Worst-case robust Omega ratio

The Omega ratio is a recent performance measure proposed to overcome the known shortcomings of the Sharpe ratio. Until recently, the Omega ratio was thought to be computationally intractable, and research was focused on heuristic optimization procedures. We have shown elsewhere that the Omega ratio optimization is equivalent to a linear program and hence can be solved exactly in polynomial time. This permits the investigation of more complex and realistic variants of the problem. The standard formulation of the Omega ratio requires perfect information for the probability distribution of the asset returns. In this paper, we investigate the problem arising from the probability distribution of the asset returns being only partially known. We introduce the robust variant of the conventional Omega ratio that hedges against uncertainty in the probability distribution. We examine the worst-case Omega ratio optimization problem under three types of uncertainty – mixture distribution, box and ellipsoidal uncertainty – and show that the problem remains tractable.     

The ratio of the numbers of odd and even cycles in outerplanar graphs

In this paper, we investigate the ratio of the numbers of odd and even cycles in outerplanar graphs. We verify that the ratio generally diverges to infinity as the order of a graph diverges to infinity. We also give sharp estimations of the ratio for several classes of outerplanar graphs, and obtain a constant upper bound of the ratio for some of them. Furthermore, we consider similar problems in graphs with some pairs of forbidden subgraphs/minors, and propose a challenging problem concerning claw-free graphs.     

A robust alternative to the ratio estimator under non-normality

In sampling theory, the traditional ratio estimator is the most common estimator of the population mean when the correlation between study and auxiliary variables is positively high. We introduce a new ratio-type estimator based on the order statistics of a simple random sample. We show that this new estimator is considerably more efficient than the traditional ratio estimator under non-normality, and remarkably robust to data anomalies such as presence of outliers in data sets.     

Model-based likelihood ratio confidence intervals for survival functions

We introduce an adjusted likelihood ratio procedure for computing pointwise confidence intervals for survival functions from censored data. The test statistic, scaled by a ratio of two variance quantities, is shown to converge to a chi-squared distribution with one degree of freedom. The confidence intervals are seen to be a neighborhood of a semiparametric survival function estimator and are shown to have correct empirical coverage. Numerical studies also indicate that the proposed intervals have smaller estimated mean lengths in comparison to the ones that are produced as a neighborhood of the Kaplan-Meier estimator. We illustrate our method using a lung cancer data set.     

The Steiner ratio of several discrete metric spaces

Steiner's Problem is the “Problem of shortest connectivity”, that means, given a finite set of points in a metric space (X,ρ), search for a network interconnecting these points with minimal length. This shortest network must be a tree and is called a Steiner Minimal Tree (SMT). It may contain vertices different from the points which are to be connected. Such points are called Steiner points. If we do not allow Steiner points, that means, we only connect certain pairs of the given points, we get a tree which is called a Minimum Spanning Tree (MST). Steiner's Problem is very hard as well in combinatorial as in computational sense, but, on the other hand, the determination of an MST is simple. Consequently, we are interested in the greatest lower bound for the ratio between the lengths of these both trees:

which is called the Steiner ratio (of (X,ρ)). We look for estimates and exact values for the Steiner ratio in several discrete metric spaces. Particularly, we determine the Steiner ratio for spaces of words, and we estimate the Steiner ratio for specific graphs.     

On the Laplacian spectral ratio of connected graphs

The Laplacian spectrum of a graph is the eigenvalues of the associated Laplacian matrix. The quotient between the largest and second smallest Laplacian eigenvalues of a connected graph, is called the Laplacian spectral ratio. Some bounds on the Laplacian spectral ratio are considered. We improve a relation on the Laplacian spectral ratio of regular graphs. Especially, the first two smallest Laplacian spectral ratios of graphs with given order are determined. And some operations on Laplacian spectral ratio are presented.     

Nonparametric likelihood ratio goodness-of-fit tests for survival data

Berk and Jones (Z. Wahrsch. Verw. Gebiete 47 (1979) 47) described a nonparametric likelihood test of uniformity that is more efficient, in Bahadur's sense, than any weighted Kolmogorov-Smirnov test at any alternative. This article shows how to obtain a nonparametric likelihood test of a general parametric family for incomplete survival data. A nonparametric likelihood ratio test process is employed to measure the discrepancy between a parametric family and the observed data. Large sample properties of the likelihood ratio test process are studied under both the null and alternative hypotheses. A Monte Carlo simulation method is proposed to estimate its null distribution. We show how to produce a likelihood ratio graphical check as well as a formal test of a parametric family based on the developed theory. Our method is developed for the right-censorship model, but can be easily extended to some other survival models. Illustrations are given using both real and simulated data.     

