Combinations Of Order Statistics Research Articles

Linear Combination of Order Statistics of Exponentiated Nadarajah–Haghighi Distribution and Their Applications

Linear Combination of Order Statistics of Exponentiated Nadarajah–Haghighi Distribution and Their Applications

Replacing pooling functions in Convolutional Neural Networks by linear combinations of increasing functions

Traditionally, Convolutional Neural Networks make use of the maximum or arithmetic mean in order to reduce the features extracted by convolutional layers in a downsampling process known as pooling. However, there is no strong argument to settle upon one of the two functions and, in practice, this selection turns to be problem dependent. Further, both of these options ignore possible dependencies among the data. We believe that a combination of both of these functions, as well as of additional ones which may retain different information, can benefit the feature extraction process. In this work, we replace traditional pooling by several alternative functions. In particular, we consider linear combinations of order statistics and generalizations of the Sugeno integral, extending the latter’s domain to the whole real line and setting the theoretical base for their application. We present an alternative pooling layer based on this strategy which we name “CombPool” layer. We replace the pooling layers of three different architectures of increasing complexity by CombPool layers, and empirically prove over multiple datasets that linear combinations outperform traditional pooling functions in most cases. Further, combinations with either the Sugeno integral or one of its generalizations usually yield the best results, proving a strong candidate to apply in most architectures.

Some limit behavior for linear combinations of order statistics

Some limit behavior for linear combinations of order statistics

Correction to: Linear combinations of order statistics to estimate the position and scale parameters of the Generalized Pareto distribution

Correction to: Linear combinations of order statistics to estimate the position and scale parameters of the Generalized Pareto distribution

Semiparametric Estimation of Treatment Effects in Randomized Experiments

We develop new semiparametric methods for estimating treatment effects. We focus on a setting where the outcome distributions may be thick tailed, where treatment effects are small, where sample sizes are large and where assignment is completely random. This setting is of particular interest in recent experimentation in tech companies. We propose using parametric models for the treatment effects, as opposed to parametric models for the full outcome distributions. This leads to semiparametric models for the outcome distributions. We derive the semiparametric efficiency bound for this setting, and propose efficient estimators. In the case with a constant treatment effect one of the proposed estimators has an interesting interpretation as a weighted average of quantile treatment effects, with the weights proportional to (minus) the second derivative of the log of the density of the potential outcomes. Our analysis also results in an extension of Huber's model and trimmed mean to include asymmetry and a simplified condition on linear combinations of order statistics, which may be of independent interest.

Confidence intervals based on L-moments for quantiles of the GP and GEV distributions with application to market-opening asset prices data

In a ground-breaking paper published in 1990 by the Journal of the Royal Statistical Society, J.R.M. Hosking defined the L-moment of a random variable as an expectation of certain linear combinations of order statistics. L-moments are an alternative to conventional moments and recently they have been used often in inferential statistics. L-moments have several advantages over the conventional moments, including robustness to the the presence of outliers, which may lead to more accurate estimates in some cases as the characteristics of distributions. In this contribution, asymptotic theory and L-moments are used to derive confidence intervals of the population parameters and quantiles of the three-parametric generalized Pareto and extreme-value distributions. Computer simulations are performed to determine the performance of confidence intervals for the population quantiles based on L-moments and to compare them to those obtained by traditional estimation techniques. The results obtained show that they perform well in comparison to the moments and maximum likelihood methods when the interest is in higher quantiles, or even best. L-moments are especially recommended when the tail of the distribution is rather heavier and the sample size is small. The derived intervals are applied to real economic data, and specifically to market-opening asset prices.

Increasing concave orderings of linear combinations of order statistics with applications to social welfare

We provide in this paper sufficient conditions for comparing, in terms of the increasing concave order, some income random variables based on linear combinations of order statistics that are relevant in the framework of social welfare. The random variables under study are weighted average incomes of the poorest and, for some particular weights, their expectations are welfare measures whose integral representations are weighted areas underneath Bonferroni curves.

