Non-degenerate Distribution Research Articles

A test for Gaussianity in Hilbert spaces via the empirical characteristic functional

AbstractLet X1,X2,… be independent and identically distributed random elements taking values in a separable Hilbert space . With applications for functional data in mind, may be regarded as a space of square‐integrable functions, defined on a compact interval. We propose and study a novel test of the hypothesis H0 that X1 has some unspecified nondegenerate Gaussian distribution. The test statistic Tn = Tn(X1,…,Xn) is based on a measure of deviation between the empirical characteristic functional of X1,…,Xn and the characteristic functional of a suitable Gaussian random element of . We derive the asymptotic distribution of Tn as n→∞ under H0 and provide a consistent bootstrap approximation thereof. Moreover, we obtain an almost sure limit of Tn and the limit distributions of Tn under fixed and contiguous alternatives to Gaussianity. Simulations show that the new test is competitive with respect to the hitherto few competitors available.

Information theory and behavior

The quantal response behavior widely observed in experiments and observations of human and animal behavior can be derived as expected payoff maximization subject to a constraint on the entropy of the subject’s behavior mixed strategy. The Lagrange multiplier corresponding to the entropy constraint is an agent’s “behavior temperatureˮ. Entropy-constrained behavior approximates payoff-maximizing behavior, but in many contexts exhibits qualitatively different outcomes. The “endowment effectˮ and other instances of “loss-aversionˮ, for example, can be seen as a consequence of entropy-constrained behavior. Identical entropy-constrained agents with the same value for a good or asset will exhibit spontaneous “noise tradingˮ. An entropy-constrained agent with a lower behavior temperature will systematically take economic surplus away from an agent with the same valuation of a good but a higher behavior temperature in bilateral transactions. The equilibrium of a standard supply-demand models with entropy-constrained agents is a non-degenerate frequency distribution of transaction prices rather than a single equilibrium price. Changes in behavior temperature can transform social interaction games from prisoners’ dilemmas to assurance games. Entropy-constrained quantal responses allow quantitative inferences about payoff changes and distribution stronger than qualitative Pareto comparisons.

Landau–Kelly representation of statistical thermodynamics of a quantum plasma and electron emission from metals

We have investigated the influence of a strong magnetic field on various aspects of a quantum Fermi plasma. Due to the strong magnetic field, the distribution function becomes anisotropic. First, we consider non-degenerate quantum, Landau and Kelly distribution function. It was found that the adiabatic equation is similar to the adiabatic equation for a Maxwell distribution function, when we include the magnetic field in the energy expression. Using the Kelly distribution for a degenerate, quantum Fermi gas, parallel and perpendicular components of the pressure were derived. It was found that perpendicular component of pressure never becomes zero and three-dimensional system always stay three-dimensional. Lastly, we investigated electron emission from metals and have shown the influence of the magnetic field. We calculated thermionic emission, the so-called Richardson effect. In addition, we investigate the influence of external electromagnetic radiation on the electron current density (Hallwachs effect) from metals.

An heterogeneous-agent New-Monetarist model with an application to unemployment

A New-Monetarist model is constructed with expenditure and unemployment risks that generates equilibria with non-degenerate distribution of money holdings. Distributional effects can overturn key insights of the model with degenerate distributions, e.g., the value of money depends on the income distribution; a one-time money injection raises aggregate real balances in the short run – price adjustments look sluggish; anticipated inflation can raise output and welfare; there can be a long-run trade-off between inflation and unemployment. Distributional effects also generate a quantitatively significant aggregate demand channel through which transfers financed with money creation can raise employment, and productivity shocks are amplified.

LIQUIDITY EXTERNALITIES AND THE WALLACE CONJECTURE

This paper presents a pure currency economy with a nondegenerate distribution of money holdings in which, as conjectured by Wallace (Quarterly Journal of Economics 129, 259–274, 2014), there are transfer schemes financed by money creation that improve ex ante welfare relative to no-intervention. Differently from what was advocated by Wallace, pecuniary-like externalities, rather than the need to share liquidity risks, are responsible for the result.

Geometric Distance Between Positive Definite Matrices of Different Dimensions

We show how the Riemannian distance on $\mathbb{S}^n_{++}$, the cone of $n\times n$ real symmetric or complex Hermitian positive definite matrices, may be used to naturally define a distance between two such matrices of different dimensions. Given that $\mathbb{S}^n_{++}$ also parameterizes $n$-dimensional ellipsoids, and inner products on $\mathbb{R}^n$, $n \times n$ covariance matrices of nondegenerate probability distributions, this gives us a natural way to define a geometric distance between a pair of such objects of different dimensions.