The degree ratio ranking method for directed graphs

One of the most famous ranking methods for digraphs is the ranking by Copeland score. The Copeland score of a node in a digraph is the difference between its outdegree (i.e. its number of outgoing arcs) and its indegree (i.e. its number of ingoing arcs). In the ranking by Copeland score, a node is ranked higher, the higher is its Copeland score. In this paper, we deal with an alternative method to rank nodes according to their out- and indegree, namely ranking the nodes according to their degree ratio, i.e. the outdegree divided by the indegree. To avoid dividing by zero, we add 1 to both the out- as well as indegree of every node. We provide an axiomatization of the ranking by degree ratio using a clone property, which says that the entrance of a clone or a copy (i.e. a node that is in some sense similar to the original node) does not change the ranking among the original nodes. We also provide a new axiomatization of the ranking by Copeland score using the same axioms except that this method satisfies a different clone property. Finally, we modify the ranking by degree ratio by taking only the out- and indegree, but by definition assume nodes with indegree zero to be ranked higher than nodes with positive indegree. We provide an axiomatization of this ranking method using yet another clone property and a maximal property. In this way, we can compare the three ranking methods by their clone property.     

Likelihood ratio tests of correlated multivariate samples

We develop methods to compare multiple multivariate normally distributed samples which may be correlated. The methods are new in the context that no assumption is made about the correlations among the samples. Three types of null hypotheses are considered: equality of mean vectors, homogeneity of covariance matrices, and equality of both mean vectors and covariance matrices. We demonstrate that the likelihood ratio test statistics have finite-sample distributions that are functions of two independent Wishart variables and dependent on the covariance matrix of the combined multiple populations. Asymptotic calculations show that the likelihood ratio test statistics converge in distribution to central Chi-squared distributions under the null hypotheses regardless of how the populations are correlated. Following these theoretical findings, we propose a resampling procedure for the implementation of the likelihood ratio tests in which no restrictive assumption is imposed on the structures of the covariance matrices. The empirical size and power of the test procedure are investigated for various sample sizes via simulations. Two examples are provided for illustration. The results show good performance of the methods in terms of test validity and power.     

Testing the Gumbel hypothesis by Galton's ratio

We study a generalization of a ratio of spacings introduced by Galton in 1902. The ratio proves to be an important building block in the construction of a large sample test for the hypothesis that a distribution from an extremal domain of attraction belongs to the domain of attraction of the Gumbel law.     

Multiparameter ratio ergodic theorems for semigroups

After one-parameter treatment of ratio ergodic theorems for semigroups, we formulate the Sucheston a.e. convergence principle of continuous parameter type. This principle plays an effective role in proving some multiparameter generalizations of Chacon?s type continuous ratio ergodic theorems for semigroups and of Jacobs? type continuous random ratio ergodic theorems for quasi-semigroups. In addition, a continuous analogue of the Brunel–Dunford–Schwartz ergodic theorem is given of sectorially restricted averages for a commutative family of semigroups. We also formulate a local a.e. convergence principle of Sucheston?s type. The local convergence principle is effective in proving multiparameter local ergodic theorems. In fact, a multiparameter generalization of Akcoglu–Chacon?s local ratio ergodic theorem for semigroups of positive linear contractions on _L1$L_1$ is proved. Moreover, some multiparameter martingale theorems are obtained as applications of convergence principles.     

A proof of independent Bartlett correctability of nested likelihood ratio tests

It is well known that likelihood ratio statistic is Bartlett correctable. We consider decomposition of a likelihood ratio statistic into 1 degree of freedom components based on sequence of nested hypotheses. We give a proof of the fact that the component likelihood ratio statistics are distributed mutually independently up to the order O(1/n) and each component is independently Bartlett correctable. This was implicit in Lawley (1956, Biometrika, 43, 295–303) and proved in Bickel and Ghosh (1990, Ann. Statist., 18, 1070–1090) using a Bayes method. We present a more direct frequentist proof.     

Poisson's ratio of foamed plastics

Six-sided and 14-sided cell models are used as a basis for calculating the Poisson ratio of foamed plastics as a function of the properties of the polymer base and the parameters of the cellular structure. The extremal nature of the dependence of the Poisson ratio on the apparent density of foamed plastics is established and explained.Vladimir Scientific-Research Institute of Synthetic Resins. Translated from Mekhanika Polimerov, No. 1, pp. 45–49, January–February, 1973.     

