Fonctions de Morse sur les espaces exotiques

Summary It is shown that on an exotique space V_2n−1 of Kervaire type, we can construct a Morse function having only four critical points, the maximum, the minimum and two points of index n−1 and n.

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Entrata in Redazione il 20 ottobre 1972.     

Measuring income inequalities beyond the Gini coefficient

Growing interest in the analysis of interrelationships between income distribution and economic growth has recently stimulated new theoretical and empirical research. Measures such as the head-count ratio for the poverty index or the widely used Gini coefficient are aggregated indicators describing the general extent of inequality without deeper insights into income distribution among households. To derive an indicator accounting for income distribution among income groups, we propose a new approach based on an output oriented DEA model where the input value is unitized to 1 for each country and weights restrictions imposed so as to favour a higher income share in the lower quantiles. We demonstrate the merits of this approach on the quintile income breakdown data of 29 European countries. Prioritizing lower income groups’ welfare, countries such as Slovenia and Slovakia can be equally favoured by the new proposed indicator while being assessed differently by the Gini index. An intertemporal analysis reveals a slight deterioration of income distribution in the majority of 29 European countries over the period of 2007–2016 in a Rawlsian sense.

     

A max-conflicts based heuristic search for the stable marriage problem with ties and incomplete lists

In this paper, we propose a heuristic search algorithm based on maximum conflicts to find a weakly stable matching of maximum size for the stable marriage problem with ties and incomplete lists. The key idea of our approach is to define a heuristic function based on the information extracted from undominated blocking pairs from the men’s point of view. By choosing a man corresponding to the maximum value of the heuristic function, we aim to not only remove all the blocking pairs formed by the man but also reject as many blocking pairs as possible for an unstable matching from the women’s point of view to obtain a solution of the problem as quickly as possible. Experiments show that our algorithm is efficient in terms of both execution time and solution quality for solving the problem.

     

Tail index estimation for heavy tails: accommodation of bias in the excesses over a high threshold

In statistics of extremes, inference is often based on the excesses over a high random threshold. Those excesses are approximately distributed as the set of order statistics associated to a sample from a generalized Pareto model. We then get the so-called “maximum likelihood” estimators of the tail index γ. In this paper, we are interested in the derivation of the asymptotic distributional properties of a similar “maximum likelihood” estimator of a positive tail index γ, based also on the excesses over a high random threshold, but with a trial of accommodation of bias in the Pareto model underlying those excesses. We next proceed to an asymptotic comparison of the two estimators at their optimal levels. An illustration of the finite sample behaviour of the estimators is provided through a small-scale Monte Carlo simulation study. Research partially supported by FCT/POCTI and POCI/FEDER.     

Projection Pursuit Indexes Based on Orthonormal Function Expansions

Abstract

Projection pursuit describes a procedure for searching high-dimensional data for “interesting” low-dimensional projections via the optimization of a criterion function called the projection pursuit index. By empirically examining the optimization process for several projection pursuit indexes, we observed differences in the types of structure that maximized each index. We were especially curious about differences between two indexes based on expansions in terms of orthogonal polynomials, the Legendre index, and the Hermite index. Being fast to compute, these indexes are ideally suited for dynamic graphics implementations.

Both Legendre and Hermite indexes are weighted L ² distances between the density of the projected data and a standard normal density. A general form for this type of index is introduced that encompasses both indexes. The form clarifies the effects of the weight function on the index's sensitivity to differences from normality, highlighting some conceptual problems with the Legendre and Hermite indexes. A new index, called the Natural Hermite index, which alleviates some of these problems, is introduced.

A polynomial expansion of the data density reduces the form of the index to a sum of squares of the coefficients used in the expansion. This drew our attention to examining these coefficients as indexes in their own right. We found that the first two coefficients, and the lowest-order indexes produced by them, are the most useful ones for practical data exploration because they respond to structure that can be analytically identified, and because they have “long-sighted” vision that enables them to “see” large structure from a distance. Complementing this low-order behavior, the higher-order indexes are “short-sighted.” They are able to see intricate structure, but only when they are close to it.

We also show some practical use of projection pursuit using the polynomial indexes, including a discovery of previously unseen structure in a set of telephone usage data, and two cautionary examples which illustrate that structure found is not always meaningful.     

