Mean, median, mode II

Summary  In a delightful note R unnenburg [2] has recently discussed the celebrated mean-median-mode inequality. The present note continues this discussion.     

Mean,median, mode IV

This note continues the discussion about the well-known mean-median-mode inequality (Runnenburg , 1978, van Zwet , 1979, and Dharmadhikari and Joag-dev , 1983). A discrete analogue of this inequality is proved under similar conditions as those used for the classical Khinchin unimodality on R .     

Mean or median? (A note on an old problem)

Abstract  The performance of the sample mean and sample median to estimate location for contaminated normal distributions is investigated. Both scale and location contaminations are considered and the problem is illustrated by the "Weight Judging Competition" example introduced by G alton in 1907.     

Mean,Median, Mode III*

Abstract Recently, Van Zwet (1979) has discussed several conditions under which the celebrated mean-median-mode inequality holds. This note points out that Van Zwet's basic condition and its variants have a simple interpretation in terms of a well-known stochastic ordering. The results are slightly more general than Van Zwet's because the definition of unimodality used here (due to Khintchine ) requires neither the existence of a density nor the uniqueness of the mode.     

Mean, Median and Mode in Binomial Distributions

Summary  While studying the median of the binomial distribution we discovered that the mean median-mode inequality, recently discussed in. Statistics Neerlandica by R unnen - burg 141 and V an Z wet [7] for continuous distributions, does not hold for the binomial distribution. If the mean is an integer, then mean = median = mode. In theorem 1 a sufficient condition is given for mode = median = rounded mean. If median and mode differ, the mean lies in between.     

The median voter and the median consumer: Local private goods and population composition

When a product's product provision entails fixed costs, it will be made available only if a sufficient number of people want it. Some products are produced and consumed locally, so that provision requires not only a large group favoring the product but a large number nearby. Just as local governments provide public goods appealing to individuals sharing the median voter's preferences for local public goods, product markets may provide an analogous benefit to individuals sharing living communities whose consumers tend to share his preferences in private goods. Using zip code level data on chain restaurants and restaurants overall, this paper documents how the mix of locally available restaurants responds to the local mix of consumers, with three findings. First, based on survey data on chain restaurant patronage, restaurant preferences differ substantially by race and education. Second, there is a strong relationship between restaurants and population at the zip code level, suggesting that restaurants' geographic markets are small. Finally, the mix of locally available chain restaurants is sensitive to the zip code demographic mix by race and by education. Hence, differentiated product markets provide a benefit—proximity to preferred restaurants—to persons in geographic markets whose customers tend to share their preferences.     

The median of the poisson distribution

The purpose of this paper is twofold: first, to provide a closed form expression for the median of the Poisson distribution and, second, to improve the known estimates of the difference between the median and the mean of the Poisson distribution. We use elementary techniques based on the monotonicity of certain sequences involving tail probabilities of the Poisson distribution and the Central Limit TheoremReceived November 2002     

Strategic participation and the median voter result

In a seminal paper on electoral equilibrium under majority rule, Ledyard (1984) demonstrates that strategic participation by voters results in an electoral equilibrium at the proposal that maximizes the utility of a randomly selected voter. Palfrey and Rosenthal (1985) limit the usefulness of this result by showing that strategic participation rates are miniscule in large electorates, and that the incentive to participate vanishes completely as the electorate grows without bound. The most reasonable modification of Ledyard’s approach that circumvents these criticisms is to allow for a negative cost of voting. We show that when voters can have even an arbitrarily small negative cost of voting, there is an electorate sufficiently large so that any proposal is defeated or tied by the median proposal. This observation raises questions about the existence of electoral equilibrium under strategic participation, and is relevant to the efficiency of elections.     

Computing median unbiased estimates in macroeconometric models

A stochastic simulation procedure is proposed in this paper for obtaining median unbiased (MU) estimates in macroeconometric models. MU estimates are computed for lagged dependent variable (LDV) coefficients in 18 equations of a macroeconometric model. The 2SLS bias for a coefficient, defined as the difference between the 2SLS estimate and the MU estimate, is on average smaller in absolute value than would be expected from Andrews exact results for an equation with only a constant term, time trend, and LDV. The results also show that in a practical sense the estimated biases are not very large because they have little effect on the overall predictive accuracy of the model and on its multiplier properties.     

Mean residual life functions of finite mixtures,order statistics and coherent systems

We study properties of the mean residual life functions of finite mixtures. Specifically, we study ordering properties, monotonicity and the limiting behaviour. We show, under some mild conditions, that the limiting behaviour is similar to that of the strongest member (in the mean residual life order) of the mixture. We also consider the case of negative mixtures (i.e., mixtures with some negative coefficients) which is applied to study the behaviour of the mean residual life of order statistics and coherent systems with possibly dependent components. Partially Supported by Ministerio de Ciencia y Tecnologí a under grant BFM2003-02947 and Fundación Séneca under grant 00698/PI/04.     

