Louis-Paul Rivest

Statistical Inference Procedures for Bivariate Archimedean Copulas

Abstract A bivariate distribution function H(x, y) with marginals F(x) and G(y) is said to be generated by an Archimedean copula if it can be expressed in the form H(x, y) = ϕ–1[ϕ{F(x)} + ϕ{G(y)}] for some convex, decreasing function ϕ defined on [0, 1] in such a way that ϕ(1) = 0. Many well-known systems of bivariate distributions belong to this class, including those of Gumbel, Ali-Mikhail-Haq-Thelot, Clayton, Frank, and Hougaard. Frailty models also fall under that general prescription. This article examines the problem of selecting an Archimedean copula providing a suitable representation of the dependence structure between two variates X and Y in the light of a random sample (X 1, Y 1), …, (X n , Y n ). The key to the estimation procedure is a one-dimensional empirical distribution function that can be constructed whether the uniform representation of X and Y is Archimedean or not, and independently of their marginals. This semiparametric estimator, based on a decomposition of Kendalls tau statistic...

On the multivariate probability integral transformation

A general formula is given for computing the distribution function K of the random variable H(X,Y) obtained by taking the bivariate probability integral transformation (BIPIT) of a random pair (X,Y) with distribution function H. Of particular interest is the behavior of the sequence (Kn) corresponding to the BIPIT of pairs (Xn,Yn) of componentwise maxima Xn=max(X1,...,Xn) and Yn=max(Y1, ..., Yn) of random samples (X1,Y1),...,(Xn,Yn) from distribution H. Illustrations are provided and the potential for statistical application is outlined. Multivariate extensions are briefly considered.

A CHARACTERIZATION OF GUMBEL'S FAMILY OF EXTREME VALUE DISTRIBUTIONS

In this note, a family of multivariate extremal distributions proposed by Gumbel (1960) is characterized among those whose dependence function is an Archimedean copula. The domains of attraction of Gumbels distributions are also determined within this class.

The low performance of forest versus rural coyotes in northeastern North America: Inequality between presence and availability of prey

Abstract Coyotes, which originate from central and southwestern North America, recently extended their range into forests of the Northeast. Forest coyotes occur in lower densities, have lower body reserves, and consume more fruits during summer than their counterparts occupying adjacent rural landscapes. We hypothesised that the forest landscape offered less animal prey to coyotes during summer than did the rural landscape. Coyote densities were higher in the rural landscape (2.7 animals 10 km-2) than in the forest landscape (0.5 animals 10 km-2) during the summer of 1997. During the summers of 1996 and 1997, coyotes in both landscapes fed mainly on wildberries (< 45% of dry matter intake), small mammals (< 10%), and snowshoe hare (< 10%). The biomass of the most abundant animal prey, snowshoe hares, was greater in the forest landscape (1.24 and 1.53 kg ha-1 in 1996 and 1997, respectively) than in the rural landscape (0.46 and 0.40 kg ha-1 in corresponding years). The biomass of the other major animal prey (small mammals), was comparable in both landscapes but irrupted during the second summer (0.09 and 0.50 kg ha-1 in 1996 and 1997, respectively). The biomass of fruits remained relatively constant in the rural landscape during the summers of 1996 and 1997 (ª 6 kg ha-1), but it tripled in the forest landscape during the second year (1.69 kg ha-1 in 1996 versus 5.30 kg ha-1 in 1997). Contrary to our prediction, the availability of animal prey in the forest landscape exceeded that in the rural landscape. Our results illustrate that the presence of prey does not correspond to its availability to predators. Coyotes appear poorly adapted for hunting in dense forest vegetation during summer and compensate for shortage of animal prey by consuming more berries.

Using orientation statistics to investigate variations in human kinematics

This paper applies orientation statistics to investigate variations in upper limb posture of human subjects drilling at six different locations on a vertical panel. Some of the drilling locations are kinematically equivalent in that the same posture could be used for these locations. Upper limb posture is measured by recording the co‐ordinates of four markers attached to the subjects hand, forearm, arm and torso. A 3×3 rotation characterizes the relative orientation of one body segment with respect to another. Replicates are available since each subject drilled at the same location five times. Upper limb postures for the six drilling locations are compared by one‐way analysis‐of‐variance tests for rotations. These tests rely on tangent space approximations at the estimated modal rotation of the sample. A parameterization of rotations in terms of unit quaternions simplifies the computations. The analysis detects significant differences in posture between all pairs of drilling locations. The smallest changes, less than 10° at all joints, are obtained for the kinematically equivalent pairs of locations. A short discussion of the biomechanical interpretation of these findings is presented.

