Probability Measure Research Articles

Beyond purchase intention in sports sponsorship: an alternative approach to measuring brand equity using best-worst scaling

PurposeThe purpose of this paper is to study the effects of sports sponsorship on brand equity using two managerially related outcomes: price premium and market share.Design/methodology/approachThis study uses a best–worst discrete choice experiment (BWDCE) and compares the outcome with that of the purchase intention scale, an established probabilistic measure of purchase intention. The total sample consists of 409 fans of three soccer teams sponsored by three different competing brands: Nike, Adidas and Puma.FindingsWith sports sponsorship, fans were willing to pay more for the sponsor’s product, with the sponsoring brand obtaining the highest market share. Prominent brands generally performed better than less prominent brands. The best–worst scaling method was also 35% more accurate in predicting brand choice than a purchase intention scale.Research limitations/implicationsFuture research could use the same method to study other types of sponsors, such as title sponsors or other product categories.Practical implicationsSponsorship managers can use this methodology to assess the return on investment in sponsorship engagement.Originality/valuePrior sponsorship studies on brand equity tend to ignore market share or fans’ willingness to pay a price premium for a sponsor’s goods and services. However, these two measures are crucial in assessing the effectiveness of sponsorship. This study demonstrates how to conduct such an assessment using the BWDCE method. It provides a clearer picture of sponsorship in terms of its economic value, which is more managerially useful.

Mean Field Approximations via Log-Concavity

Abstract We propose a new approach to deriving quantitative mean field approximations for any probability measure $P$ on $\mathbb {R}^{n}$ with density proportional to $e^{f(x)}$, for $f$ strongly concave. We bound the mean field approximation for the log partition function $\log \int e^{f(x)}dx$ in terms of $\sum _{i \neq j}\mathbb {E}_{Q^{*}}|\partial _{ij}f|^{2}$, for a semi-explicit probability measure $Q^{*}$ characterized as the unique mean field optimizer, or equivalently as the minimizer of the relative entropy $H(\cdot \,|\,P)$ over product measures. This notably does not involve metric-entropy or gradient-complexity concepts which are common in prior work on nonlinear large deviations. Three implications are discussed, in the contexts of continuous Gibbs measures on large graphs, high-dimensional Bayesian linear regression, and the construction of decentralized near-optimizers in high-dimensional stochastic control problems. Our arguments are based primarily on functional inequalities and the notion of displacement convexity from optimal transport.

Entropic taming of the Look Elsewhere Effect

To mitigate the Look Elsewhere Effect in multiple hypothesis testing using p-values, the paper suggests an “entropic correction” of the significance level at which the null hypothesis is rejected. The proposed correction uses the entropic uncertainty associated with the probability measure that expresses the prior-to-test probabilities expressing how likely the confirming evidence may occur at values of the parameter. When the prior-to-test probability is uniform (embodying maximal uncertainty) the entropic correction coincides with the Bonferroni correction. When the prior-to-test probability embodies maximal certainty (is concentrated on a single value of the parameter at which the evidence is obtained), the entropic correction overrides the Look Elsewhere Effect completely by not requiring any correction of significance. The intermediate situation is illustrated by a simple hypothetical example. Interpreting the prior-to-test probability subjectively allows a Bayesian spirit enter the frequentist multiple hypothesis testing in a disciplined manner. If the prior-to-test probability is determined objectively, the entropic correction makes possible to take into account in a technically explicit way the background theoretical knowledge relevant for the test.

Multivariable versions of a lemma of Kaluza's

AbstractLet and be a convergent multivariable power series in . In this paper, we present two conditions on the positive coefficients that imply that for nonnegative coefficients . If , then both of our results reduce to a lemma of Kaluza's. For , we present examples to show that our two conditions are independent of one another. It turns out that functions of the type satisfy one of our conditions, whenever is a product of probability measures on . Our results have applications to the theory of Nevanlinna–Pick kernels.

Arbitrage problems with reflected geometric Brownian motion

Contrary to the claims made by several authors, a financial market model in which the price of a risky security follows a reflected geometric Brownian motion is not arbitrage-free. In fact, such models violate even the weakest no-arbitrage condition considered in the literature. Consequently, they do not admit numéraire portfolios or equivalent risk-neutral probability measures, which makes them unsuitable for contingent claim valuation. Unsurprisingly, the published option pricing formulae for such models violate classical no-arbitrage bounds.

