Quantiles, Expectiles and Splines

A time-varying quantile can be fitted to a sequence of observations by formulating a time series model for the corresponding population quantile and iteratively applying a suitably modified state space signal extraction algorithm. It is shown that such time-varying quantiles satisfy the defining property of fixed quantiles in having the appropriate number of observations above and below. Expectiles are similar to quantiles except that they are defined by tail expectations. Like quantiles, time varying expectiles can be estimated by a state space signal extraction algorithm and they satisfy properties that generalize the moment conditions associated with fixed expectiles. Time-varying quantiles and expectiles provide information on various aspects of a time series, such as dispersion and asymmetry, while estimates at the end of the series provide the basis for forecasting. Because the state space form can handle irregularly spaced observations, the proposed algorithms can be easily adapted to provide a viable means of computing spline-based non-parametric quantile and expectile regressions

Variance estimation for a low-income proportion

Proportions below a given fraction of a quantile of an income distribution are often estimated from survey data in poverty comparisons. We consider the estimation of the variance of such a proportion, estimated from Family Expenditure Survey data. We show how a linearization method of variance estimation may be applied to this proportion, allowing for the effects of both a complex sampling design and weighting by a raking method to population controls. We show that, for 1998-99 data, the estimated variances are always increased when allowance is made for the design and raking weights, the principal effect arising from the design. We also study the properties of a simplified variance estimator and discuss extensions to a wider class of poverty measures

Time-Varying Quantiles

A time-varying quantile can be fitted to a sequence of observations by formulating a time series model for the corresponding population quantile and iteratively applying a suitably modified state space signal extraction algorithm. Quantiles estimated in this way provide information on various aspects of a time series, including dispersion, asymmetry and, for financial applications, value at risk. Tests for the constancy of quantiles, and associated contrasts, are constructed using indicator variables; these tests have a similar form to stationarity tests and, under the null hypothesis, their asymptotic distributions belong to the Cramér von Mises family. Estimates of the quantiles at the end of the series provide the basis for forecasting. As such they offer an alternative to conditional quantile autoregressions and, at the same time, give some insight into their structure and potential drawbacks

Inference on sets in finance

In this paper we introduce various set inference problems as they appear in finance and propose practical and powerful inferential tools. Our tools will be applicable to any problem where the set of interest solves a system of smooth estimable inequalities, though we will particularly focus on the following two problems: the admissible mean-variance sets of stochastic discount factors and the admissible mean-variance sets of asset portfolios. We propose to make inference on such sets using weighted likelihood-ratio and Wald type statistics, building upon and substantially enriching the available methods for inference on sets.

Small area estimation of the homeless in Los Angeles: An application of cost-sensitive stochastic gradient boosting

In many metropolitan areas efforts are made to count the homeless to ensure proper provision of social services. Some areas are very large, which makes spatial sampling a viable alternative to an enumeration of the entire terrain. Counts are observed in sampled regions but must be imputed in unvisited areas. Along with the imputation process, the costs of underestimating and overestimating may be different. For example, if precise estimation in areas with large homeless c ounts is critical, then underestimation should be penalized more than overestimation in the loss function. We analyze data from the 2004--2005 Los Angeles County homeless study using an augmentation of $L_1$ stochastic gradient boosting that can weight overestimates and underestimates asymmetrically. We discuss our choice to utilize stochastic gradient boosting over other function estimation procedures. In-sample fitted and out-of-sample imputed values, as well as relationships between the response and predictors, are analyzed for various cost functions. Practical usage and policy implications of these results are discussed briefly.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS328 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Uniform Bahadur Representation for Nonparametric Censored Quantile Regression: A Redistribution-of-Mass Approach

Censored quantile regressions have received a great deal of attention in the literature. In a linear setup, recent research has found that an estimator based on the idea of “redistribution-of-mass” in Efron (1967, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 4, pp. 831–853, University of California Press) has better numerical performance than other available methods. In this paper, this idea is combined with the local polynomial kernel smoothing for nonparametric quantile regression of censored data. We derive the uniform Bahadur representation for the estimator and, more importantly, give theoretical justification for its improved efficiency over existing estimation methods. We include an example to illustrate the usefulness of such a uniform representation in the context of sufficient dimension reduction in regression analysis. Finally, simulations are used to investigate the finite sample performance of the new estimator