Coefficients of ergodicity for Markov chains with uncertain parameters

One of the central considerations in the theory of Markov chains is their convergence to an equilibrium. Coefficients of ergodicity provide an efficient method for such an analysis. Besides giving sufficient and sometimes necessary conditions for convergence, they additionally measure its rate. In this paper we explore coefficients of ergodicity for the case of imprecise Markov chains. The latter provide a convenient way of modelling dynamical systems where parameters are not determined precisely. In such cases a tool for measuring the rate of convergence is even more important than in the case of precisely determined Markov chains, since most of the existing methods of estimating the limit distributions are iterative. We define a new coefficient of ergodicity that provides necessary and sufficient conditions for convergence of the most commonly used class of imprecise Markov chains. This so-called weak coefficient of ergodicity is defined through an endowment of the structure of a metric space to the class of imprecise probabilities. Therefore we first make a detailed analysis of the metric properties of imprecise probabilities.     

Best constants in Chebyshev inequalities with various applications

In this article we describe some ways to significantly improve the Markov-Gauss-Camp-Meidell inequalities and provide specific applications. We also describe how the improved bounds are extendable to the multivariate case. Applications include explicit finite sample construction of confidence intervals for a population mean, upper bounds on a tail probability P(X>k) by using the density at k, approximation of P-values, simple bounds on the Riemann Zeta function, on the series , improvement of Minkowski moment inequalities, and construction of simple bounds on the tail probabilities of asymptotically Poisson random variables. We also describe how a game theoretic argument shows that our improved bounds always approximate tail probabilities to any specified degree of accuracy. Received: April 1999     

On the uniqueness of local equilibria

The purpose of this paper is to define a new notion of local equilibrium in an exchange economy, where the consumers face lower bounds on net trades. Then, we show that the local equilibrium is unique if the lower bounds are closed enough to 0. By the way, we also provide a convergence result of local equilibrium price toward Walras equilibrium price of a suitable tangent linear economy.     

Ein allgemeines Konstruktionsverfahren für Konfidenzbereiche

Summary A theorem ofTakács concerning interchangeable random variables is used to derive a simple method for the construction of confidence regions. Applying this method to a location parametera we get a.s. convergence of the confidence interval toa if the sample sizen increases while its probability is (n–1)/(n+1). Under certain conditions the interval contains always the maximum-likelihood estimate and another estimate which results from a least squares postulate. Lower bounds are given for the probability that our intervals become shorter than the intervals we would get relying on the central limit theorem. In order to avoid an assumption of finite support needed first to derive the a.s. convergence, we modify our method omitting extreme values.The modified intervals converge forn with probability 1 to the true parameter value under weaker conditions. A lower bound for the probability and-using a result due toRényi-theasymptotic probability of the modified interval is given. For the two kinds of intervals a formula concerning the velocity of their convergence to the length 0 is derived.Finally, the results are extended to a shift parameter in the two-sample case. Here we derive for equal sample sizes the exact probability of the modified interval and give upper and lower bounds fir its asymptotic probability. The method is practicable also if one sample size is an integer multiple of the other.     

Are we there yet? On the convergence of financial reforms

Since the 1970s, there have been substantial financial reforms across the world, however little research has been devoted to studying the convergence path of these reforms. While Abiad and Mody (Am Econ Rev 95:66–88, 2005) find that there is movement towards a global ‘norm’ of financial reform, their findings are based upon a cross-sectional approach that has been widely criticized in the literature. In this paper we offer new time-series evidence on the convergence of financial reforms both across and within regions, which can also act as a metric to measure the degree of globalization among countries’ financial systems. We find that some regions of the world have fully converged, but the advanced economies and Sub-Saharan Africa are not converging. In addition, while most countries have fully converged within their own region, notable exceptions are also identified. These results suggest that while financial reforms have largely become homogenized, important distinctions still remain.     

