patients suffering from disorder of consciousness led by Lionel Naccache (alphabetic order): Athena Demertzi; Denis Engemann; Frédéric

Funding: This work was supported by: « Amicale des Anciens Internes des Hôpitaux de Paris & Syndicat des Chefs de Cliniques et Assistants des Hôpitaux de Paris » (AAIHP -SCCAHP; BR), Assistance Publique -Hôpitaux de Paris (AP-HP; BR, LN), Institut National de la Santé et de la Recherche Médicale (Inserm; BH, JS, LN), Sorbonne Université (LN), the James S. McDonnell Foundation (LN), Académie des Sciences-Lamonica Prize 2016 (LN) and Philippe Foundation (BR). The research leading to these results has received funding from the program "Investissements d'avenir" ANR-10-IAIHU-06.

All the authors report no competing interest related to any of the previously mentioned funders.

Authors Contributions (alphabetic order):

Study concept and design: BR, GG, JB, JDS, KC, LD, LN, LRG, SD and SL. Data collection: BH, GG, KC, LD, MV and SL. Analysis and interpretation of data: BH, BR. Drafting of the manuscript: BH, BR and LN. Critical revision of the manuscript for important intellectual content: BH, BR, JDS, LN and SD. Statistical analysis: BH, BR and LN. Study supervision: BH, BR, GG, KC and MV had full access to all the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis. BH, GG and KC contributed equally to this work.

Key words: disorders of consciousness, diagnosis, coma recovery scale, minimally conscious state, vegetative state / unresponsive wakefulness syndrome, group decision making.

Reference test

One-hundred and forty-seven CRS-R assessments were performed, with a median of 3 [START_REF] Tim | Reforming the taxonomy in disorders of consciousness[END_REF][START_REF] Bernat | Nosologic considerations in disorders of consciousness[END_REF][START_REF] Luauté | Long-term outcomes of chronic minimally conscious and vegetative states[END_REF] per patient. According to the best CRS-R (reference standard), 27 patients (57%) were diagnosed as being in a MCS (target condition) and 20 (43%) were classified as being in a UWS. Patients with MCS less frequently had suffered from anoxia and had a longer delay between the acute brain injury and the study assessment (see Table 1). No differences were found in the number of CRS-R assessment per patient or brain imaging explorations between UWS and MCS patients.

Index test

Six-hundred and ninety-two DoC-feeling individual ratings were obtained (median of 12 [START_REF] Giacino | Disorders of consciousness after acquired brain injury: the state of the science[END_REF][START_REF] Giacino | The JFK Coma Recovery Scale-Revised: measurement characteristics and diagnostic utility[END_REF][START_REF] Schnakers | Diagnostic accuracy of the vegetative and minimally conscious state: Clinical consensus versus standardized neurobehavioral assessment[END_REF][START_REF] Rohaut | Neurology of consciousness impairments[END_REF][START_REF] Wannez | The repetition of behavioral assessments in diagnosis of disorders of consciousness[END_REF][START_REF] Kurvers | Boosting medical diagnostics by pooling independent judgments[END_REF][START_REF] Bang | Making better decisions in groups[END_REF][START_REF] Wijdicks | Validation of a new coma scale: The FOUR score[END_REF][START_REF] Development | R: A language and environment for statistical computing[END_REF][START_REF] Pinheiro | nlme: Linear and Nonlinear Mixed Effects Models[END_REF][START_REF] Robin | pROC: an open-source package for R and S+ to analyze and compare ROC curves[END_REF] ratings per patient). Eighty-three caregivers, 57 Nurses and 26 NAs (47 neuro-ICU staff members and 36 float staff members) participated to the study. Each nursing staff member filled a median of 4 [START_REF] Naccache | Minimally conscious state or cortically mediated state?[END_REF][START_REF] Tim | Reforming the taxonomy in disorders of consciousness[END_REF][START_REF] Bernat | Nosologic considerations in disorders of consciousness[END_REF][START_REF] Luauté | Long-term outcomes of chronic minimally conscious and vegetative states[END_REF][START_REF] Noé | Behavioral recovery in disorders of consciousness: a prospective study with the Spanish version of the Coma Recovery Scale-Revised[END_REF][START_REF] Bruno | Prognosis of patients with altered state of consciousness[END_REF][START_REF] Klein | Rehabilitation outcome of unconscious traumatic brain injury patients[END_REF][START_REF] Faugeras | Survival and consciousness recovery are better in the minimally conscious state than in the vegetative state[END_REF][START_REF] Giacino | Disorders of consciousness after acquired brain injury: the state of the science[END_REF][START_REF] Giacino | The JFK Coma Recovery Scale-Revised: measurement characteristics and diagnostic utility[END_REF][START_REF] Schnakers | Diagnostic accuracy of the vegetative and minimally conscious state: Clinical consensus versus standardized neurobehavioral assessment[END_REF][START_REF] Rohaut | Neurology of consciousness impairments[END_REF] evaluations. Median delay between the first and the last individual rating was 6 days [START_REF] Noé | Behavioral recovery in disorders of consciousness: a prospective study with the Spanish version of the Coma Recovery Scale-Revised[END_REF][START_REF] Bruno | Prognosis of patients with altered state of consciousness[END_REF][START_REF] Klein | Rehabilitation outcome of unconscious traumatic brain injury patients[END_REF][START_REF] Faugeras | Survival and consciousness recovery are better in the minimally conscious state than in the vegetative state[END_REF][START_REF] Giacino | Disorders of consciousness after acquired brain injury: the state of the science[END_REF]. No statistical differences were found between UWS and MCS in the number of DoC-feeling ratings per patient, number of rater per patient or in the number of ratings per rater.