Constrained ordered weighted averaging aggregation with multiple comonotone constraints

The constrained ordered weighted averaging (OWA) aggregation problem arises when we aim to maximize or minimize a convex combination of order statistics under linear inequality constraints that act on the variables with respect to their original sources. The standalone approach to optimizing the OWA under constraints is to consider all permutations of the inputs, which becomes quickly infeasible when there are more than a few variables, however in certain cases we can take advantage of the relationships amongst the constraints and the corresponding solution structures. For example, we can consider a land-use allocation satisfaction problem with an auxiliary aim of balancing land-types, whereby the response curves for each species are non-decreasing with respect to the land-types. This results in comonotone constraints, which allow us to drastically reduce the complexity of the problem.In this paper, we show that if we have an arbitrary number of constraints that are comonotone (i.e., they share the same ordering permutation of the coefficients), then the optimal solution occurs for decreasing components of the solution. After investigating the form of the solution in some special cases and providing theoretical results that shed light on the form of the solution, we detail practical approaches to solving and give real-world examples.

Calibrated bootstrap and saddlepoint approximations of finite population L-statistics

We propose two methods to approximate the distribution function of a Studentized linear combination of order statistics for a simple random sample drawn without replacement from a finite population. Using auxiliary data available for the population units, the first method modifies a nonparametric bootstrap approximation, and the second one corrects an empirical saddlepoint approximation based on the bootstrap. We conclude from simulations that, on the tails of distribution of interest, both approximations improve their initial versions and alternative Edgeworth approximations.

Calibrated Edgeworth expansions of finite population L-statistics

ABSTRACTA short Edgeworth expansion is approximated for the distribution function of a Studentized linear combination of order statistics computed on a random sample drawn without replacement from a finite population, and using auxiliary data available for the population units. Simulations show an improvement over the usual Gaussian approximation and previous empirical Edgeworth expansions. Naive synthetic estimates of the distribution function, based on the auxiliary data only, yield accurate results when the auxiliary variable is well correlated with the study variable.

Linear Detection of a Weak Signal in Additive Cauchy Noise

The detection of a weak signal in additive Cauchy noise is of great importance in many applications. A locally optimum detector (LOD) exists for such a scenario; however, it is non-linear in nature. In general, implementation of non-linear detectors is difficult in practice, and linear detectors with good properties, such as high asymptotic relative efficiency (ARE) with respect to the LOD, are often desirable. In this paper, we propose a linear detector for a weak signal in additive Cauchy noise. The proposed test statistic is a linear combination of order statistics. For the special case of a constant signal in additive Cauchy noise, we prove the asymptotic normality of the trimmed linear detector, and show that the ARE of the trimmed linear detector with respect to the LOD is unity. Extensive simulation results are provided to demonstrate that the loss in the performance of the linear detector is very small compared with the non-linear LOD. We also discuss the hardware complexities of the LOD and the linear detector, and demonstrate the advantages of the linear detector over the LOD, in terms of hardware implementation.

Order statistics and their concomitants from multivariate normal mean–variance mixture distributions with application to Swiss Markets Data

ABSTRACTIn this article, by considering a multivariate normal mean–variance mixture distribution, we derive the exact joint distribution of linear combinations of order statistics and their concomitants. From this general result, we then deduce the exact marginal and conditional distributions of order statistics and their concomitants arising from this distribution. We finally illustrate the usefulness of these results by using a Swiss markets dataset.

Comparación de la regresión GINI con la regresión de mínimos cuadrados ordinarios y otros modelos de regresión lineal robustos

En este trabajo se compara la regresión de Gini con la regresión OLS y otros métodos de regresión robustos, del tipo L (LAV, combinaciones lineales de estadísticos de orden), del tipo M (M de Huber, basado en el concepto de máxima verosimilitud) y del tipo MM (basado en la minimización de un estimador M). La comparación de los métodos se realiza vía simulación bajo diferentes escenarios. Los resultados encontrados vía simulación muestran que la regresión de Gini tiene un mayor grado de robustez en comparación con la regresíon OLS al estimar los coeficientes de regresión ante la presencia de datos atípicos, pero su robustez es menor que la de los métodos de estimación robustos LAV, M de Huber y MM

On Inferring Income Inequality Measures Using L-moments

Abstract L-moments are expectations of linear combinations of order statistics, their use in reliability analysis has been established. In the present paper we study L-moments in relation to income inequality measures. Models characterized by functional relationships between L-moments and income inequality measures are studied. Finally, we define the ordering based on L-moments and study its implications on other existing orderings.