Extracting Sustainable Earnings from Profit Margins

Revenues and expenses are fundamentally proportional to one another, but are likely to be disproportionally affected by transitory items or economic shocks. We build on this observation and propose a new measure of sustainable earnings based on deviations from normal profit margins. While some other sustainable earnings metrics attempt to identify transitory components on a line-by-line basis, our measure, referred to as the intensity of core earnings (ICE), uses ratio analysis to extract the transitory portion of earnings from all line items. We find that the ICE, as measured here, is positively associated with earnings persistence, better earnings predictability, and stronger market reaction to unexpected earnings. We also find that our measure is positively associated with post-earnings announcement excess stock returns. Comparing our measure with an accrual-based measure of earnings quality, we find that, in general, the two metrics provide distinct incremental information relative to one another and in some instances our measure is better than an accrual-based measure in assessing earnings quality.     

Embedded in context: How time and distance affect the convergence of personnel selection practices

The debate on convergence versus divergence or stasis in human resource management (HRM) practices over time is still ongoing. We look at configurations of organisations' personnel selection practices and empirically analyse the role of geographic, cultural and regulatory institutional distance between countries for emerging similarity or dissimilarity in these practices. We also examine whether convergence occurred between 1995 and 2015. Based on the Cranet data of 25,869 organisations from 42 countries and statistical tests using energy distance, we find a pattern over time, moving from stasis to divergence. In addition, personnel selection configurations relate to cultural and regulatory institutional differences in the sense that smaller distances lead to higher similarity. This is not the case, however, for geographic distance. Our study adds to the debate on HRM convergence and offers a new method of analysis for other areas of HRM research where configurations instead of single HRM practices play a role.     

Finite-sample exact tests for linear regressions with bounded dependent variables

We introduce tests for finite-sample linear regressions with heteroskedastic errors. The tests are exact, i.e., they have guaranteed type I error probabilities when bounds are known on the range of the dependent variable, without any assumptions about the noise structure. We provide upper bounds on probability of type II errors, and apply the tests to empirical data.     

Solving the income fluctuation problem with unbounded rewards

This paper studies the income fluctuation problem without imposing bounds on utility, assets, income or consumption. We prove that the Coleman operator is a contraction mapping over the natural class of candidate consumption policies when endowed with a metric that evaluates consumption differences in terms of marginal utility. We show that this metric is complete, and that the fixed point of the operator coincides with the unique optimal policy. As a consequence, even in this unbounded setting, policy function iteration always converges to the optimal policy at a geometric rate.     

The impact of inventory dynamics on long-term stock returns – An empirical investigation of U.S. manufacturing companies

This paper investigates the relationship between the inventory dynamics and long-term stock returns of a large panel of U.S. manufacturing firms over the time period from 1991 to 2010. We propose two measures of inventory dynamics: one metric to assess the fluctuations of quarterly inventories within the year and a second metric to quantify relative year-over-year inventory growth. Our results indicate that within-year inventory volatility (IV) and abnormal year-over-year inventory growth (ABI) are associated with abnormal stock returns. Both metrics cannot be entirely explained by common risk factors. We find that firms with high IV and low ABI have the best long-term stock returns, and that stock performance decreases monotonically with higher ABI values. Our results are robust to various control variables including size, book-to-market value, industry and prior performance. We therefore conclude that changes in inventory levels provide valuable insights into the risks and opportunities faced by a company.     

A general framework for approximate sampling with an application to generating points on the boundary of bounded convex regions

We consider the problem of generating a sample of points according to some given probability distribution over some region. We give a general framework for constructing approximate sampling algorithms based on the theory of Markov chains. In particular, we show how it can be proven that a Markov chain has a limiting distribution. We apply these results to prove convergence for a class of so-called Shake-and-Bake algorithms, which can be used to approximate any absolutely continuous distribution over the boundary of a full-dimensional convex body.     

Selecting the best regression equation via the P-value of F-test

In this article, we establish a method of selecting the best regression equation on the basis of F-test. The basic idea is to select the most significant regression equation, which corresponds to the minimum P-value of F-test. We also present upper and lower bounds for the P-value together with approximations for these bounds which are simple to compute. Using this method, not only can we find out the best regression equation, we also obtain its significance probability. When the method is applied to the well-known data from Hald (1952) and Gorman & Toman (1966), the results are satisfactory.     