Analysis of individual DoC-feeling ratings

Inspection of the 692 DoC-feeling ratings' distribution revealed higher values for MCS than for UWS patients but with an important variability of ratings for a given patient (Figure 3 and supplementary material). The linear mixed model (LMM) analysis revealed a strong significant association between DoC-feeling individual ratings and the state of consciousness (t-value=6.47, df=45, p<0.001).

Diagnostic accuracy of DoC-feeling scores

Overall, patients underwent 12 [START_REF] Giacino | Disorders of consciousness after acquired brain injury: the state of the science[END_REF][START_REF] Giacino | The JFK Coma Recovery Scale-Revised: measurement characteristics and diagnostic utility[END_REF][START_REF] Schnakers | Diagnostic accuracy of the vegetative and minimally conscious state: Clinical consensus versus standardized neurobehavioral assessment[END_REF][START_REF] Rohaut | Neurology of consciousness impairments[END_REF][START_REF] Wannez | The repetition of behavioral assessments in diagnosis of disorders of consciousness[END_REF][START_REF] Kurvers | Boosting medical diagnostics by pooling independent judgments[END_REF][START_REF] Bang | Making better decisions in groups[END_REF][START_REF] Wijdicks | Validation of a new coma scale: The FOUR score[END_REF][START_REF] Development | R: A language and environment for statistical computing[END_REF][START_REF] Pinheiro | nlme: Linear and Nonlinear Mixed Effects Models[END_REF][START_REF] Robin | pROC: an open-source package for R and S+ to analyze and compare ROC curves[END_REF] DoC-feeling individual ratings performed by 7 [START_REF] Noé | Behavioral recovery in disorders of consciousness: a prospective study with the Spanish version of the Coma Recovery Scale-Revised[END_REF][START_REF] Bruno | Prognosis of patients with altered state of consciousness[END_REF][START_REF] Klein | Rehabilitation outcome of unconscious traumatic brain injury patients[END_REF][START_REF] Faugeras | Survival and consciousness recovery are better in the minimally conscious state than in the vegetative state[END_REF][START_REF] Giacino | Disorders of consciousness after acquired brain injury: the state of the science[END_REF][START_REF] Giacino | The JFK Coma Recovery Scale-Revised: measurement characteristics and diagnostic utility[END_REF] different raters. All DoC-feeling ratings obtained for a given patient were summarized using the median to obtain a pooled metric called DoC-feeling score (index test). Median DoC-feeling scores were smaller for UWS than for MCS patients (7.2 mm [2.4-11.4] vs. 59.2 mm [27. 3-77.3] respectively; p<0.001; Figure 4). ROC curve revealed excellent accuracy at detecting MCS (AUC=0.92 [95% CI 0.84-0.99];

Figure 4) with a sensitivity of 89% [95% CI 71-98] and a specificity of 85% [95% CI 62-97] when using a DoC-feeling score cut-off at 16.7 mm.


Jean-Marc Azaïs
email: jean-marc.azais@math.univ-toulouse.fr

Stéphane Mourareau
email: stephane.mourareau@gmail.com

How sharp are classical approximations for statistical applications?

Keywords: MSC 2010 subject classifications: Primary 60G15, 60-04; secondary 62-04 Gaussian Process, Kac-Rice formula, Pickands method, Euler characteristic method

This paper aims at comparing theoretical approximations of the tail of the maximum of stochastic processes and the corresponding numerical evaluations. More particularly, we focus on the Pickands or double sum method, the Rice method, the Euler Characteristic method and a new one called the Poisson method. The numerical evaluation, performed using mainly Quasi Monte-Carlo integration and adaptations of the programs of Genz, show the domains of validity of each method.

Introduction

The distribution of the maximum of a Gaussian vector, process or fields has many application in Statistics and Spatial Modeling (see Worsley et al. [START_REF] Keith | A unified statistical approach for determining significant signals in images of cerebral activation[END_REF] or Cressie [START_REF] Cressie | Statistics for spatial data[END_REF]). For instance, in the "signal plus noise model" on the real line, we assume that the observation Y = (Y (t)) t is the sum of a signal µ = (µ(t)) t and a noise χ = (X(t)) t which is a centered stationary Gaussian process. If we want to test the null hypothesis H 0 : µ ≡ 0, a natural statistic is

M T = max t∈[0,T ] X(t),
the observed maximum over the interval [0, T ]. Note that the maximum of the absolute value is also a relevant statistics, but for the sake of simplicity, we will not consider this last case which is very similar. Computing thresholds and power in our case involve some computations on the distribution of M T . Unfortunately, it is a largely unsolved problem and exact closed formulas exist only in some particular cases. In the other cases, some bound or asymptotic expansions have been constructed starting from the pioneering work of Rice [START_REF] Stephen O Rice | Mathematical analysis of random noise[END_REF] then Pickands [START_REF] Pickands | Asymptotic properties of the maximum in a stationary Gaussian process[END_REF] and Adler [START_REF] Adler | The geometry of random fields[END_REF].