Convergence of random elements via fuzzy integrals

Can convergence of random variables, random vectors or, more generally, random elements of a metric space be studied using fuzzy integrals, in particular the seminormed integral? Among the positive results we present, it is shown that for large families of t-seminorms convergence in distribution and convergence in probability can be characterized via seminormed integrals. This suggests that, while fuzzy integrals were devised for working with non-additive fuzzy measures, some of their theoretical properties within the realm of probability measures can be comparable to those of the Lebesgue integral.

New Ways to Calculate the Probability in the Bertrand Problem

We give two new ways of calculating the probability of a chord of circumference randomly selected being larger than the side of an equilateral triangle inscribed in the circumference (this problem is known as the Bertrand paradox). The first one employs an immersion in R4, and the second one uses a direct probability measure over the set of chords.

Novel deterministic and probabilistic forecasting methods for crude oil price employing optimized deep learning, statistical and hybrid models

In this paper, individual and hybrid methods are proposed employing optimized statistical and deep learning (DL) models for deterministic (point) and probabilistic (interval) forecasting of crude oil price time series. The statistical models are optimized using the Forecast package of R. To enhance the performance of DL models, a novel pruning DE-DL method is proposed, which employs the differential evolution (DE) algorithm to optimize architecture and continuous and discrete-valued hyper-parameters. The proposed DE-DL method is so generic that it can be applied to optimize different DL models for any supervised learning problem. Five DL models (LSTM, BiLSTM, GRU, CNN, and ConvLSTM) are optimized for forecasting monthly crude oil prices and hybridized with an optimized ARIMA model for developing optimized additive and multiplicative hybrid forecasting models. The effectiveness of the proposed methods is evaluated through deterministic and probabilistic forecasting measures, comparing the results with six optimized statistical models, thirteen machine learning models, five optimized DL models, and ten optimized hybrid models. It is observed from the simulation results that the proposed optimized Additive-ARIMA-GRU hybrid model provides statistically superior forecasts, and the t Location Scale distribution is more suitable than the Gaussian distribution for computing reliable prediction intervals with different significance levels.

The smallest nanodrop describable via macroscopic interfacial concepts: Testing classical heterogeneous nucleation theory with perfect wetting down to 3 nm

HypothesisThe smallest nanodrop tractable with macroscopic notions such as the interfacial energy could be determined by comparing heterogeneous nucleation observations and capillary theory predictions at decreasing drop diameters dp. AnalysisThis is done here for the condensation of n-butanol vapors on polyethylene glycol nano-globules (3 nm ≤ dp ≤ 9 nm). We use published activation probability measurements P(w,dp), where w is the accurately controlled saturation ratio of n-butanol vapor in a gas stream exiting a saturator. The maximal saturation ratio achieved in the nucleation region by cooling this gas-vapor stream in the apparatus of Gallar et al. satisfies Smax = Cw. The key unknown constant C and the preexponential term K governing the nucleation rate are determined by assuming that classical theory applies to the largest particles used. This yields P(Smax,dp) data, directly comparable with capillary theory with perfect wetting. FindingsExcellent agreement is found above 5 nm for the critical dependence Smax(dp) resulting from the constraint P(Smax,dp) = 0.5. The entire P(Smax,dp) curves also agree closely between 5 and 7 nm. Smaller particles depart only slightly from theory, even at dp = 3 nm. Capillary theory hence describes accurately the heterogeneous nucleation process above 3–5 nm, provides a reliable method to determine Smax, and yields experimentally the nucleation rate constant K.

A prelude to statistics in Wasserstein metric spaces

PurposeThis paper aims mainly at introducing applied statisticians and econometricians to the current research methodology with non-Euclidean data sets. Specifically, it provides the basis and rationale for statistics in Wasserstein space, where the metric on probability measures is taken as a Wasserstein metric arising from optimal transport theory.Design/methodology/approachThe authors spell out the basis and rationale for using Wasserstein metrics on the data space of (random) probability measures.FindingsIn elaborating the new statistical analysis of non-Euclidean data sets, the paper illustrates the generalization of traditional aspects of statistical inference following Frechet's program.Originality/valueBesides the elaboration of research methodology for a new data analysis, the paper discusses the applications of Wasserstein metrics to the robustness of financial risk measures.

