Inclusion of a decoy alternative dominated by a target option, but not its competitor, typically leads to increased choice for the target over the competitor, known as the attraction effect. However, the reverse sometimes occurs, known as the repulsion effect. This research tested factors that moderate the repulsion effect in preferential choice scenarios with numerical attributes. Experiment 1 used a between-subjects design with a small set of consumer products and demonstrated robust repulsion effects that did not depend on the relative similarity of the decoy and target. Experiments 2 and 4 used a more powerful within-subjects design along with an expanded set of products and showed that repulsion effects were generally enhanced when the decoy and target had more similar attributes; however, the moderating effect of decoy–target similarity appeared to be fragile and sensitive to stimulus presentation factors. These findings provided mixed support for the hypothesis that the target is tainted by its proximity to the decoy. Experiments 3 and 5 tested whether the extremity of values on the attribute favoring the target moderates the repulsion effect. The results demonstrated that repulsion is more likely when all the alternatives have extremely high values on the target’s better attribute. Extremity of attribute values on the dimension favoring the target may result in a categorical assessment along that dimension and shift focus to the attribute favoring the competitor as one way to foster the repulsion effect.

We investigate the large values of the derivatives of the Riemann zeta function $\zeta (s)$ on the 1-line. We give a larger lower bound for $\max _{t\in [T,2T]}|\zeta ^{(\ell )}(1+{i} t)|$, which improves the previous result established by Yang [‘Extreme values of derivatives of the Riemann zeta function’, Mathematika 68 (2022), 486–510].

An asymptotic model for the extreme behavior of certain Markov chains is the ‘tail chain’. Generally taking the form of a multiplicative random walk, it is useful in deriving extremal characteristics, such as point process limits. We place this model in a more general context, formulated in terms of extreme value theory for transition kernels, and extend it by formalizing the distinction between extreme and nonextreme states. We make the link between the update function and transition kernel forms considered in previous work, and we show that the tail chain model leads to a multivariate regular variation property of the finite-dimensional distributions under assumptions on the marginal tails alone.

The P.O.T. method (Peaks Over Threshold) consists in using the generalized Pareto distribution (GPD) as an approximation for the distribution of excesses over a high threshold. In this work, we use a refinement of this approximation in order to estimate second order parameters of the model using the method of probability-weighted moments (PWM): in particular, this leads to the introduction of a new estimator for the second order parameter ρ, which will be compared to other recent estimators through some simulations. Asymptotic normality results are also proved. Our new estimator of ρ looks especially competitive when  |ρ|  is small.

In this paper we propose an additive mixture model, where one component is the Generalized Pareto distribution (GPD) that allows us to estimate extreme quantiles. GPD plays an important role in modeling extreme quantiles for the wide class of distributions belonging to the maximum domain of attraction of an extreme value model. One of the main difficulty with this modeling approach is the choice of the threshold u, such that all observations greater than u enter into the likelihood function of the GPD model. Difficulties are due to the fact that GPD parameter estimators are sensible to the choice of u. In this work we estimate u, and other parameters, using suitable priors in a Bayesian approach. In particular, we propose to model all data, extremes and non-extremes, using a semiparametric model for data below u, and the GPD for the exceedances over u. In contrast to the usual estimation techniques for u, in this setup we account for uncertainty on all GPD parameters, including u, via their posterior distributions. A Monte Carlo study shows that posterior credible intervals also have frequentist coverages. We further illustrate the advantages of our approach on two applications from insurance.

It is generally agreed that cephalometric analyses of the Americans Ricketts, Steiner, and Tweed are of questionable utility. The C.R.A.N.I.O.M group has formulated a new method of using cephalometric measurements that it believeswill be moderately helpful to orthodontists in making their diagnoses. We analyzed 83 young adults in Class I occlusion who had had no orthodontic treatment.

The most interesting new formulation of our plan was to examine the extremes of the variables that we considered in this population. These figures constitutedlimits that differed widely fromeach other: there wasmore than a 30°gap betweenthe measurement of the incisor inclined furthest buccally and the one inclined mostlingually (or palatally).

Accordingly, for these lower anterior teeth inclinations to Downs’s mandibularplane ranging from 78 to 114°were acceptable. And for the maxillary incisorsinclinations to the Frankfort plane varying from 97.5 to 130.1° are considered to be ina standard range. The need to reposition incisors in order to conform towhat we cannow see as an abusively rigid normal accordingly occurs much less frequently.

For this reason, orthodontists will find far fewer indications for the extraction of bicuspid teeth than they would in rigidly adhering to the standards imposed by the Ricketts, Steiner, and Tweed cephalometric analyses. We believe that measurements of skeletal, osseous structures merely differentiate between various typologies and do not describe forms that constitute anomalies.

The C.R.A.N.I.O.M group affirms that cephalometrics occupies a position in thearray of orthodontic diagnostic tools that comes after the assessment of esthetic,periodontal status, and muscular equilibrium.

Today there is near universal agreement that the cephalometric analyses presented by the Americans Ricketts, Steiner, and Tweed need to be re-evaluated and brought up to date. To this end, our C.R.A.N.I.O.M group has prepared a new method for using cephalometric numerical and angular readings in the preparation of a reasonable and well-reasoned aid to diagnosis.

We analyzed the data from a sample of 83 young adults with Class I occlusions, none of whom had ever received orthodontic treatment.

In our work the most striking new consideration that we employed was to use the extreme values for each variable of this sample. These figures constitute extreme parameter limits very distant from each other: for example there is a gap of 30° between the most buccally inclined incisor and the most lingually, or palatally, inclined.