These expansions have generally good properties when T is fixed and u is large (u → ∞), but in statistical applications as computation of 0.95 fractile for M T , T and u are linked and both can be "rather large". In such a case nothing is known on the properties of the considered expansions. For example, if we focus

INTRODUCTION

on the Pickands approximation given by equations ( 2) or (3), a numerical study in the case of the Ornstein-Uhlenbeck process show a very interesting behaviour. Indeed, Figure 1 suggests that the Pickands approximations are sharp for large values of T . In this example the approximation derived from equation ( 2) is a little more precise that the one of equation (3) but it is not always the case, see Section 4.4.1. We can also see that the approximation are unsuitable for small-size intervals. . Exact level of a test constructed for a nominal level of α = 5% using the Pickands method and the Ornstein-Uhlenbeck process. On the left: small intervals, on the right: large intervals (see Section 2). The level corresponding to equation (2) is given in brown while the one of equation (3) is given in green. Both are computed thanks to the pou program (see Section 3.1.1) {figure1}

Moreover, if we want to compute now the median of M T , and if we use the Rice method defined in Section 2.2, Figure 2 shows that the method does not perform accurately for large values of T .

{figure2}

In light of the above examples, the main contribution of this paper will be threefold.

• In Section 2.3 we present a new approximation based on a Poisson approximation.

• In Section 3.2.1 and 3.2.2, we present an extension of the MGP program of Genz (see [START_REF] Azaïs | Computation of the distribution of the maximum of stationary Gaussian processes[END_REF]), firstly to non-stationary case, secondly to random fields. • In Section 4, we use these programs and MGP to address the quality of the above-mentioned expansions for two precise problems.

-Computing a critical value at level α for statistical purpose.

-Computing the median of the distribution of the maximum.

As far as we know, it is the first time that such a numerical experiment is conducted.

In addition, Section 2 describes the approximations widely encountered in the literature as well as the new Poisson approximation. Section 3 describes the numerical methods that, except for those of Section 3.1, are modifications or improvements of the seminal work of [START_REF] Genz | Numerical computation of multivariate normal probabilities[END_REF].

Description of the approximations {s:approx}

We provide here a brief summary of approximation methods for P(M > u) where M is the maximum of a Gaussian process or random field on sets detailed below. Before that we defined our assumption and notation.

• χ = (X(t)) t is a Gaussian process defined on [0, T ] or a Gaussian field defined on the rectangle [0, T 1 ] × [0, T 2 ] (excepting the Bessel process of Section 3.1.1). Its expectation is m(t), its covariance is r(s, t) in the general case and ρ(h

) = r(t, t + h) if χ is stationary. • M T and M T1,T2 are respectively the maximum of X(t) on [0, T ] and [0, T 1 ] × [0, T 2 ].
• φ and Φ denote respectively the probability density function and the cumulative distribution function of a standard normal variable. Moreover, set Φ = 1 -Φ. • In case of stationarity, we suppose that χ is centered and reduced, i.e. ρ(0) = 1. • In case of stationarity and differentiability, we suppose that χ is centered, reduced with unit speed, i.e. Var(X ′ (t)) = -ρ ′′ (0) = 1. • x + = max(x, 0) is the positive part.

• P Z (z) denotes the density of the random variable Z at point z when it exists.

Pickands method.

{s: pic} Let 0 < β ≤ 2 and χ be a β-process i.e. stationary Gaussian process with covariance function ρ satisfying

ρ(t) = 1 -C|t| β + o(|t| β ) as t → 0, (1)
{cov:pic} {cov:pic} then the Pickands approximations, as u tends to infinity, are given by

P{M T > u} ≃ T C 1/β u (2/β)-1 φ(u)H β , (2)
{Pickands:born1} {Pickands:born1}

see Pickands [START_REF] Pickands | Asymptotic properties of the maximum in a stationary Gaussian process[END_REF], Piterbarg [START_REF] Vi Piterbarg | Asymptotic methods in the theory of Gaussian processes and fields[END_REF] or

P{M T > u} ≃ T C 1/β u (2/β) Φ(u)H β , (3)
{Pickands:born2} {Pickands:born2}

see Pickands [START_REF] Pickands | Asymptotic properties of the maximum in a stationary Gaussian process[END_REF], Albin and Choi [START_REF] Jmp Albin | A new proof of an old result by Pickands[END_REF] or Michna [START_REF] Michna | Remarks on Pickands theorem[END_REF]. More precisely, the H β are the Pickands constants and T > 0.

The main drawback of the method is that the constants H β are known explicitly only in the cases H 1 = 1 and H 2 = (π) -1/2 (see Leadbetter, Lindgren and Rootzén [START_REF] Malcolm R Leadbetter | Extremes and related properties of random sequences and processes[END_REF] or Albin and Choi [START_REF] Jmp Albin | A new proof of an old result by Pickands[END_REF]). In the other cases, some numerical approximation must be used as in Dieker and Yakir [START_REF] Ab Dieker | On asymptotic constants in the theory of extremes for Gaussian processes[END_REF] where the conjecture

H β = Γ(β -1 ) -1 ,
with Γ the Gamma function, is discussed. Indeed, this conjecture has been proved to be false by Harper [START_REF] Adam | Pickands' constant H_{α} does not equal 1/Γ(1/α) , for small α[END_REF] for small values of β.