Non-Uniform Bounds in the Poisson Approximation With Applications to Informational Distances I

We explore asymptotically optimal bounds for deviations of Bernoulli convolutions from the Poisson limit in terms of the Shannon relative entropy and the Pearson $\chi^2$-distance. The results are based on proper non-uniform estimates for densities. They deal with models of non-homogeneous, non-degenerate Bernoulli distributions.

Self-confirming price dispersion in monetary economies

In a monetary economy, we show that price dispersion arises as an equilibrium outcome without the need for costly simultaneous search or any heterogeneity in preferences, production costs, or search technologies. A distribution of money holdings among buyers makes sellers indifferent across a set of posted prices, leading to a non-degenerate price distribution. This price distribution, in turn, makes buyers indifferent across a range of money balances, rationalizing the non-degenerate distribution of money holdings. We completely characterize the distribution of posted prices and money holdings in any equilibrium. Equilibria with price dispersion admit higher maximum prices than observed in any single-price equilibrium and feature an increasing density of prices, in contrast to existing theories. Price dispersion reduces welfare by creating mismatch between posted prices and money balances. Inflation exacerbates this welfare loss by shifting the distribution towards higher prices.

Bootstrapping complex time-to-event data without individual patient data, with a view toward time-dependent exposures.

We consider nonparametric and semiparametric resampling of multistate event histories by simulating multistate trajectories from an empirical multivariate hazard measure. One advantage of our approach is that it does not necessarily require individual patient data, but may be based on published information. This is also attractive for both study planning and simulating realistic real‐world event history data in general. The concept extends to left‐truncation and right‐censoring mechanisms, nondegenerate initial distributions, and nonproportional as well as non‐Markov settings. A special focus is on its connection to simulating survival data with time‐dependent covariates. For the case of qualitative time‐dependent exposures, we demonstrate that our proposal gives a more natural interpretation of how such data evolve over the course of time than many of the competing approaches. The multistate perspective avoids any latent failure time structure and sampling spaces impossible in real life, whereas its parsimony follows the principle of Occam's razor. We also suggest empirical simulation as a novel bootstrap procedure to assess estimation uncertainty in the absence of individual patient data. This is not possible for established procedures such as Efron's bootstrap. A simulation study investigating the effect of liver functionality on survival in patients with liver cirrhosis serves as a proof of concept. Example code is provided.

A New Ratio Statistic to Test Mean Change Point in Long Memory Time Series

In this paper, we propose a new ratio statistic to test mean change point in long memory time series. We derive the test statistic converges to a non-degenerate distribution under the null hypothesis and that it diverges to infinity under the alternative of a change-point with constant height. Furthermore, we estimate the long memory parameters and approximate the critical values of the statistic by the Sieve Bootstrap method. We show the size and power properties of our test through numerical simulations and the performance is inspiring. Finally, we illustrate the effectiveness of this method through a set of actual data.

On the long-run wealth distribution in a simple Ramsey model with heterogeneous households

This paper analyzes the long-run wealth distribution in a Ramsey model where individuals have a common rate of time preference but different intertemporal elasticities of substitution. As a result, it is shown that heterogeneity among households in intertemporal substitution is sufficient for the existence of a non-degenerate long-run wealth distribution. We also investigate the properties of the long-run wealth distribution and the transition of capital and consumption using the phase diagram.

Land Reform and Productivity: A Quantitative Analysis with Micro Data

We assess the effects of a major land policy change on farm size and agricultural productivity using a quantitative model and micro-level data. We study the 1988 land reform in the Philippines that imposed a ceiling on land holdings, redistributed above-ceiling lands to landless and smallholder households, and severely restricted the transferability of the redistributed farmlands. We study this reform in the context of an industry model of agriculture with a nondegenerate distribution of farm sizes featuring an occupation decision and a technology choice of farm operators. In this model, the land reform can reduce agricultural productivity not only by misallocating resources across farmers but also by distorting farmers’ occupation and technology decisions. The model, calibrated to prereform farm-level data in the Philippines, implies that on impact, the land reform reduces average farm size by 34 percent and agricultural productivity by 17 percent. The government assignment of land and the ban on its transfer are key for the magnitude of the results since a market allocation of the above-ceiling land produces about one-third of the size and productivity effects. These results emphasize the potential role of land market efficiency for misallocation and productivity in the agricultural sector. (JEL D24, O11, O13, Q12, Q15, Q18, Q24, Q28)

Recursive allocations and wealth distribution with multiple goods: Existence, survivorship, and dynamics

We characterize the equilibrium of a complete markets economy with multiple agents featuring a preference for the timing of the resolution of uncertainty. Utilities are defined over an aggregate of two goods. We provide conditions under which the solution of the planner's problem exists, and it features a nondegenerate invariant distribution of Pareto weights. We also show that perturbation methods replicate the salient features of our recursive risk‐sharing scheme, provided that higher‐order terms are included. Recursive preferences multiple agents equilibrium C62 F37