A uniqueness result for [formula omitted]-estimators, with applications to L-moments

We show that if a linear combination of expectations of order statistics has mean zero across all random variables that have finite mean, then the linear combination is identically zero. A consequence of this result is that any functional of a probability distribution can have essentially only one unbiased L-estimator (i.e., an estimator that has the form of a linear combination of order statistics): if two such linear combinations have the same expectation then they must be algebraically identical. We use this result to prove the equivalence of two statistics that have been proposed as estimators of the L-moments introduced by Hosking (1990), and to provide alternative means of computing estimators of the trimmed L-moments introduced by Elamir and Seheult (2003). We also make comparisons of the speed of various methods for computing estimators of L-moments and trimmed L-moments.

A New Estimator for a Population Proportion Using Group Testing

The maximum likelihood estimator is widely used in estimating the population proportion using group testing. However, it is positive biased and some alternatives have been raised in literatures. In this study, we propose a new estimator by weighted combination of order statistics. Two rules are supplied to determine the unknown weight. Using the rule of minimizing the absolute bias, our estimator is almost unbiased in most cases shown by simulations. Using the rule of minimizing the mean square error, a simple estimator with weight 1 is recommended for its good performance.

Kernel quantile estimator with ICI adaptive bandwidth selection technique

In this article, we consider a class of kernel quantile estimators which is the linear combination of order statistics. This class of kernel quantile estimators can be regarded as an extension of some existing estimators. The exact mean square error expression for this class of estimators will be provided when data are uniformly distributed. The implementation of these estimators depends mostly on the bandwidth selection. We then develop an adaptive method for bandwidth selection based on the intersection confidence intervals (ICI) principle. Monte Carlo studies demonstrate that our proposed approach is comparatively remarkable. We illustrate our method with a real data set.

Efficient Estimation of Parameters of the Extreme Value Distribution

The problem of efficient estimation of the parameters of the extreme value distribution has not been addressed in the literature. Our paper attempts to obtain efficient estimators of the parameters without solving the likelihood equations. The paper provides for the first time simple expressions for the elements of the information matrix for type II censoring. We consider the problem of estimation of θ and σ in the model (1/σ)f((x − θ)/σ) when f is the standard form of the pdf of type I (maximum) extreme value distribution. We construct efficient estimators of θ and σ using linear combinations of order statistics of a random sample drawn from the population. We derive explicit formulas for the information matrix for this problem for type II censoring and construct efficient estimators of the parameters using linear combinations of available order statistics with additional weights to the smallest and the largest order statistics. We consider numerical examples to illustrate the applications of the estimators. We also consider a Monte Carlo simulation study to examine the performance of the estimators for small samples. Using the asymptotic covariance matrix of the estimators in the above model we then construct an efficient estimator of θ in the reduced model (1/cθ)f((x − θ)/cθ) where θ ( > 0) is unknown and c ( > 0) is known for complete and censored samples.

On the asymptotic normality of finite population $$L$$ L -statistics

We give sufficient conditions for the asymptotic normality of linear combinations of order statistics ( $$L$$ -statistics) in the case of simple random samples without replacement. In the first case, restrictions are imposed on the weights of $$L$$ -statistics. The second case is on trimmed means, where we introduce a new finite population smoothness condition.

$$L$$ L -statistics from multivariate unified skew-elliptical distributions

We study here the distributions of order statistics and linear combinations of order statistics from a multivariate unified skew-elliptical distribution. We show that these distributions can be expressed as mixtures of unified skew-elliptical distributions, and then use these mixture forms to study some distributional properties and moments.