In addition, the approximations given by ( 2) and (3) are obviously not efficient when T is small. An heuristic improvement would be to add the factor Φ(u) that appears in the Rice method hereunder leading to four versions of the Pickands approximation and raising the question: Which method is the best one? This question will be addressed in Section 4. Finally, note that the Pickands method, also called double sum method, has received a lot of generalization to various situations. The interested reader may consult Debicki, Hashorva, Ji et al. [START_REF] Debicki | Extremes of a class of nonhomogeneous Gaussian random fields[END_REF] and references therein.

Rice method.

{s:riz}

Let χ be a stationary Gaussian differentiable process then the Rice bound is given by

P{M T > u} ≤ Φ(u) + T φ(u) √ 2π . (4)
{chap5:approxi:rice} {chap5:approxi:rice}

Note that both Rice and Pickands apply when β = 2 in (1). In that case, Rice is asymptotically more precise as u → +∞ (see Azaïs and Wschebor [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF], chapter 4). Moreover, the Rice bound, under some regularity conditions, is superexponentially sharp in the sense that the error is bounded by

C exp - u 2 (1 + δ) 2 ,
for some positive constants C and δ > 0.Finally, there exists a non stationary version of formula (4) given by

P(M T > u) ≤ P(X(0) > u) + T 0 E (X ′ (t)) + |X(t) = u p X(t) (u)dt,
where the conditional expectation may be easily computed in the Gaussian case.

Poisson method :A new approximation {section:poiss}

Let us focus on stationary differentiable Gaussian processes. In that case, we already know that the Rice method, see [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF], gives the following bound

P{M T > u} ≤ Φ(u) + T φ(u) √ 2π . ( 5
)
{f:rice} {f:rice}

This bound is very efficient for small intervals and rather high levels. But of course it cannot be very accurate if u is fixed and T is very large, as it can be even larger than 1, so, almost every time, a better approximation is given by using the Poisson Clamping heuristic of Aldous [START_REF] Aldous | Probability approximations via the Poisson clumping heuristic[END_REF]. Basically, it assumes that the number U u (χ, [0, T ]) of up-crossings of the level u on [0, T ] follows a Poisson distribution so that the probability of having at least one crossing is approximatively 1 -exp(E(U u (χ, [0, T ]))). Note that when u and T tend jointly to infinity so that E(U u (χ, [0, T ])) is constant, the Volkonskii Rozanov theorem (see [START_REF] Va | Some limit theorems for random functions[END_REF]) implies that the limit is actually Poisson. Taking into account the fact that the limit as T goes to 0 should be Φ(u), we are conducted to use the approximation

P{M T > u} ≃ Φ(u) + Φ(u) 1 -exp - T φ(u) √ 2πΦ(u) . ( 6
) {e:poiss} {e:poiss}
Theoretically, a Taylor expansion show that the bound from the Rice method and the Poisson approximation are equivalent when u → +∞. A discussion concerning it's numerical validity and performance is given in Section 4.

Euler characteristic method.

The Euler Characteristic method is defined in its full generality for random fields defined on stratified manifold of R d having constant variance (say equal to one). However, the formula is cumbersome and no numerical application exists. For simplicity we limit our attention to a particular case.

Let χ be a Gaussian centered differentiable isotropic random field on the rectangle [0,

T 1 ] × [0, T 2 ]. Denote ρ( t -s ) = Cov(X(s), X(t))
and suppose that χ is centered, reduced and with unit speed (this can always be obtained by a scaling) then the Euler characteristic approximation is given by

P{M T1,T2 > u} ≃ Φ(u) + (T 1 + T 2 ) φ(u) √ 2π + T 1 T 2 u φ(u) 2π ,
see Adler and Taylor [START_REF] Adler | Random fields and geometry[END_REF] or Azaïs and Delmas [START_REF] Azaïs | Asymptotic expansions for the distribution of the maximum of Gaussian random fields[END_REF] for more details. In particular, the Euler Characteristic method is the natural generalization of the Rice method to dimension 2.

Numerical methods and programs {section:num}

In this section we present the juges of the numerical comparison of Section 4.

Existing programs {section:existing:programs}

The simplest cases are the few situations cases where the exact result is known by a closed formula and when this formula is simple enough so that a numerical program is available (the second condition being more restrictive that the first one).

Exact results

{s:exact}

Among the few classical cases where an explicit formula exists, see [START_REF] Azaïs | A toolbox on the distribution of the maximum of Gaussian process[END_REF], some are rather easy to compute and have been incorporated to the Maxgbbox toolbox Ṁore precisely, the following cases are covered:

• The Brownian motion or Wiener process : functions pbm, qbm, pnbm and qnbm. • The Brownian bridge: pbb, qbb, pnbb, qnbb.

• The sine-cosine process: psc, qsc, pnsc and qnsc.

• The Ornstein-Uhlenbeck process: pou, qou, pnou, qnou.

• The Bessel process: pbp, qbp, pnbp, qnbp.

The prefix p is for the cumulative distribution, q is for the inverse distribution function or fractile function and n is for the norm or the absolute value.

A first approach for Gaussian vector {s:genz}

The first program QSIMVN written by Genz [START_REF] Genz | Numerical computation of multivariate normal probabilities[END_REF] deals with integration of Gaussian densities over hyper-rectangles. It uses Quasi Monte-Carlo integration. In particular, considering an hyper rectangle in R n which is the product n times of (-∞, u), it allow us to compute numerically the distribution of the maximum of a Gaussian vector. A complementary version named QSIMVNEF computes the expectation of a function with respect to the Gaussian density.