Calving photocensus of the Rivière George Caribou Herd and comparison with an independent census

Vertical photographs of the calving grounds have been used since 1984 to estimate the caribou (Rangifer tarandus) population of the Riviere George Caribou Herd (RGCH) in Northern Quebec and Labrador. In spite of large confidence intervals, the 1984 and 1988 estimates suggested that the herd stabilized at more than 650 000 caribou (fall estimate including calves) making the RGCH the largest caribou herd in the world. Between 1984 and 1990, studies suggested that the former rapid growth of the herd deteriorated the calving and summer habitats. This poor habitat quality affected physical condition, pregnancy rate and calf survival. It was important to have a valid estimate of the herd size and a photocensus was done in June 1993. Contrary to previous censuses, a slightly different sampling design was applied in 1993. Two methods were used to estimate the number of females in the June population. In the first method, the number of females was derived from the estimated number of calves on the photographs and from the June female/calf ratio. The second method was used in the previous census and is based on the number of adults on the photos and on the June female/adult ratio. It is suggested that the first method of estimating female abundance in June is better due to sampling problems associated with a strong adult sex segregation during calving. From the first method, the herd size in October 1993 was estimated at 583 829 adults (±33.79%) and at 749 869 caribou including calves (±33.15%) while the second method provided estimates of 764 221 adults (±23.55%) and 981 565 caribou including calves (±22.64%). It was possible to compare those population estimates with an independent census. In July 1993, an oblique photocensus of the post-calving aggregations was conducted by Russell et al. (1996). A new analysis of their raw data provided an estimate of 608 384 adults (±14.35%). Both estimates from the June and July photocensus were combined. From the first and second method respectively, combined herd size estimates were 775 891 (±13.40%) and 823 375 (±12.36%) caribou including calves. The management implications are discussed and it is emphasized that the herd is still underharvested.

Outlier Resistant Alternatives to the Ratio Estimator

Abstract Many techniques have been suggested to lower the impact of outliers on sample survey estimates. Outliers can be downweighted by winsorization; that is, by replacing extreme data points by a data-dependent or a predetermined value before calculating estimates. Another approach is to reduce the weight of outliers, from the inverse sampling fraction to 1, in the estimation of population characteristics. In this article the problem of estimating the population mean using auxiliary information in the presence of outliers is considered. A resistant version of the ratio estimator is introduced. It is constructed with a M or a GM estimator of the slope of the regression model through the origin, which is implicitly called on when considering the ratio estimator. The asymptotic biases and asymptotic variances of the proposed alternatives to the ratio estimator are calculated with respect to the randomization of the sampling plan. The selection of a resistant estimator is seen to involve a trade-off betwee...

A distribution for dependent unit vectors

A bivariate generalization of the Fisher-von Mises distribution is introduced. The moments and limiting forms of the new distribution are derived. Then procedures to calculate consistent and efficient estimates of the parameters are proposed. New tests of independence are constructed and a numerical example is presented. A special attention is given to instances where the individual samples are highly clustered since they tend to occur often in applications. For such cases simple approximations to the maximum likelihood estimates and reliable small sample tests of independence are provided.

Redefining the maximum sustainable yield for the Schaefer population model including multiplicative environmental noise

The focus of this article is to investigate the biological reference points, such as the maximum sustainable yield (MSY), in a common Schaefer (logistic) surplus production model in the presence of a multiplicative environmental noise. This type of model is used in fisheries stock assessment as a first-hand tool for biomass modelling. Under the assumption that catches are proportional to the biomass, we derive new conditions on the environmental noise distribution such that stationarity exists and extinction is avoided. We then get new explicit results about the stationary behavior of the biomass distribution for a particular specification of the noise, namely the biomass distribution itself and a redefinition of the MSY and related quantities that now depend on the value of the variance of the noise. Consequently, we obtain a more precise vision of how less optimistic the stochastic version of the MSY can be than the traditionally used (deterministic) MSY. In addition, we give empirical conditions on the error variance to approximate our specific noise by a lognormal noise, the latter being more natural and leading to easier inference in this context. These conditions are mild enough to make the explicit results of this paper valid in a number of practical applications. The outcomes of two case-studies about northwest Atlantic haddock [Spencer, P.D., Collie, J.S., 1997. Effect of nonlinear predation rates on rebuilding the Georges Bank haddock (Melanogrammus aeglefinus) stock. Can. J. Fish. Aquat. Sci. 54, 2920-2929] and South Atlantic albacore tuna [Millar, R.B., Meyer, R., 2000. Non-linear state space modelling of fisheries biomass dynamics by using Metropolis-Hastings within-Gibbs sampling. Appl. Stat. 49, 327-342] are used to illustrate the impact of our results in bioeconomic terms.

Aerial surveys for white-tailed deer with the double-count technique in Québec: two 5-year plans completed

Abstract Aerial surveys can provide direct density estimates for ungulates over large areas such as hunting zones, a prerequisite for intensive management. In 1991 Québec implemented double-count aerial surveys as part of its white-tailed deer (Odocoileus virginianus) management program, on a hunting-zone basis. This technique involves 2 independent observers located on the same side of an aircraft who simultaneously count animals in sample plots. Two 5-year survey plans have been successfully completed for the whole province. Deer densities for most zones could be estimated with a ±20% CI (P = 0.90), while the survey provided less precise evaluations for subunits or individual wintering areas. A typical survey for a hunting zone with 200 plots (5 km × 60 m) required approximately 30–40 helicopter hours and cost