Nonparametric estimation of the cross ratio function

The cross ratio function (CRF) is a commonly used tool to describe local dependence between two correlated variables. Being a ratio of conditional hazards, the CRF can be rewritten in terms of (first and second derivatives of) the survival copula of these variables. Bernstein estimators for (the derivatives of) this survival copula are used to define a nonparametric estimator of the cross ratio, and asymptotic normality thereof is established. We consider simulations to study the finite sample performance of our estimator for copulas with different types of local dependency. A real dataset is used to investigate the dependence between food expenditure and net income. The estimated CRF reveals that families with a low net income relative to the mean net income will spend less money to buy food compared to families with larger net incomes. This dependence, however, disappears when the net income is large compared to the mean income.

     

UMVU estimation of the ratio of powers of normal generalized variances under correlation

We consider estimation of the ratio of arbitrary powers of two normal generalized variances based on two correlated random samples. First, the result of Iliopoulos [Decision theoretic estimation of the ratio of variances in a bivariate normal distribution, Ann. Inst. Statist. Math. 53 (2001) 436-446] on UMVU estimation of the ratio of variances in a bivariate normal distribution is extended to the case of the ratio of any powers of the two variances. Motivated by these estimators’ forms we derive the UMVU estimator in the multivariate case. We show that it is proportional to the ratio of the corresponding powers of the two sample generalized variances multiplied by a function of the sample canonical correlations. The mean squared errors of the derived UMVU estimator and the maximum likelihood estimator are compared via simulation for some special cases.     

Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio

We develop a pricing rule for life insurance under stochastic mortality in an incomplete market by assuming that the insurance company requires compensation for its risk in the form of a pre-specified instantaneous Sharpe ratio. Our valuation formula satisfies a number of desirable properties, many of which it shares with the standard deviation premium principle. The major result of the paper is that the price per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting price as an expectation with respect to an equivalent martingale measure. Via this representation, one can interpret the instantaneous Sharpe ratio as a market price of mortality risk. Another important result is that if the hazard rate is stochastic, then the risk-adjusted premium is greater than the net premium, even as the number of contracts approaches infinity. Thus, the price reflects the fact that systematic mortality risk cannot be eliminated by selling more life insurance policies. We present a numerical example to illustrate our results, along with the corresponding algorithms.     

Eliciting density ratio classes

The probability distributions of uncertain quantities needed for predictive modelling and decision support are frequently elicited from subject matter experts. However, experts are often uncertain about quantifying their beliefs using precise probability distributions. Therefore, it seems natural to describe their uncertain beliefs using sets of probability distributions. There are various possible structures, or classes, for defining set membership of continuous random variables. The Density Ratio Class has desirable properties, but there is no established procedure for eliciting this class. Thus, we propose a method for constructing Density Ratio Classes that builds on conventional quantile or probability elicitation, but allows the expert to state intervals for these quantities. Parametric shape functions, ideally also suggested by the expert, are then used to bound the nonparametric set of shapes of densities that belong to the class and are compatible with the stated intervals. This leads to a natural metric for the size of the class based on the ratio of the total areas under upper and lower bounding shape functions. This ratio will be determined by the characteristics of the shape functions, the scatter of the elicited values, and the explicit expert imprecision, as characterized by the width of the stated intervals. We provide some examples, both didactic and real, and conclude with recommendations for the further development and application of the Density Ratio Class.     

Likelihood ratio tests for goodness-of-fit of a nonlinear regression model

We propose likelihood and restricted likelihood ratio tests for goodness-of-fit of nonlinear regression. The first-order Taylor approximation around the MLE of the regression parameters is used to approximate the null hypothesis and the alternative is modeled nonparametrically using penalized splines. The exact finite sample distribution of the test statistics is obtained for the linear model approximation and can be easily simulated. We recommend using the restricted likelihood instead of the likelihood ratio test because restricted maximum-likelihood estimates are not as severely biased as the maximum-likelihood estimates in the penalized splines framework.     

The cross ratio and Clifford algebras

Motivated by recent quaternionic approach of Bobenko and Pinkall to the complex cross ratio we presenta simple method to eva]uate the cross ratio in the Euclidean space? ⁿ identifying the space with vectors generating the Clifford algebraC(n). We apply the Clifford cross ratio to describe discrete analogues of orthogonal nets in? ⁿ .