Record method and Quasi Montecarlo integration for Gaussian processes

The method relies on the record method (see Mercadier [START_REF] Mercadier | Numerical bounds for the distributions of the maxima of some one-and two-parameter Gaussian processes[END_REF] and Azaïs Wschebor [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF]) which is a refinement of the Rice method.

{record:dim:1} Theorem 1. Suppose that the process χ is Gaussian with C 1 paths and satisfies:

• ∀(s, t) ∈ [0, T ] 2 , s < t, the distribution of (X(s), X(t)) does not degenerate, • ∀t ∈ [0, T ], the distribution of (X(t), X ′ (t)) does not degenerate, then P(M T > u) = P(X(0) > u) + T 0 E X ′ (t) + 1 X(s)≤u∀s<t X(t) = u p X(t) (u)dt. ( 7
)
{e:r2} {e:r2}

The formula [START_REF] Azaïs | A toolbox on the distribution of the maximum of Gaussian process[END_REF] is in fact an implicit formula which has its own theoretical interest but is not directly usable for a numerical purpose. The condition X(s) ≤ u∀s < t has to be discretized using a grid on the set [0, T ] giving now the following upper-bound: [START_REF] Azaïs | [END_REF] {record:dim:11} {record:dim:11} where D n (T ) := {t 0 = 0, t 1 = T /n, . . . , t n = T } is, for example, the regular grid of [0, T ]. To complete this study, a lower bound is obtained simply by discretization of time and using the routine QSIMVN of Section 3.1.2.

P(X(0) > u) + T 0 E X ′ (t) + 1 X(s)≤u∀s<t∈Dn(T ) X(t) = u p X(t) (u)dt,
After the pioneering work of Mercadier [START_REF] Mercadier | Numerical bounds for the distributions of the maxima of some one-and two-parameter Gaussian processes[END_REF] with the toolbox MAGP, this method has been fully detailed in the program MGP written by Genz and described in [START_REF] Azaïs | Computation of the distribution of the maximum of stationary Gaussian processes[END_REF], including a majoration of the error.

New programs

Record method for non stationary processes {section:MAXNSGP}

We present here a generalization of MGP called MAXNSGP. At cost of being rather slower, it is adapted to non-centered non-stationary differentiable Gaussian processes.

This program provides an upper bound and a lower bound for the cumulative distribution function of the maximum. The lower bound is based on a time discretisation similar to the one of MGP. As for the upper bound, we consider the equation ( 8) and

• Use a trapezoidal rule to compute the integral using the same grid D n (T ) so that we limit our attention to the computation of integrands at points

t i = i/n, i ≤ n. • For each t i ∈ D n (T ), set Y = (X ′ (t i ), X(t 0 ), . . . , X(t i )) = (Y 1 , . . . , Y i+2 )
and note that

E X ′ (t i ) + 1 X(s)≤u∀s<t∈Dn(T ) ) X(t i ) = u p X(ti) (u) = +∞ 0 y 1 ]-∞,u] i+1 p Y (y 1 , . . . , y i+1 , u)dy 1 . . . dy i+1 ,
where p Y denotes the density of the Gaussian vector Y and let Σ i be its variance-covariance matrix.

• Use the change of variable

Y = Σ 1/2 i Z where Σ 1/2 i
is the square root of Σ i obtained via the Cholesky decomposition.

• Use the change of variable U j = Φ(Z j ) where the U j are independant and follow a uniform distribution for all choice of j. • Reduce the problem to the hypercube [0, 1] i . • Finally use MCQMC (Monte-Carlo Quasi Monte-Carlo) integration method on [0, 1] i . By this last method we mean that in general Quasi Monte-Carlo are very precise but do not give a reliable estimation of their error. To get this estimation we add an extra Monte-Carlo layer. See [START_REF] Genz | Numerical computation of multivariate normal probabilities[END_REF] and [START_REF] Azaïs | Computation of the distribution of the maximum of stationary Gaussian processes[END_REF] for more details.

Contrary to MGP, the non stationarity of χ do not allow to construt recursively the numerous variance-covariance matrices Σ i as i varies. By consequence, MAXNSGP is slower but more general. An exemple of application of MAXNSGP, available at [START_REF] Mourareau | [END_REF], is given in Section A.

Record method in dimension 2

{section:MAXNSGP2}

In dimension two, the record method is adapted to a Gaussian field defined on compact, convex subsets having a.s. a piecewise C 1 parametrization of the boundary. Here we limit our presentation to the case where the parameter set is the rectangle [0, T 1 ] × [0, T 2 ]. We use the lexicographic order defined by

s = (s 1 , s 2 ) ⊳ (t 1 , t 2 ) = t ⇐⇒ {s 2 < t 2 } or {s 2 = t 2 ; s 1 < t 1 }
and define the record set similarly to the one dimensional case with the relation ⊳ instead of <. Our aim is to apply a Kac-Rice formula on the rectangle, which imply the application of Kac-Rice formula in dimension higher than 1. To do so, set X ′ 1 = ∂X ∂t1 and define the following mild hypotheses for χ:

(A1) The sample paths of the random field Z = (X, X ′ 1 ) are a.s. C 1 . (A2) The distribution of Z(t) does not degenerate.

(A3) For every t ∈ R 2 , P(∃s ∈ S; Z(s) = t, det(Z ′ (s)) = 0) = 0.