Random and mean Lyapunov exponents for

Abstract We consider orthogonally invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices with the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu and Shub [On random and mean exponents for unitarily invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {C})$ . Astérisque287 (2003), xvii, 1–18]. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.

Stochastic filtering under model ambiguity

In this paper, we study a non-linear filtering problem in the presence of signal model uncertainty. The model ambiguity is characterized by a class of probability measures from which the true one is taken. After interchanging the order of extremum problems by using the min-max theorem, we find that the uncertain filtering problem can be converted to a weighted conditional mean-field optimal control problem. Further, we characterize the ambiguity filter and prove its unique existence.

Monte Carlo Sensitivities Using the Absolute Measure-Valued Derivative Method

Measure-valued differentiation (MVD) is a relatively new method for computing Monte Carlo sensitivities, relying on a decomposition of the derivative of transition densities of the underlying process into a linear combination of probability measures. In computing the sensitivities, additional paths are generated for each constituent distribution and the payoffs from these paths are combined to produce sample estimates. The method generally produces sensitivity estimates with lower variance than the finite difference and likelihood ratio methods, and can be applied to discontinuous payoffs in contrast to the pathwise differentiation method. However, these benefits come at the expense of an additional computational burden. In this paper, we propose an alternative approach, called the absolute measure-valued differentiation (AMVD) method, which expresses the derivative of the transition density at each simulation step as a single density rather than a linear combination. It is computationally more efficient than the MVD method and can result in sensitivity estimates with lower variance. Analytic and numerical examples are provided to compare the variance in the sensitivity estimates of the AMVD method against alternative methods.

Topologies on unparameterised path space

The signature of a path, introduced by K.T. Chen [10] in 1954, has been extensively studied in recent years. The fundamental 2010 paper [20] of Hambly and Lyons showed that the signature is an injective function on the space of continuous, finite-variation paths up to a general notion of reparameterisation called tree-like equivalence. This result has been extended to geometric rough paths by Boedihardjo et al. [5]. More recently, the approximation theory of the signature has been widely used in the literature in applications. The archetypal instance of these results, see e.g. [24], guarantees uniform approximation, on compact sets, of a continuous function by a linear functional on the (extended) tensor algebra acting on the signature.In this paper we study in detail, and for the first time, the properties of three natural candidate topologies on the set of unparameterised paths, i.e. the tree-like equivalence classes. These are obtained by privileging different properties of the signature and are: (1) the product topology, obtained by equipping the range of the signature with the (subspace topology of the) product topology in the extended tensor algebra and then requiring S to be an embedding, (2) the quotient topology derived from the 1-variation topology on the underlying path space, and (3) the metric topology associated to d([γ],[σ]):=||γ⁎−σ⁎||1 using the (constant-speed) tree-reduced representatives γ⁎ and σ⁎ of the respective equivalence classes. We evaluate these spaces from the point of view of their suitability when it comes to studying (probability) measures on them. We prove that the respective collections of open sets are ordered by strict inclusion, (1) being the weakest and (3) the strongest. Our other conclusions can be summarised as follows. All three topological spaces are separable and Hausdorff, (1) being both metrisable and σ-compact, but not a Baire space and hence being neither Polish nor locally compact. The completion of (1), in any metric inducing the product topology, is the subspace G⁎ of group-like elements. The quotient topology (2) is not metrisable and the metric d is not complete. We also discuss some open problems related to these spaces.We consider finally the implications of the selection of the topology for uniform approximation results involving the signature. A stereotypical model for a continuous function on (unparameterised) path space is the solution of a controlled differential equation. We thus prove, for a broad class of these equations, well-definedness and measurability of the (fixed-time) solution map with respect to the Borel sigma-algebra of each topology. Under stronger regularity assumptions, we further show continuity of this same map on explicit compact subsets of the product topology (1). We relate these results to the expected signature model of [24].

Absolutely continuous and BV-curves in 1-Wasserstein spaces

We extend the result of Lisini (Calc Var Partial Differ Equ 28:85–120, 2007) on the superposition principle for absolutely continuous curves in p-Wasserstein spaces to the special case of p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document}. In contrast to the case of p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document}, it is not always possible to have lifts on absolutely continuous curves. Therefore, one needs to relax the notion of a lift by considering curves of bounded variation, or shortly BV-curves, and replace the metric speed by the total variation measure. We prove that any BV-curve in a 1-Wasserstein space can be represented by a probability measure on the space of BV-curves which encodes the total variation measure of the Wasserstein curve. In particular, when the curve is absolutely continuous, the result gives a lift concentrated on BV-curves which also characterizes the metric speed. The main theorem is then applied for the characterization of geodesics and the study of the continuity equation in a discrete setting.