On powers of likelihood functions of random walks on ℤͩ

Abstract Let {Xi}i≥1 be independent, identically distributed random vectors in ℤd,d≥1. Let LLn(x)≡ℙ(Sn=x),n≥1,x∈ℤd, be the likelihood function for Sn=∑i=1nXi. For integers j≥2 and n≥1, let an(j)≡∑x∈ℤd(Ln(x))j. We show that if X1-X2 has a nondegenerate aperiodic distribution in ℤd and 𝔼(∥X1∥2)>∞, then limn→∞n(j-1)d∕2an(j)≡a(j,d) exists and 0<a(j,d)<∞. Some extensions and open problems are also outlined.

On the limiting distribution of sample central moments

We investigate the limiting behavior of sample central moments, examining the special cases where the limiting (as the sample size tends to infinity) distribution is degenerate. Parent (non-degenerate) distributions with this property are called singular, and we show in this article that the singular distributions contain at most three supporting points. Moreover, using the delta-method, we show that the (second-order) limiting distribution of sample central moments from a singular distribution is either a multiple, or a difference of two multiples of independent Chi-square random variables with one degree of freedom. Finally, we present a new characterization of normality through the asymptotic independence of the sample mean and all sample central moments.

Markov-perfect risk sharing, moral hazard and limited commitment

We define, characterize and compute Markov-perfect risk-sharing contracts in a dynamic stochastic economy with endogenous asset accumulation and simultaneous limited commitment and moral hazard frictions. We prove that Markov-perfect insurance contracts preserve standard properties of optimal insurance with private information and are not more restrictive than a long-term contract with one-sided commitment. Markov-perfect contracts imply a determinate asset time-path and a non-degenerate long-run stationary wealth distribution. Quantitatively, we show that Markov-perfect risk-sharing contracts provide sizably more consumption smoothing relative to self-insurance and that the welfare gains from resolving the commitment friction are larger than the gains from resolving the moral hazard friction at low asset levels, while the opposite holds for high asset levels.

Self-confirming Price Dispersion in Monetary Economies

In a monetary economy, we show that price dispersion arises as an equilibrium outcome without the need for costly simultaneous search or any heterogeneity in preferences, production costs, or search technologies. A distribution of money holdings among buyers makes sellers indifferent across a set of posted prices, leading to a non-degenerate price distribution. This price distribution, in turn, makes buyers indifferent across a range of money balances, rationalizing the non-degenerate distribution of money holdings. We completely characterize the distribution of posted prices and money holdings in any equilibrium. Equilibria with price dispersion admit higher maximum prices than observed in any single-price equilibrium. Also, price dispersion reduces welfare by creating mismatch between posted prices and money balances. Inflation exacerbates this welfare loss by shifting the distribution towards higher prices.

The Shortest Possible Return Time of <inline-formula> <tex-math notation="LaTeX">$\beta$ </tex-math> </inline-formula>-Mixing Processes

We consider a stochastic process and a given $n$ -string. We study the shortest possible return time (or shortest return path) of the string over all the realizations of process starting from this string. For a $\beta $ -mixing process having complete grammar, and for each size $n$ of the strings, we approximate the distribution of this short return (properly re-scaled) by a non-degenerated distribution. Under mild conditions on the $\beta $ coefficients, we prove the existence of the limit of this distribution to a non-degenerated distribution. We also prove that ergodicity is not enough to guaranty this convergence. Finally, we present a connection between the shortest return and the Shannon entropy, showing that maximum of the re-scaled variables grow as the matching function of Wyner and Ziv.

Algebras of Probability Distributions on Finite Sets

We consider the problems of transforming random variables over finite sets by discrete functions. We describe the problems of exact and approximate expression of random variables as functions of other random variables from the point of view of universal algebra and provide a review of results in the area. Sufficient conditions are obtained for a system of transforming functions to allow the approximation of an arbitrary probability distribution on a finite set using a given nondegenerate initial distribution.

Einstein-Hilbert type action on spacetimes

The mixed gravitational field equations have been recently introduced for codimension one foliated manifolds, e.g. stably causal and globally hyperbolic spacetimes. These Euler-Lagrange equations for the total mixed scalar curvature (as analog of Einstein-Hilbert action) involve a new kind of Ricci curvature (called the mixed Ricci curvature). In the work, we derive Euler-Lagrange equations of the action for any spacetime, in fact, for a pseudo-Riemannian manifold endowed with a non-degenerate distribution. The obtained equations are presented in the classical form of Einstein field equation with the new Ricci type curvature instead of Ricci curvature