Sufficient conditions for (A1)-(A3) are generic. They are almost always satisfied in the case of a process χ which is stationary. In that context, Mercadier [START_REF] Mercadier | Numerical bounds for the distributions of the maxima of some one-and two-parameter Gaussian processes[END_REF] proved the following theorem.

{record:dim:2} Theorem 2. With assumption above, for every u ∈ R, P(M T1,T2 > u) = P(X(0) > u)

+ 3 i=1 Bi E |X ′ T (t)|1 X(s)<X(t),∀s⊳t X(t) = u p X(t) (u)dσ(t) (9)
+ T1 0 T2 0 E X ′′ - 11 X ′ + 2 (t)1 X(s)<X(t),∀s⊳t (X, X ′ 1 )(t) = (u, 0) p X(t),X ′ 1 (t) (u, 0)dt, (10)
{record:dim2:2} {record:dim2:2}

where

• B 1 = [(0, 0), (T 1 , 0)], B 2 = [(0, 0), (0, T 2 )], B 3 = [(T 1 , 0), (T 1 , T 2 )
] are the three relevant boundaries of the rectangle: the fourth one does not contribute because of the conditions. • σ is the surfacic measure (here the length) on the boundary.

• X ′ i (t) = ∂X ∂ti , X ′′ 11 (t) = ∂ 2 X ∂t 2 1
and X ′ T denote the derivative of X along B i for i = 1, 2, 3.

From formula [START_REF] Cressie | Statistics for spatial data[END_REF], we can derive a numerical routine to bound P(M T1,T2 > u) for non-stationary non-centered differentiable Gaussian fields. This program uses the same tricks than MAXNSGP, the only main difference is that we don't consider the same grids to compute the two dimensional integral over [0, T 1 ] × [0, T 2 ] and to discretize the condition 1 X(s)<X(t), ∀s⊳t . As a matter of fact, this last discretization is time consuming and can be faster performed on a coarser grid without impacting the precision.

More precisely, let

G n1,n2 (T 1 , T 2 ) = t 1 i , t 2 j ; t 1 i = i n 1 T 1 , t 2 j = j n 2 , 0 ≤ i ≤ n 1 , 0 ≤ j ≤ n 2
be the grid for the trapezoidal rule and let m 1 dividing n 1 and m 2 dividing n 2 , then the subgrid of G n1,n2 (T 1 , T 2 ) defined for the discretization of the condition

X(s) < X(t), ∀s ⊳ t is D m1,m2 (T 1 , T 2 ) = s 1 k , s 2 l ; s 1 k = k m 1 T 1 , s 2 l = l m 2 T 2 , 0 ≤ k ≤ m 1 , 0 ≤ l ≤ m 2 .
In particular, for each point t of G n1,n2 (T 1 , T 2 ), we define the past of t (with respect to the order ⊳) as the set

P t = {s ; s ⊳ t} ∩ D m1,m2 (T 1 , T 2 )
in order to construct our numerical routine MAXNSGP2 similarly to MAXNSGP.

To do so, we consider the one dimensional algorithm of Section 3.2.1 on the vectors

• X ′ 1 t 1 i , 0 , P (t 1 i ,0) , X t 1 i , 0 0≤i≤n1 for B 1 , • X ′ 2 0, t j 2 , P (0,t j 2 ) , X 0, t j 2 0≤j≤n2 for B 2 , • X ′ 2 T 1 , t j 2 , P (T1,t j 2 ) , X T 1 , t j 2 0≤j≤n2
for B 3 , while the computation of the interior is based on a trapezoidal rule on the grid G n1,n2 (T 1 , T 2 ) combined with the previous algorithm on the vector

X ′ 1 t 1 i , t 2 j , X ′ 2 t 1 i , t 2 j , X ′′ 11 t 1 i , t 2 j , P (t 1 i ,t 2 j ) , X t 1 i , t 2 j 0≤i≤n1,0≤j≤n2
.

An exemple of application of MAXNSGP2, available at [START_REF] Mourareau | [END_REF], is given in Section B. Finally, note that MAGP by Mercadier [START_REF] Mercadier | MAGP Toolbox[END_REF][START_REF] Mercadier | Numerical bounds for the distributions of the maxima of some one-and two-parameter Gaussian processes[END_REF] is another version that uses the WAFO toolbox and that does not give an estimation of the error. As far as we can tell, MAGP was until recently the only program that concerns two-dimensional stationary Gaussian fields.

A program for the Poisson method

Finally, the Poisson approximation is illustrated by functions pPOISSON and qPOISSON, that can be found at [START_REF] Mourareau | [END_REF]. An example of utilization is given by p = qPoisson(0.05,10,1)

where α = 0.05 is the level fixed by the user, T = 10 is the length of the interval and γ 2 = 1.

Applications to Statistics {section:appli:stat}

In this section we want to consider the three classical approximations (with their variants) given in introduction and the Poisson approximation of Section 2.3.

Our aim is to compare these tools in various scenarii.

Simulation procedure

Our computation scheme is the following. For a given size of interval (or rectangle), we use the relevant approximation of Section 2 to compute a α approximate threshold u α and then we use the relevant numerical program to compute the corresponding actual level ᾱ. The approximation is said satisfactory is there is a small discrepancy between α and ᾱ.