FMC2 model based perception grading for dark insurgent network analysis.

The burgeoning role of social network analysis (SNA) in various fields raises complex challenges, particularly in the analysis of dark and dim networks involved in illicit activities. Existing models like the stochastic block model (SBM), exponential graph model (EGM), and latent space model (LSM) are limited in scope, often only suitable for one-mode networks. This article introduces a novel fuzzy multiple criteria multiple constraint model (FMC2) tailored for community detection in two-mode networks, which are particularly common in dark networks. The proposed method quantitatively determines the relationships between nodes based on a probabilistic measure and uses distance metrics to identify communities within the network. Moreover, the model establishes fuzzy boundaries to differentiate between the most and least influential nodes. We validate the efficacy of FMC2 using the Noordin Terrorist dataset and conduct extensive simulations to evaluate performance metrics. The results demonstrate that FMC2 not only effectively identifies communities but also ranks influential nodes within them, contributing to a nuanced understanding of complex networks. The method promises broad applicability and adaptability, particularly in intelligence and security domains where identifying influential actors within covert networks is critical.

Fractional higher order thin film equation with linear mobility: gradient flow approach

We prove existence of weak solutions of a fractional thin film type equation with linear mobility in any space dimension and for any order of the equation. The proof is based on a gradient flow technique in the space of Borel probability measures endowed with the Wasserstein distance.

On the ergodicity of interacting particle systems under number rigidity

In this paper, we provide relations among the following properties: the tail triviality of a probability measure mu on the configuration space {varvec{Upsilon }};the finiteness of a suitable L^2-transportation-type distance bar{textsf {d} }_{varvec{Upsilon }};the irreducibility of local {mu }-symmetric Dirichlet forms on {varvec{Upsilon }}. As an application, we obtain the ergodicity (i.e., the convergence to the equilibrium) of interacting infinite diffusions having logarithmic interaction and arising from determinantal/permanental point processes including text {sine}_{2}, text {Airy}_{2}, text {Bessel}_{alpha , 2} (alpha ge 1), and text {Ginibre} point processes. In particular, the case of the unlabelled Dyson Brownian motion is covered. For the proof, the number rigidity of point processes in the sense of Ghosh–Peres plays a key role.

Research on decision method based on probability hesitation fuzzy comprehensive distance measure

Aiming at the defects of the existing probabilistic hesitation fuzzy distance measures, which require the number of membership degree to be consistent and the order to be rearranged, a probabilistic hesitation fuzzy multi-attribute decision making method based on the comprehensive characteristic distance measure is proposed. First, a new law of probability hesitant fuzzy number comparison is defined. Then four characteristics of aggregation, discreteness, fuzziness and consistency are defined, and a new comprehensive distance measure is defined based on definitions above. Finally, on the basis of traditional TODIM method and prospect theory, the validity and rationality of the distance measure and new probability hesitant fuzzy recognition method in this paper are verified by examples and comparative analysis.

Improved mean-field dynamical equations are able to detect the two-step relaxation in glassy dynamics at low temperatures

We study the stochastic relaxation dynamics of the Ising p-spin model on a random graph, which is a well-known model with glassy dynamics at low temperatures. We introduce and discuss a new closure scheme for the master equation governing the continuous-time relaxation of the system, which translates into a set of differential equations for the evolution of local probabilities. The solution to these dynamical mean-field equations describes the out-of-equilibrium dynamics at high temperatures very well, notwithstanding the key observation that the off-equilibrium probability measure contains higher-order interaction terms not present in the equilibrium measure. In the low-temperature regime, the solution to the dynamical mean-field equations shows the correct two-step relaxation (a typical feature of glassy dynamics), but with a too-short relaxation timescale. We propose a solution to this problem by identifying the range of energies where entropic barriers play a key role and defining a renormalized microscopic timescale for the dynamical mean-field solution. The final result perfectly matches the complex out-of-equilibrium dynamics computed through extensive Monte Carlo simulations.