Therefore, we found incisor inclination to be acceptable within a range of 78 to 114° to Downs’s mandibular plane on the lower arch and from 97.5 to 130.1° to the Frankfort plane for the upper arch. So in our reasoning there is far less need to reposition these teeth in conformity to an arbitrary cephalometric norm.

And, for the same reason, we find far fewer cases requiring extraction in order to allow orthodontists to achieve results that satisfy abusive standards established by Ricketts, Steiner, and Tweed. The osseous measurements that we report simply describe the forms we observed, none of which constitute anomalies but, instead, describe a particular type.

The C.R.A.N.I.O.M group believes that cephalometric analysis occupies a place in orthodontic record taking that comes after those occupied by esthetics, periodontal status, and muscular balance.

In this paper, we give sufficient conditions to establish central limittheorems and moderate deviation principle for a class of support estimates ofempirical and Poisson point processes. The considered estimates are obtained bysmoothing some bias corrected extreme values of the point process. We show howthe smoothing permits to obtain Gaussian asymptotic limits and thereforepointwise confidence intervals. Some unidimensional and multidimensionalexamples are provided.

We present a method for estimating the edge of a two-dimensionalbounded set, given a finite random set of points drawn from the interior.The estimator is based both on a Parzen-Rosenblatt kernel and extreme values of point processes. We give conditionsfor various kinds of convergence and asymptotic normality.We propose a method of reducing the negative bias and edge effects,illustrated by some simulations.

The focus of our attention is the limit distribution of the sum of independent and identically distributed random vectors from which all the extreme summands are removed. The problem is rather trivial if the summands are ordered by their norms. It is of much more interest when the vertices of the convex hull generated by the vectors are taken as the extremes.

We study estimation of the tail-decay parameter of the marginal distribution corresponding to a discrete-time, real-valued stationary stochastic process. Assuming that the underlying process is short-range dependent, we investigate properties of estimators of the tail-decay parameter which are based on the maximal extreme value of the process observed over a sampled time interval. These estimators only assume that the tail of the marginal distribution is roughly exponential, plus some modest ‘mixing’ conditions. Consistency properties of these estimators are established, as well as minimax convergence rates. We also provide some discussion on estimating the pre-exponent, when a more refined tail asymptotic is assumed. Properties of a certain moving-average variant of the extremal-based estimator are investigated as well. In passing, we also characterize the precise dependence (mixing) assumptions that support almost-sure limit theory for normalized extreme values and related first-passage times in stationary sequences.

The P.O.T. (Peaks-Over-Threshold) approachconsists of using the Generalized ParetoDistribution (GPD)to approximate the distribution of excesses over a threshold.We use the probability-weighted momentsto estimate the parameters of the approximating distribution.We study the asymptotic behaviour ofthese estimators (in particular their asymptotic bias) and also thefunctional bias of the GPD as an estimate of thedistribution function of the excesses. We adapt penultimateapproximation results to the case where parameters are estimated.

Empirical point processes of exceedances play an important role in extreme value theory, and their limiting behaviour has been extensively studied. Here, we provide explicit bounds on the accuracy of approximating an exceedance process by a compound Poisson or Poisson cluster process, in terms of a Wasserstein metric that is generally more suitable for the purpose than the total variation metric. The bounds only involve properties of the finite, empirical sequence that is under consideration, and not of any limiting process. The argument uses Bernstein blocks and Lindeberg's method of compositions.

Let {Xn, n ≥ 1} be independent identically distributed random variables with a common non-degenerate distribution function F. For each n ≥ 1, denote Mn = max {X1,…, Xn}. Under certain conditions on F, there exist constants an > 0 and bn ∈ R such that . In this paper, we shall show that {(Mn – bn)/an} exhibits ergodic behaviour under additional conditions of F.

A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.

This paper uses the Rice method [18] to give bounds tothe distribution of the maximum of a smooth stationary Gaussianprocess. We give simpler expressions of the first two terms ofthe Rice series [3,13] for the distribution of the maximum. Our main contribution is a simpler form of the second factorial momentof the number of upcrossings which is in some sense a generalizationof Steinberg et al.'s formula ([7] p. 212). Then, we present a numerical application and asymptotic expansionsthat give a new interpretation of a result by Piterbarg [15].

Let n points be placed independently in ν-dimensional space according to the standard ν-dimensional normal distribution. Let Mn be the longest edge-length of the minimal spanning tree on these points; equivalently let Mn be the infimum of those r such that the union of balls of radius r/2 centred at the points is connected. We show that the distribution of (2 log n)1/2Mn - bn converges weakly to the Gumbel (double exponential) distribution, where bn are explicit constants with bn ~ (ν - 1)log log n. We also show the same result holds if Mn is the longest edge-length for the nearest neighbour graph on the points.

Extreme value results for a class of shot noise processes with heavy tailed amplitudes are considered. For a process of the form, , where {τ k} are the points of a renewal process and {Ak} are i.i.d. with d.f. having a regularly varying tail, the limiting behavior of the maximum is determined. The extremal index is computed and any value in (0, 1) is possible. Two-dimensional point processes of the form are shown to converge to a compound Poisson point process limit. As a corollary to this result, the joint limiting distribution of high local maxima is obtained.

Shot noise processes form an important class of stochastic processes modeling phenomena which occur as shocks to a system and with effects that diminish over time. In this paper we present extreme value results for two cases — a homogeneous Poisson process of shocks and a non-homogeneous Poisson process with periodic intensity function. Shocks occur with a random amplitude having either a gamma or Weibull density and dissipate via a compactly supported impulse response function. This work continues work of Hsing and Teugels (1989) and Doney and O'Brien (1991) to the case of random amplitudes.