APPLICATIONS TO STATISTICS

Such a kind of study has been performed in the numerical study of the domain of validity of the classical χ 2 test. Indeed, the well-known condition "the expected size of each cell must be greater than 5 " corresponds to conditions that ensures that the actual level ᾱ differs from the nominal level α by less than 20 % in relative variation (see [START_REF] Greenwood | A guide to chi-squared testing[END_REF]). In other words, for a nominal level α of 5% (resp 1%) we demand the actual level to be in (4%, 6%) (resp. (0.8%, 1.2%)).

In the interpretation of our numerical experiments we will generally consider a more restrictive rule of relative varaition of 5% instead of 20%. In ther words we will demand, for a nominal level of 5 %, (resp. 1%) the true level to belong to [4.75%, 5%] (resp.[0.95%, 1.05%] .

The processes {section:processes}

For the Pickands approximation,

• the stationary Gaussian process with covariance ρ(t) = exp(-t 2 /2) for which the result is known from the MGP program,

• the stationary Gaussian process with covariance ρ(t) = exp(-t 1.95 ) for which the result is approximatively known from the lower bound of the MGP program.

• the Ornstein-Uhlenbeck process with covariance ρ(t) = exp(-|t|) for which the result is exactly known from the pou function.

These are the few case where we are able to perform a numerical study for β processes.

For the Rice and the Poisson approximation, we consider three stationary processes with covariance

• ρ(t) = exp(-t 2 /2) , • ρ(t) = 1/ cosh(t) , • ρ(t) = sinc( √ 3t) = sin( √ 3t) √ 3t .
For the Euler characteristic, we consider isotropic and stationary Gaussian fields with covariances

• ρ(t) = exp(-t 2 /2), • ρ(t) = (1 + t ) exp(-t ) which is a Matern covariance with param- eter 3/2.
To give a precise meaning to the interpretation, we will always normalize our processes with ρ(0) = 1 and "unit speed transformation" -ρ ′′ (0) = 1.

General presentation of the figures

In a first part we consider the problem of determination of a threshold for test of level α = 0.01 and α = 0.05. In the second part we consider the problem of the determination of the median α = 0.5. Let us consider the example of Figure 3, the others being very similar. As a function of the length T of the set, in abscissa, a threshold u has been computed using the Rice bound given by (4) for α = 0.05. This bound is only approximative. So the true level is not really α and is measured using a numerical tool here MGP. Recall that the upper bound is in general closer to the reality. From left to right, the three processes of Section 4.2 are considered.

Results and comments

We give a list of the main points • Figures 3 and4 show that the Rice method is, in general, satisfactory for statistical application: the upper-bound in red which is the more precise is very close to the nominal value. • Figures 5, 6 and7 show that the form 2 or 3 of the Pickands methods perform for large time intervals but neither is preferable to the other in the general case. Moreover, for small intervals, both give too low levels.

If the heruristic correction for small intervals is performed, the method performs well in every conditions. Note that there is some instability for very small intervals which is studied Figure 8: a "zoom" of Figure 5. It shows that the local increase around T = 0 of the actual level is not that important. • Both Rice (Figure 2) and Pickands (not displayed) methods are not relevant to compute the median as soon as T is not very small. • In case of smooth processes, the Poisson method performs well in every situation. • The problem of computing the median of M T for β processes β = 2 remains open. • The Euler characteristic method performs rather well, except for α = 0.5 and very small rectangles. In our interpretation this is due to the fact that in almost every considered case the level u is high.

Conclusion

The present paper has shown that the classical bounds are efficient for statistical application as soon as the good version of Pickands method is used. However, they are not efficient for computing the median of the maximum. In that case, the proposed Poisson method performs well and is equivalent to the Rice method in the previous situations. In addition, Figure 14 show that it performs well for the computation of others fractiles. Finally, the Euler characteristic method is efficient in almost every cases. {toto}

Figure 1

 1 Figure1. Exact level of a test constructed for a nominal level of α = 5% using the Pickands method and the Ornstein-Uhlenbeck process. On the left: small intervals, on the right: large intervals (see Section 2). The level corresponding to equation (2) is given in brown while the one of equation (3) is given in green. Both are computed thanks to the pou program (see Section 3.1.1)

Figure 4 .Figure 5 .

 45 Figure 4. Same as Figure 3 except that α = 0.01.

Figure 6 .Figure 7 .Figure 8 .

 678 Figure 6. The process considered here has covariance ρ(t) = exp(-t1.95 ); Exact level of a test constructed for a nominal level of α = 1% (left) or α = 5% (right) using the Pickands method. Our numerical computation uses the lower bound of MGP, thus it provides an under estimation. The meaning of the colours and dashed lin is the same as in Figure5..

Figure 11 .

 11 Figure 11. Exact level of a test constructed for a nominal level of α = 5% using the Euler characteristic method (see Section 2) on subsets of [0, 50] 2 . The upper bound presented here is computed by the program MAXNSGP2 (see Section 3.1.1) and the processes considered have covariance function ρ(t) = exp(-t 2 /2) (left) and ρ(t) = (1 + t ) exp(-t ) (right).

Figure 12 .Figure 13 .

 1213 Figure 12. Same as Figure 11 except that : The covariance is always ρ(t) = exp(-t 2 /2) and α = 0.01 (left) and α = 0.5 (right) {figure10}

  {figure11}

Figure 14 .

 14 Figure 14. Exact probability of M T to be above the approximate 1 -0.40 (left top); 1 -0.70 (right top) 1 -0.30 (left bottom) 1 -0.60 (right bottom) fractile computed with the Poisson method (see Section 2) on subsets of [0, 50]. An upper bound (red line) and a lower bound (blue line) of the level are computed by the program MGP (see Section 3.1.1). The process considered has covariance function ρ(t) = exp(-t 2 /2).

