Bringing game theory to hypothesis testing: Establishing finite sample bounds on inference

Small sample properties are of fundamental interest when only limited data is avail-able. Exact inference is limited by constraints imposed by speci.c nonrandomizedtests and of course also by lack of more data. These e¤ects can be separated as we propose to evaluate a test by comparing its type II error to the minimal type II error among all tests for the given sample. Game theory is used to establish this minimal type II error, the associated randomized test is characterized as part of a Nash equilibrium of a .ctitious game against nature.We use this method to investigate sequential tests for the di¤erence between twomeans when outcomes are constrained to belong to a given bounded set. Tests ofinequality and of noninferiority are included. We .nd that inference in terms oftype II error based on a balanced sample cannot be improved by sequential sampling or even by observing counter factual evidence providing there is a reasonable gap between the hypotheses.

Exact and approximate stepdown methods for multiple hypothesis testing

Consider the problem of testing k hypotheses simultaneously. In this paper,we discuss finite and large sample theory of stepdown methods that providecontrol of the familywise error rate (FWE). In order to improve upon theBonferroni method or Holm's (1979) stepdown method, Westfall and Young(1993) make eective use of resampling to construct stepdown methods thatimplicitly estimate the dependence structure of the test statistics. However,their methods depend on an assumption called subset pivotality. The goalof this paper is to construct general stepdown methods that do not requiresuch an assumption. In order to accomplish this, we take a close look atwhat makes stepdown procedures work, and a key component is a monotonicityrequirement of critical values. By imposing such monotonicity on estimatedcritical values (which is not an assumption on the model but an assumptionon the method), it is demonstrated that the problem of constructing a validmultiple test procedure which controls the FWE can be reduced to the problemof contructing a single test which controls the usual probability of a Type 1error. This reduction allows us to draw upon an enormous resamplingliterature as a general means of test contruction.

Lag-one autocorrelation in short series: Estimation and hypothesis testing

In the first part of the study, nine estimators of the first-order autoregressive parameter are reviewed and a new estimator is proposed. The relationships and discrepancies between the estimators are discussed in order to achieve a clear differentiation. In the second part of the study, the precision in the estimation of autocorrelation is studied. The performance of the ten lag-one autocorrelation estimators is compared in terms of Mean Square Error (combining bias and variance) using data series generated by Monte Carlo simulation. The results show that there is not a single optimal estimator for all conditions, suggesting that the estimator ought to be chosen according to sample size and to the information available of the possible direction of the serial dependence. Additionally, the probability of labelling an actually existing autocorrelation as statistically significant is explored using Monte Carlo sampling. The power estimates obtained are quite similar among the tests associated with the different estimators. These estimates evidence the small probability of detecting autocorrelation in series with less than 20 measurement times.

A data-dependent skeleton estimate and a scale-sensitive dimension for classification

The classical binary classification problem is investigatedwhen it is known in advance that the posterior probability function(or regression function) belongs to some class of functions. We introduceand analyze a method which effectively exploits this knowledge. The methodis based on minimizing the empirical risk over a carefully selected``skeleton'' of the class of regression functions. The skeleton is acovering of the class based on a data--dependent metric, especiallyfitted for classification. A new scale--sensitive dimension isintroduced which is more useful for the studied classification problemthan other, previously defined, dimension measures. This fact isdemonstrated by performance bounds for the skeleton estimate in termsof the new dimension.