Mean value theorem for convex functions

Closed proper convex functions have many properties in common with differentiable functions such as continuity and one-sided directional derivatives. In this paper it is shown that there exists a mean value theorem for such functions with the gradient vector in the differentiable case replaced by an element of the subdifferential in the convex case.Under compactness of the isoquant sets, (minus) the cost functions are convex and with the theorem above it is easy to show that under incomplete specialization and constant returns to scale the prices of international goods determine uniquely the prices of domestic goods if the isoquant sets are linearly independent.     

日本大公司的小家子气

日本是一个高消费的社会,但仍有一些别具眼光的企业家保持着朴素节俭的处世风格,对改善企业经营管理起到潜移默化的促进作用.     

胡伯林的中庸之学

一位跨国公司中国区的领导,如何以中庸之道寻找西方商业规范与东方本土文化之间的平衡?     

Bayesian median autoregression for robust time series forecasting

We develop a Bayesian median autoregressive (BayesMAR) model for time series forecasting. The proposed method utilizes time-varying quantile regression at the median, favorably inheriting the robustness of median regression in contrast to the widely used mean-based methods. Motivated by a working Laplace likelihood approach in Bayesian quantile regression, BayesMAR adopts a parametric model bearing the same structure as autoregressive models by altering the Gaussian error to Laplace, leading to a simple, robust, and interpretable modeling strategy for time series forecasting. We estimate model parameters by Markov chain Monte Carlo. Bayesian model averaging is used to account for model uncertainty, including the uncertainty in the autoregressive order, in addition to a Bayesian model selection approach. The proposed methods are illustrated using simulations and real data applications. An application to U.S. macroeconomic data forecasting shows that BayesMAR leads to favorable and often superior predictive performance compared to the selected mean-based alternatives under various loss functions that encompass both point and probabilistic forecasts. The proposed methods are generic and can be used to complement a rich class of methods that build on autoregressive models.     

Measuring housing affordability: Looking beyond the median

We draw a distinction between the concepts of purchase affordability (whether a household is able to borrow enough funds to purchase a house) and repayment affordability (the burden imposed on a household of repaying the mortgage). We operationalize this distinction in the context of a new methodology for constructing affordability measures that draws on the value-at-risk concept and takes account of the whole distribution of household income and house prices rather than just the median. Empirically we find that the distinction between purchase and repayment affordability can be pronounced. In the Sydney prime mortgage market over the period 1996–2006, repayment affordability deteriorated very significantly while purchase affordability remained quite stable. This difference can be attributed to the loosening of credit constraints in the mortgage market which it seems has carried through primarily into higher house prices rather than an improvement in purchase affordability. We also show how median house-price-to-income ratio measures of affordability can be extended to take account of the whole distribution of income and house prices, and how as a result of differential skewness in the house price and income distributions the housing affordability problem may be significantly worse for lower income households than suggested by standard median measures.     

Properties of a distribution-free two-stage two-sample median test

Abstract  In this paper we consider a class of distribution-free two-stage two-sample median tests. These procedures are based on a fixed size sample from one population, while the observations from the second population are collected in a truncated sequential fashion. Necessary tables are provided enabling us to select a particular member from the class such that arbitrarily preset power levels are approximately obtained. We also present asymptotic expressions and tables for the expected number of observations from the second population. Finally, comparisons between the asymptotic version of this two-stage median test and the binomial sequential probability ratio test due to W ald (1947) are discussed.     

Mean square error estimation in multi-stage sampling

Summary: Suppose for a homogeneous linear unbiased function of the sampled first stage unit (fsu)-values taken as an estimator of a survey population total, the sampling variance is expressed as a homogeneous quadratic function of the fsu-values. When the fsu-values are not ascertainable but unbiased estimators for them are separately available through sampling in later stages and substituted into the estimator, Raj (1968) gave a simple variance estimator formula for this multi-stage estimator of the population total. He requires that the variances of the estimated fsu-values in sampling at later stages and their unbiased estimators are available in certain `simple forms'. For the same set-up Rao (1975) derived an alternative variance estimator when the later stage sampling variances have more ‘complex forms’. Here we pursue with Raj's (1968) simple forms to derive a few alternative variance and mean square error estimators when the condition of homogeneity or unbiasedness in the original estimator of the total is relaxed and the variance of the original estimator is not expressed as a quadratic form.  We illustrate a particular three-stage sampling strategy and present a simulation-based numerical exercise showing the relative efficacies of two alternative variance estimators. Received: 19 February 1999