Jürg Hüsler

Extremes of a certain class of Gaussian processes

We consider the extreme values of fractional Brownian motions, self-similar Gaussian processes and more general Gaussian processes which have a trend -ct[beta] for some constants c,[beta]>0 and a variance t2H. We derive the tail behaviour of these extremes and show that they occur mainly in the neighbourhood of the unique point t0 where the related boundary function (u+ct[beta])/tH is minimal. We consider the case that H

On multivariate Gaussian tails

Let {Xn, n≥1} be a sequence of standard Gaussian random vectors in ℝd,d ≥ 2. In this paper we derive lower and upper bounds for the tail probabilityP{Xn>tn} withtn ∈ ℝd some threshold. We improve for instance bounds on Mills ratio obtained by Savage (1962,J. Res. Nat. Bur. Standards Sect. B,66, 93–96). Furthermore, we prove exact asymptotics under fairly general conditions on bothXn andtn, as ‖tn‖→∞ where the correlation matrix Σn ofXn may also depend onn.

Failures and complications in patients with birth defects restored with fixed dental prostheses and single crowns on teeth and/or implants

OBJECTIVES To assess retrospectively, over at least 5 years, the incidences of technical and biological complications and failures in young adult patients with birth defects affecting the formation of teeth. MATERIAL AND METHODS All insurance cases with a birth defect that had crowns and fixed dental prostheses (FDPs) inserted more than 5 years ago were contacted and asked to participate in a reexamination. RESULTS The median age of the patients was 19.3 years (range 16.6-24.7 years) when prosthetic treatment was initiated. Over the median observation period of 15.7 years (range 7.4-24.9 years) and considering the treatment needs at the reexamination, 19 out of 33 patients (58%) with reconstructions on teeth remained free from all failures or complications. From the patients with FDPs and single unit crowns (SCs) on implants followed over a median observation period of 8 years (range 4.6-15.3 years), eight out of 17% or 47% needed a retreatment or repair at some point due to a failure or a complication. From the three groups of patients, the cases with amelogenesis/dentinogenesis imperfecta demonstrated the highest failure and complication rates. In the cases with cleft lip, alveolus and palate (CLAP) or hypodontia/oligodontia, 71% of the SCs and 73% of the FDPs on teeth (FDP T) remained complication free over a median observation period of about 16 years. Sixty-two percent of the SCs and 64% of the FDPs on implants remained complication free over 8 years. Complications occurred earlier with implant-supported reconstructions. CONCLUSIONS Because healthy, pristine teeth can be left unprepared, implant-supported SCs and FDPs are the treatment choice in young adults with birth defects resulting in tooth agenesis and in whom the edentulous spaces cannot be closed by means of orthodontic therapy. However, the trend for earlier and more frequent complications with implant-supported reconstructions in young adults, expecting many years of function with the reconstructions, has to be weighed against the benefits of keeping teeth unprepared. In cases with CLAP in which anatomical conditions render implant placement difficult and in which teeth adjacent to the cleft require esthetic corrections, the conventional FDP T still remains the treatment of choice.

Asymptotics of a boundary crossing probability of a Brownian bridge with general trend

Let us consider a signal-plus-noise model γh(z)+B0(z), z ∈ [0,1], where γ > 0, h: [0,1] → ℝ, and B0 is a Brownian bridge. We establish the asymptotics for the boundary crossing probability of the weighted signal-plus-noise model for γ→∞, that is P (supzε [0,1]w(z)(γ h(z)+B0(z))>c), for γ→∞, (1) where w: [0,1]→ [0,∞ is a weight function and c>0 is arbitrary. By the large deviation principle one gets a result with a constant which is the solution of a minimizing problem. In this paper we show an asymptotic expansion that is stronger than large deviation. As byproduct of our result we obtain the solution of the minimizing problem occurring in the large deviation expression. It is worth mentioning that the probability considered in (1) appears as power of the weighted Kolmogorov test applied to the test problem H0: h≡ 0 against the alternative K: h>0 in the signal-plus-noise model.

Exact asymptotics for Boundary crossings of the brownian bridge with trend with application to the Kolmogorov test

We consider a boundary crossing probability of a Brownian bridgeB0 and a piecewise linear boundary functionu(t)−γh(t). The main result of this paper is an asymptotic expansion for γ→∞ of the boundary crossing probability thatB0(t) is larger than the piecewise linear boundary functionu(t)−γh(t) for somet. Such probabilities occur for instance in the context of change point problems when the Kolmogorov test is used. Examples are discussed showing that the approximation is rather accurate even for small positive γ values.