Sequential Maximum Likelihood Estimation for the Squared Radial Ornstein-Uhlenbeck Process

In this paper, we study the properties of a sequential maximum likelihood estimator of the unknown parameter for the squared radial Ornstein-Uhlenbeck process. The estimator is proved to be closed, unbiased, normally distributed and strongly consistent. Lastly a simulation study is presented to illustrate the efficiency of the estimators.

     

Numerical Stochastic Model of Non-stationary Time Series of the Wind Chill Index

A numerical stochastic model of the high-resolution time series of the wind chill index is considered. The model is constructed under the assumption that time series of the wind chill index are non-stationary non-Gaussian random processes with time-dependent one-dimensional distributions. This assumption makes possible to take into account both daily and seasonal variations of real meteorological processes. Data of the long-term real observations at weather stations were used for estimating the model parameters and for the verification of the model. Based on the simulated trajectories, some statistical properties of rare and adverse weather events, like long periods of time with a low wind chill index, are studied. The model is also used to study the dependence of the statistical properties of the wind chill index time series on a climate change.

     

Efficient and robust estimation for autoregressive regression models using shape mixtures of skewt normal distribution

Multiple linear regression model based on normally distributed and uncorrelated errors is a popular statistical tool with application in various fields. But these assumptions of normality and no serial correlation are hardly met in real life. Hence, this study considers the linear regression time series model for series with outliers and autocorrelated errors. These autocorrelated errors are represented by a covariance-stationary autoregressive process where the independent innovations are driven by shape mixture of skew-t normal distribution. The shape mixture of skew-t normal distribution is a flexible extension of the skew-t normal with an additional shape parameter that controls skewness and kurtosis. With this error model, stochastic modeling of multiple outliers is possible with an adaptive robust maximum likelihood estimation of all the parameters. An Expectation Conditional Maximization Either algorithm is developed to carryout the maximum likelihood estimation. We derive asymptotic standard errors of the estimators through an information-based approximation. The performance of the estimation procedure developed is evaluated through Monte Carlo simulations and real life data analysis.

     

A differential-equation-based model of the glass ceiling in career progression

ABSTRACT

We introduce a model based on Ordinary Differential Equations to describe how two mutually exclusive groups progress through a career hierarchy, whether in a single organization, or in an entire economic sector. The intended application is to gender imbalance at the top of the academic hierarchy in European Universities; however, the model is entirely generic and may be applied in other contexts also. Previous research on gender imbalance in European universities has focused on large-scale statistical studies. Our model represents a point of departure, as it is deterministic (i.e., based on Ordinary Differential Equations). The model requires a precise definition of the progression rates for the different groups through the hierarchy; these are key parameters governing the dynamics of career progression. The progression rate for each group can be decomposed into a product: the proportion of group members at a low level in the hierarchy who compete for promotion to the next level a given year, multiplied by the in-competition success rate for the group in question. Either of these two parameters can differ across the groups under consideration; this introduces a group asymmetry into the organization’s composition. We introduce a glass-ceiling index to summarize this asymmetry succinctly. Using case studies from the literature, we demonstrate how the mathematical framework can pinpoint the proximate cause of the glass ceiling in European academia.     

Perpetual American Double Lookback Options on Drawdowns and Drawups with Floating Strikes

We present closed-form solutions to the problems of pricing of the perpetual American double lookback put and call options on the maximum drawdown and the maximum drawup with floating strikes in the Black-Merton-Scholes model. It is shown that the optimal exercise times are the first times at which the underlying risky asset price process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum as well as the maximum drawdown or maximum drawup. The proof is based on the reduction of the original double optimal stopping problems to the appropriate sequences of single optimal stopping problems for the three-dimensional continuous Markov processes. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state spaces. We show that the optimal exercise boundaries are determined as either the unique solutions of the associated systems of arithmetic equations or the minimal and maximal solutions of the appropriate first-order nonlinear ordinary differential equations.

