Serial and nonserial sign-and-rank statistics: asymptotic representation and asymptotic normality

The classical theory of rank-based inference is entirely based either on ordinary ranks, which do not allow for considering location (intercept) parameters, or on signed ranks, which require an assumption of symmetry. If the median, in the absence of a symmetry assumption, is considered as a location parameter, the maximal invariance property of ordinary ranks is lost to the ranks and the signs. This new maximal invariant thus suggests a new class of statistics, based on ordinary ranks and signs. An asymptotic representation theory \`{a} la H\'{a}jek is developed here for such statistics, both in the nonserial and in the serial case. The corresponding asymptotic normality results clearly show how the signs add a separate contribution to the asymptotic variance, hence, potentially, to asymptotic efficiency. As shown by Hallin and Werker [Bernoulli 9 (2003) 137--165], conditioning in an appropriate way on the maximal invariant potentially even leads to semiparametrically efficient inference. Applications to semiparametric inference in regression and time series models with median restrictions are treated in detail in an upcoming companion paper.Comment: Published at http://dx.doi.org/10.1214/009053605000000769 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

An Alternative Asymptotic Analysis of Residual-Based Statistics

This paper presents an alternative method to derive the limiting distribution of residual-based statistics. Our method does not impose an explicit assumption of (asymptotic) smoothness of the statistic of interest with respect to the model's parameters. and, thus, is especially useful in cases where such smoothness is difficult to establish. Instead, we use a locally uniform convergence in distribution condition, which is automatically satisfied by residual-based specification test statistics. To illustrate, we derive the limiting distribution of a new functional form specification test for discrete choice models, as well as a runs-based tests for conditional symmetry in dynamic volatility models.Le Cam's third lemma, Local Asymptotic Normality (LAN)

The asymptotic structure of nearly unstable non-negative integer-valued AR(1) models

This paper considers non-negative integer-valued autoregressive processes where the autoregression parameter is close to unity. We consider the asymptotics of this `near unit root' situation. The local asymptotic structure of the likelihood ratios of the model is obtained, showing that the limit experiment is Poissonian. To illustrate the statistical consequences we discuss efficient estimation of the autoregression parameter and efficient testing for a unit root.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ153 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

On Bounded Completeness and the L1L_1-Denseness of Likelihood Ratios

The classical concept of bounded completeness and its relation to sufficiency and ancillarity play a fundamental role in unbiased estimation, unbiased testing, and the validity of inference in the presence of nuisance parameters. In this short note, we provide a direct proof of a little-known result by \cite{Far62} on a characterization of bounded completeness based on an $L^1$ denseness property of the linear span of likelihood ratios. As an application, we show that an experiment with infinite-dimensional observation space is boundedly complete iff suitably chosen restricted subexperiments with finite-dimensional observation spaces are

Health cost risk:A potential solution to the annuity puzzle

We find that health cost risk lowers optimal annuity demand at retirement. If medical expenses can be sizeable early in retirement, full annuitisation at retirement is no longer optimal because agents do not have enough time to build a liquid wealth buffer. Furthermore, large deviations from optimal annuitisation levels lead to small utility differences. Our results suggest that health cost risk can explain a large proportion of empirically observed annuity choices. Finally, allowing additional annuitisation after retirement results in welfare gains of at most 2.5% when facing health cost risk, and negligible gains without this risk

Linear Factor Models and the Estimation of Expected Returns

Linear factor models of asset pricing imply a linear relationship between expected returns of assets and exposures to one or more sources of risk. We show that exploiting this linear relationship leads to statistical gains of up to 31% in variances when estimating expected returns on individual assets over historical averages. When the factors are weakly correlated with assets, i.e. β’s are small, and the interest is in estimating expected excess returns, that is risk premiums, on individual assets rather than the prices of risk, the Generalized Method of Moment estimators of risk premiums does lead to reliable inference, i.e. limiting variances suffer from neither lack of identification nor unboundedness. If the factor model is misspecified in the sense of an omitted factor, we show that factor model-based estimates may be inconsistent. However, we show that adding an alpha to the model capturing mispricing only leads to consistent estimators in case of traded factors. Moreover, our simulation experiment documents that using the more precise estimates of expected returns based on factor{models rather than the historical averages translates into significant improvements in the out-of-sample performances of the optimal portfolios

Efficient Estimation in Semiparametric Time Series: the ACD Model

In this paper we consider efficient estimation in semiparametric ACD models. We consider a suite of model specifications that impose less and less structure. We calculate the corresponding efficiency bounds, discuss the construction of efficient estimators in each case, and study tvide a simulation study that shows the practical gain from using the proposed semiparametric procedures. We find that, although one does not gain as much as theory suggests, these semiparametric procedures definitely outperform more classical procedures. We apply the procedures to model semiparametrically durations observed on the Paris Bourse for the Alcatel stock in July and August 1996.

Semiparametrically point-optimal hybrid rank tests for unit roots

We propose a new class of unit root tests that exploits invariance properties in the Locally Asymptotically Brownian Functional limit experiment associated to the unit root model. The invariance structures naturally suggest tests that are based on the ranks of the increments of the observations, their average and an assumed reference density for the innovations. The tests are semiparametric in the sense that they are valid, that is, have the correct (asymptotic) size, irrespective of the true innovation density. For a correctly specified reference density, our test is point-optimal and nearly efficient. For arbitrary reference densities, we establish a Chernoff–Savage-type result, that is, our test performs as well as commonly used tests under Gaussian innovations but has improved power under other, for example, fat-tailed or skewed, innovation distributions. To avoid nonparametric estimation, we propose a simplified version of our test that exhibits the same asymptotic properties, except for the Chernoff–Savage result that we are only able to demonstrate by means of simulations

On quadratic expansions of log likelihoods and a general asymptotic linearity result

Irrespective of the statistical model under study, the derivation of limits, in the Le Cam sense, of sequences of local experiments (see [7]-[10]) often follows along very similar lines, essentially involving differentiability in quadratic mean of square roots of (conditional) densities. This chapter establishes two abstract and very general results providing sufficient and nearly necessary conditions for (i) the existence of a quadratic expansion, and (ii) the asymptotic linearity of local log-likelihood ratios (asymptotic linearity is needed, for instance, when unspecified model parameters are to be replaced, in some statistic of interest, with some preliminary estimator). Such results have been established, for locally asymptotically normal (LAN) models involving independent and identically distributed observations, by, e.g., [1], [11] and [12]. Similar results are provided here for models exhibiting serial dependencies which, so far, have been treated on a case-by-case basis (see [4] and [5] for typical examples) and, in general, under stronger regularity assumptions. Unlike their i.i.d. counterparts, our results extend beyond the context of LAN experiments, so that non-stationary unit-root time series and cointegration models, for instance, also can be handled (see [6])