Multivariate extreme values in stationary random sequences

The limit distributions of multivariate extreme values of stationary random sequences are associated under mild mixing conditions. Also sufficient conditions are given such that the limit has independent components. Examples indicate that these results do not hold for general stationary sequences.

On asymptotics of multivariate integrals with applications to records

Let { X n ,n≥1} be a sequence of iid. Gaussian random vectors in R d , d≥2, with nonsingular distribution function F. In this paper the asymptotics for the sequence of integrals I F,n (G n )≔n∫ R d G n n−1( X ) dF( X ) is considered with G n some distribution function on R d . In the case G n =F the integral I F,n (F)/n is the probability that a record occurs in X 1,…, X n at index n. [1] obtained lower and upper asymptotic bounds for this case, whereas [2] showed the rate of convergence if d=2. In this paper we derive the exact rate of convergence of I F,n (G n ) for d≥2 under some restrictions on the distribution function G n . Some related results for multivariate Gaussian tails are discussed also.

Cumulative costs for the prosthetic reconstructions and maintenance in young adult patients with birth defects affecting the formation of teeth

OBJECTIVES To assess retrospectively the cumulative costs for the long-term oral rehabilitation of patients with birth defects affecting the development of teeth. METHODS Patients with birth defects who had received fixed reconstructions on teeth and/or implants > or =5 years ago were asked to participate in a comprehensive clinical, radiographic and economic evaluation. RESULTS From the 45 patients included, 18 were cases with a cleft lip and palate, five had amelogenesis/dentinogenesis imperfecta and 22 were cases with hypodontia/oligodontia. The initial costs for the first oral rehabilitation (before the age of 20) had been covered by the Swiss Insurance for Disability. The costs for the initial rehabilitation of the 45 cases amounted to 407,584 CHF (39% for laboratory fees). Linear regression analyses for the initial treatment costs per replaced tooth revealed the formula 731 CHF+(811 CHF x units) on teeth and 3369 CHF+(1183 CHF x units) for reconstructions on implants (P<.001). Fifty-eight percent of the patients with tooth-supported reconstructions remained free from failures/complications (median observation 15.7 years). Forty-seven percent of the patients with implant-supported reconstructions remained free from failures/complications (median observation 8 years). The long-term cumulative treatment costs for implant cases, however, were not statistically significantly different compared with cases reconstructed with tooth-supported fixed reconstructions. Twenty-seven percent of the initial treatment costs were needed to cover supportive periodontal therapy as well as the treatment of technical/biological complications and failures. CONCLUSION Insurance companies should accept to cover implant-supported reconstructions because there is no need to prepare healthy teeth, fewer tooth units need to be replaced and the cumulative long-term costs seem to be similar compared with cases restored on teeth.

Estimation of Tails and Related Quantities Using the Number of Near-Extremes

Abstract In an insurance context, consider {Xn, n ≥ 1} random claim sizes with common distribution function F and {N(t), t ≥ 0} an integer valued stochastic process tfiat counts the number of claims occurring during the time interval [0, t]. Based on the number of near-extremes which are the observations Xi near the largest or the mth largest observation, we derive in this article a strongly consistent estimator of upper tails of X1. Furthermore, estimators for both the tail index and the upper endpoint are introduced when F is a generalized Pareto distribution. Asymptotic normal law for the proposed estimators is additionally presented.

Extremes of Gaussian Processes with Maximal Variance near the Boundary Points

Let X(t), t∈[0,1], be a Gaussian process with continuous paths with mean zero and nonconstant variance. The largest values of the Gaussian process occur in the neighborhood of the points of maximum variance. If there is a unique fixed point t0 in the interval [0,1], the behavior of P{supt∈[0,1] X(t)>u} is known for u→∞. We investigate the case where the unique point t0 = tu depends on u and tends to the boundary. This is reasonable for a family of Gaussian processes Xu(t) depending on u, which have for each u such a unique point tu tending to the boundary as u→∞. We derive the asymptotic behavior of P{supt∈[0,1] X(t)>u}, depending on the rate as tu tends to 0 or 1. Some applications are mentioned and the computation of a particular case is used to compare simulated probabilities with the asymptotic formula. We consider the exceedances of such a nonconstant boundary by a Ornstein-Uhlenbeck process. It shows the difficulties to simulate such rare events, when u is large.