A nonlinear threshold model for the dependence of extremes of stationary sequences

One of the main implications of the efficient market hypothesis (EMH) is that expected future returns on financial assets are not predictable if investors are risk neutral. In this paper we argue that financial time series offer more information than that this hypothesis seems to supply. In particular we postulate that runs of very large returns can be predictable for small time periods. In order to prove this we propose a TAR(3,1)-GARCH(1,1) model that is able to describe two different types of extreme events: a first type generated by large uncertainty regimes where runs of extremes are not predictable and a second type where extremes come from isolated dread/joy events. This model is new in the literature in nonlinear processes. Its novelty resides on two features of the model that make it different from previous TAR methodologies. The regimes are motivated by the occurrence of extreme values and the threshold variable is defined by the shock affecting the process in the preceding period. In this way this model is able to uncover dependence and clustering of extremes in high as well as in low volatility periods. This model is tested with data from General Motors stocks prices corresponding to two crises that had a substantial impact in financial markets worldwide; the Black Monday of October 1987 and September 11th, 2001. By analyzing the periods around these crises we find evidence of statistical significance of our model and thereby of predictability of extremes for September 11th but not for Black Monday. These findings support the hypotheses of a big negative event producing runs of negative returns in the first case, and of the burst of a worldwide stock market bubble in the second example. JEL classification: C12; C15; C22; C51 Keywords and Phrases: asymmetries, crises, extreme values, hypothesis testing, leverage effect, nonlinearities, threshold models

Statistical Analysis of Eco-Toxicologic Data of Mammals from a Polluted Area

Several eco-toxicological studies have shown that insectivorous mammals, due to theirfeeding habits, easily accumulate high amounts of pollutants in relation to other mammal species. To assess the bio-accumulation levels of toxic metals and their in°uenceon essential metals, we quantified the concentration of 19 elements (Ca, K, Fe, B, P,S, Na, Al, Zn, Ba, Rb, Sr, Cu, Mn, Hg, Cd, Mo, Cr and Pb) in bones of 105 greaterwhite-toothed shrews (Crocidura russula) from a polluted (Ebro Delta) and a control(Medas Islands) area. Since chemical contents of a bio-indicator are mainly compositional data, conventional statistical analyses currently used in eco-toxicology can givemisleading results. Therefore, to improve the interpretation of the data obtained, weused statistical techniques for compositional data analysis to define groups of metalsand to evaluate the relationships between them, from an inter-population viewpoint.Hypothesis testing on the adequate balance-coordinates allow us to confirm intuitionbased hypothesis and some previous results. The main statistical goal was to test equalmeans of balance-coordinates for the two defined populations. After checking normality,one-way ANOVA or Mann-Whitney tests were carried out for the inter-group balances

Altruism, Equity, and Reciprocity in a Gift-Exchange Experiment: An Encompassing Approach

Considerable experimental evidence suggests that non-pecuniary motives must be addressed when modeling behavior in economic contexts. Recent models of non-pecuniary motives can be classified as either altruism- based, equity-based, or reciprocity-based. We estimate and compare leading approaches in these categories, using experimental data. We then offer a flexible approach that nests the above three approaches, thereby allowing for nested hypothesis testing and for determining the relative strength of each of the competing theories. In addition, the encompassing approach provides a functional form for utility in different settings without the restrictive nature of the approaches nested within it. Using this flexible form for nested tests, we find that intentional reciprocity, distributive concerns, and altruistic considerations all play a significant role in players' decisions.

A new method for constructing exact tests without making any assumptions

We present a new method for constructing exact distribution-free tests (and confidence intervals) for variables that can generate more than two possible outcomes.This method separates the search for an exact test from the goal to create a non-randomized test. Randomization is used to extend any exact test relating to meansof variables with finitely many outcomes to variables with outcomes belonging to agiven bounded set. Tests in terms of variance and covariance are reduced to testsrelating to means. Randomness is then eliminated in a separate step.This method is used to create confidence intervals for the difference between twomeans (or variances) and tests of stochastic inequality and correlation.

Strong minimax lower bounds for learning

Minimax lower bounds for concept learning state, for example, thatfor each sample size $n$ and learning rule $g_n$, there exists a distributionof the observation $X$ and a concept $C$ to be learnt such that the expectederror of $g_n$ is at least a constant times $V/n$, where $V$ is the VC dimensionof the concept class. However, these bounds do not tell anything about therate of decrease of the error for a {\sl fixed} distribution--concept pair.\\In this paper we investigate minimax lower bounds in such a--stronger--sense.We show that for several natural $k$--parameter concept classes, includingthe class of linear halfspaces, the class of balls, the class of polyhedrawith a certain number of faces, and a class of neural networks, for any{\sl sequence} of learning rules $\{g_n\}$, there exists a fixed distributionof $X$ and a fixed concept $C$ such that the expected error is larger thana constant times $k/n$ for {\sl infinitely many n}. We also obtain suchstrong minimax lower bounds for the tail distribution of the probabilityof error, which extend the corresponding minimax lower bounds.