Turnpike property and convergence rate for an investment model with general utility functions

In this paper we aim to address two questions faced by a long-term investor with a power-type utility at high levels of wealth: one is whether the turnpike property still holds for a general utility that is not necessarily differentiable or strictly concave, the other is whether the error and the convergence rate of the turnpike property can be estimated. We give positive answers to both questions. To achieve these results, we first show that there is a classical solution to the HJB equation and give a representation of the solution in terms of the dual function of the solution to the dual HJB equation. We demonstrate the usefulness of that representation with some nontrivial examples that would be difficult to solve with the trial and error method. We then combine the dual method and the partial differential equation method to give a direct proof to the turnpike property and to estimate the error and the convergence rate of the optimal policy when the utility function is continuously differentiable and strictly concave. We finally relax the conditions of the utility function and provide some sufficient conditions that guarantee the turnpike property and the convergence rate in terms of both primal and dual utility functions.     

Efficiency comparisons between two estimators based on matrix determinant Kantorovich-type inequalities

We first establish two matrix determinant Kantorovich-type inequalities. Then, based on these two and other inequalities, we introduce new efficiency criteria and present their upper bounds to make efficiency comparisons between the ordinary least squares estimator and the best linear unbiased estimator in the general linear model. We provide numerical examples to examine the upper bounds of some new and old efficiency criteria. Received: June 1999     

Walras degrees and probability of a blocking coalition at pareto allocations

We attempt to determine the probability of a blocking coalition from a notion of being non- Walrasian which does not rely on the explicit use of norms. The key concept introduced is that of Walras degrees. Theorem 1 gives the bounds of the probability of blocking in terms of these degrees. Theorem 2 gives the asymptotic conditional probability of a blocking coalition given that coalitions are losing. The relationship between the degrees of the conventional norms on allocations is also investigated.     

Weighted distances between preferences

Individual disagreements are assumed to be reflected in the preferences. Distance functions, e.g., the well-known Kemeny (1959) metric, are used to measure these disagreements. However, a disagreement on how to rank the top two alternatives may be perceived more (or less) than a disagreement on how to rank the bottom two alternatives. We propose two conditions on functions which characterize a class of weighted semi-metric functions. This class of semi-metrics allows to quantify disagreements according to where they occur in preferences. It turns out one of these functions, “the path minimizing function”, is the only metric which generalizes the Kemeny metric.     

Socio‐economic distance and spatial patterns in unemployment

This paper examines the spatial patterns of unemployment in Chicago between 1980 and 1990. We study unemployment clustering with respect to different social and economic distance metrics that reflect the structure of agents' social networks. Specifically, we use physical distance, travel time, and differences in ethnic and occupational distribution between locations. Our goal is to determine whether our estimates of spatial dependence are consistent with models in which agents' employment status is affected by information exchanged locally within their social networks. We present non‐parametric estimates of correlation across Census tracts as a function of each distance metric as well as pairs of metrics, both for unemployment rate itself and after conditioning on a set of tract characteristics. Our results indicate that there is a strong positive and statistically significant degree of spatial dependence in the distribution of raw unemployment rates, for all our metrics. However, once we condition on a set of covariates, most of the spatial autocorrelation is eliminated, with the exception of physical and occupational distance. Racial and ethnic composition variables are the single most important factor in explaining the observed correlation patterns. Copyright © 2002 John Wiley & Sons, Ltd.     

Upper and lower bounds on infinite time ruin probabilities in case of constraints on claim size distributions

Probabilities of ruin (or non-ruin) are solutions of differential or integro-differential equations. Solving these kinds of equations analytically and/or numerically causes a lot of mathematical difficulties. In addition there exists a practical problem of determining, estimating or guessing the distribution of the risk. A realistic way to deal with this problem consists in deriving upper and lower bounds for the ruin probability in case of incomplete information on the distribution F. The present contribution is inspired by and generalises a result of G. Taylor who uses the concept of ordering of risks to order ruin probabilities. We show how some of the results obtained by F. De Vylder for deriving sharp bounds on the stop-loss premium in case of incomplete information can be applied to the evaluation of practical bounds on infinite time ruin probabilities.     

The algebra of Bonferroni bounds: discrete tubes and extensions

Bonferroni, or inclusion-exclusion, bounds and identities have a rich history. They concern the indicator function, and hence the probability content, of the union of sets. In previous work, the authors defined a discrete tube which yields upper and lower bounds which are at least as tight as the standard bounds obtained by truncating inclusion-exclusion identities at particular depths. Here, some connections to other fields are made, based particularly on the algebra of indicator functions. These leads to the consideration of the complexity of more general Boolean statements.