  Figure 10. Exact probability of M T to be above the approximate median computed with the Poisson method (see Section 2). An upper bound (red line) and a lower bound (blue line) of the probability are computed by MGP (see Section 3.1.1). From left to right, the processes considered have covariance function ρ(t) = exp(-t 2 /2), ρ(t) = 1/ cosh(t) and ρ(t) =

							0.53													0.55												0.55
																				0.54												0.54
							0.52
																				0.53												0.53
																				0.52												0.52
							0.51
																				0.51												0.51
							0.5													0.5												0.5
																				0.49												0.49
							0.49
																				0.48												0.48
																				0.47												0.47
							0.48
																				0.46												0.46
							0.47	0	5	10	15	20	25	30	35		40	45	50	0.45	0	5	10	15	20	25	30	35	40	45	50	0.45	0	5	10	15	20	25	30	35	40	45	50
							sinc( √	3t).
	4.4.2. Computing the median of M T
	0.56													0.56													0.56
	0.54													0.54													0.54
	0.52													0.52													0.52
	0.5													0.5													0.5
	0.48													0.48													0.48
	0.46													0.46													0.46
	0.44													0.44													0.44
	0.42													0.42													0.42
	0.4													0.4													0.4
	0.38	0	5	10	15	20	25	30	35	40	45	50	0.38	0	5	10	15	20	25	30	35	40	45	50	0.38	0	5	10	15	20	25	30	35	40	45	50
	Figure 9. Exact level of a test constructed for a nominal level of α = 50% using the Rice
	method (see Section 2). An upper bound (red line) and a lower bound (blue line) of the level
	are computed by the program MGP (see Section 3.1.1). From left to right, the processes con-sidered have covariance function ρ(t) = exp(-t 2 /2), ρ(t) = 1/ cosh(t) and ρ(t) = sinc( √ 3t).	{figure6}

Appendix A: MAXNSGP {section:annex1}

The syntax of MGP and MAXNSGP are very similar. For instance, to compute the probability that the maximum of the Gaussian process satisfying m(t) = 4 and r(s, t) = e st -1

exceed the level u = 1 on the set [START_REF] Adler | The geometry of random fields[END_REF][START_REF] Jmp Albin | A new proof of an old result by Pickands[END_REF], it reads as r = @(t,s) (exp(s.*t) -1)./sqrt(exp(s.^2)-1)./sqrt(exp(t.^2)-1)

[p_low,p_upp,e_low,e_upp] = MAXNSGP(12,10,5000,1,@(t) sqrt(t), @(t,s) r(t,s),1,3) where • 12 is the typical value for the number of Monte-Carlo (MC) replications in the MCQMC integration; • 10 is the number of discretization point of the interval [START_REF] Adler | The geometry of random fields[END_REF][START_REF] Jmp Albin | A new proof of an old result by Pickands[END_REF]; • 5000 is the number of points used for the QMC routine;

• 1 is the level of comparison; • @(t) sqrt(t) is the mean function of the process; • @(t,s) r(t,s) is the covariance function of the process; • 1 is the beginning of the interval; • 3 is the end of the interval.

Note that this kind of processes appear in the asymptotic theory of Likelihood Ratio Test in the case of a mixture, see [START_REF] Delmas | On likelihood ratio tests in Gaussian mixture models[END_REF] for more details. In case of discrepancy, the upper-bound is, in general, more precise. Finally, the upper-bound is very efficient even for small values of n.

Appendix B: MAXNSGP2 {section:annex2}

Again, the syntax is very similar to the one of MGP and MAXNSGP. For instance, to compute the probability that the maximum of the Gaussian field satisfying m(t) = 0 and ∀(s, t) ∈ (R 2 ) 2 , r(s, t) = exp(-t -s 2

2 ) exceed the level u = 2 on [0, 5] × [0, 10], it reads as

[p_low,p_upp,e_low,e_upp] = MAXNSGP2 [START_REF] Delmas | On likelihood ratio tests in Gaussian mixture models[END_REF][START_REF] Michna | Remarks on Pickands theorem[END_REF]40,[START_REF] Adler | Random fields and geometry[END_REF][START_REF] Aldous | Probability approximations via the Poisson clumping heuristic[END_REF],1000,2,@(t,s) 0,r,5,10) where • 12 is the typical value for the number of Monte-Carlo (MC) replications in the MCQMC integration; • 20 is the number of discretization points for trapezoidal rule on the x-axis; • 40 is the number of discretization points for trapezoidal rule on the y-axis; • 2 is the number of points used for the record set on the x-axis; • 4 is the number of points used for the record set on the y-axis; • 1000 is the number of points used for the QMC routine;

• 2 is the level of comparison; • @(t,s) 0 is the mean function of the process; • r is the covariance function of the process; • 5 and 10 denote the length of each side of the rectangle, here [0, 5]× [0, 10]; Note again that in case of discrepancy, the upper-bound is, in general, more precise.

Appendix C: Maxgpbox {section:annex3}

Maxgpbox is a collection of matlab programs available at [START_REF] Azaïs | [END_REF] that performs exact calculations for the few cases for which an exact formula is known. It includes in particular the pou function devoted to the Ornstein-Uhlenbeck process.