The minimax distortion redundancy in empirical quantizer design

We obtain minimax lower and upper bounds for the expected distortionredundancy of empirically designed vector quantizers. We show that the meansquared distortion of a vector quantizer designed from $n$ i.i.d. datapoints using any design algorithm is at least $\Omega (n^{-1/2})$ awayfrom the optimal distortion for some distribution on a bounded subset of${\cal R}^d$. Together with existing upper bounds this result shows thatthe minimax distortion redundancy for empirical quantizer design, as afunction of the size of the training data, is asymptotically on the orderof $n^{1/2}$. We also derive a new upper bound for the performance of theempirically optimal quantizer.

Exact tests for correlation and for the slope in simple linear regressions without making assumptions

We present an exact test for whether two random variables that have known bounds on their support are negatively correlated. The alternative hypothesis is that they are not negatively correlated. No assumptions are made on the underlying distributions. We show by example that the Spearman rank correlation test as the competing exact test of correlation in nonparametric settings rests on an additional assumption on the data generating process without which it is not valid as a test for correlation.We then show how to test for the significance of the slope in a linear regression analysis that invovles a single independent variable and where outcomes of the dependent variable belong to a known bounded set.

A note on robust detection

We introduce a simple new hypothesis testing procedure, which,based on an independent sample drawn from a certain density, detects which of $k$ nominal densities is the true density is closest to, under the total variation (L_{1}) distance. Weobtain a density-free uniform exponential bound for the probability of false detection.

Altruism, equity and reciprocity in a gift-exchange experiment: An encompassing approach

Considerable experimental evidence suggests that non-pecuniary motivesmust be addressed when modeling behavior in economic contexts. Recentmodels of non-pecuniary motives can be classified as either altruism-based, equity-based, or reciprocity-based. We estimate and compareleading approaches in these categories, using experimental data. Wethen offer a flexible approach that nests the above three approaches,thereby allowing for nested hypothesis testing and for determiningthe relative strength of each of the competing theories. In addition,the encompassing approach provides a functional form for utility in different settings without the restrictive nature of the approaches nested within it. Using this flexible form for nested tests, we findthat intentional reciprocity, distributive concerns, and altruisticconsiderations all play a significant role in players' decisions.

Is the matching function Cobb-Douglas?

[cat] Estudiem les propietats teòriques que una funció d.emparellament ha de satisfer per tal de representar un mercat laboral amb friccions dins d'un model d'equilibri general amb emparellament aleatori. Analitzem el cas Cobb-Douglas, CES i altres formes funcionals per a la funció d.emparellament. Els nostres resultats estableixen restriccions sobre els paràmetres d'aquests formes funcionals per assegurar que l.equilibri és interior. Aquestes restriccions aporten raons teòriques per escollir entre diverses formes funcionals i permeten dissenyar tests d'error d'especificació de model en els treballs empírics.

Is the matching function Cobb-Douglas?

[cat] Estudiem les propietats teòriques que una funció d.emparellament ha de satisfer per tal de representar un mercat laboral amb friccions dins d'un model d'equilibri general amb emparellament aleatori. Analitzem el cas Cobb-Douglas, CES i altres formes funcionals per a la funció d.emparellament. Els nostres resultats estableixen restriccions sobre els paràmetres d'aquests formes funcionals per assegurar que l.equilibri és interior. Aquestes restriccions aporten raons teòriques per escollir entre diverses formes funcionals i permeten dissenyar tests d'error d'especificació de model en els treballs empírics.