     

On Torsion-Free Minimal Abelian Groups

ABSTRACT

An abelian group is said to be minimal if it is isomorphic to all its subgroups of finite index. In this article we show that torsion-free groups which are complete in their ?-adic topology or are of p-rank not greater than 1, for all primes p, are minimal. A criterion is found for the minimality of all finite rank and for large classes of infinite rank completely decomposable groups. Separable minimal groups are also considered.     

On the Finite Group with the Index of Maximal Subgroups of the Monster

Abstract

We study the problem concerning the influence of the index of maximal subgroup or the degree of primitive permutation representation of the finite simple groups on the structure of a group. Let G be a finite group and s be the index of maximal subgroup of the Monster M. In this paper, we prove that there exists an epimorphism from G to M or A _s if G has the primitive permutation representation of degree s, and as a consequence we prove that the Monster is determined by every s.     

Drift Estimation of Multiscale Diffusions Based on Filtered Data

We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure.

     

Tools for Two-Dimensional Exploratory Projection Pursuit

Abstract

In this article we examine the current status of exploratory projection pursuit. This involves a large comparison of all two-dimensional projection indexes and optimization algorithms that have been proposed in the literature. We show that the efficacy of the exploration depends to a large extent on the optimization routine. We also stress the importance of studying the behavior of the empirical projection indexes rather than the underlying theoretical distances they estimate. Indeed, indexes based on orthonormal polynomial expansions differ greatly in their behavior from the theoretical performance of the weighted L ²-distances they estimate. This study reveals three universal indexes, namely the Legendre index (Friedman), the Hermite index (Hall), and the chi-squared index (Posse), which are sensitive to any kind of departure from normality in the core of the distribution, and two indexes ideal for catching clusters, namely the Laguerre-Fourier index (Morton) and the Natural Hermite index (Cook, Buja, and Cabrera).     

Mean-field stochastic control with elephant memory in finite and infinite time horizon

ABSTRACT

Our purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case.

• In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem.

• For infinite horizon, we derive sufficient and necessary maximum principles.

  As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.

     

Stochastic Model of Conditional Non-stationary Time Series of the Wind Chill Index in West Siberia

In this paper, we propose a stochastic model of the conditional time series of the wind chill index. The model is based on the inverse distribution function method and on the normalization method for simulation of the non-Gaussian non-stationary random processes as well as on the method of conditional distributions for simulation of the conditional Gaussian processes. In the framework of the approach considered, two types of conditions (point conditions and interval conditions) are imposed on the time series. The model in question was verified using the real data collected at the weather stations located in West Siberia (Russia). It is shown that the simulated trajectories are close in their statistical properties to the real time series. The model proposed was used for stochastic forecasting of the wind chill index and the results of the numerical experiments have shown that the accuracy of the short-term forecasts is high enough.

     

Logspline Density Estimation for Censored Data

Abstract

Logspline density estimation is developed for data that may be right censored, left censored, or interval censored. A fully automatic method, which involves the maximum likelihood method and may involve stepwise knot deletion and either the Akaike information criterion (AIC) or Bayesian information criterion (BIC), is used to determine the estimate. In solving the maximum likelihood equations, the Newton–Raphson method is augmented by occasional searches in the direction of steepest ascent. Also, a user interface based on S is described for obtaining estimates of the density function, distribution function, and quantile function and for generating a random sample from the fitted distribution.     

On the Chermak–Delgado lattice of a finite group

Abstract

By imposing conditions upon the index of a self-centralizing subgroup of a group, and upon the index of the center of the group, we are able to classify the Chermak-Delgado lattice of the group. This is our main result. We use this result to classify the Chermak-Delgado lattices of dicyclic groups and of metabelian p-groups of maximal class.     

Using the generalized maximum covering location model to control a project’s progress

Project control consists of monitoring a project’s progress at so called control points, finding possible deviations from the baseline schedule and if necessary, making adjustments to the deviated schedule subject to the available control budget, the adjusting strategies and also other technical and environmental possibilities in order to bring the schedule back on the right track. In this study, we adapt for the first time the generalized maximum covering location model to determine the adjusting strategies such that the maximum control coverage is achieved, i.e. under the given constraints, a schedule that is globally as close to the baseline schedule as possible is obtained. Numerical examples are given to illustrate the intricacies of the proposed method and also to demonstrate its applicability.