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We consider conditional distribution and conditional density functionals in the space of generalized functions. The approach follows Phillips (1985, 1991, 1995) who employed generalized functions to overcome non-differentiability in order to develop expansions. We obtain the limit of the kernel estimators for weakly dependent data, even under non-differentiability of the distribution function; the limit Gaussian process is characterized as a stochastic random functional (random generalized function) on the suitable function space. An alternative simple to compute estimator based on the empirical distribution function is proposed for the generalized random functional. For test statistics based on this estimator, limit properties are established. A Monte Carlo experiment demonstrates good finite sample performance of the statistics for testing logit and probit specification in binary choice models.

There has been growing interest in recent years in modelling various poverty‐related issues. However, there have not been many attempts at empirical estimation of best‐fit income distribution functions with an objective of subsequent use in poverty focused models. The purpose of this paper is to fill this gap by empirically estimating best‐fit income distribution functions for different household income groups and computing poverty and inequality indices for Sri Lanka.

The authors empirically estimated a number of popular distribution functions found in the income distribution literature to find the best‐fit income distribution using household income and expenditure survey data for Sri Lanka and subsequently estimated various poverty and inequality measures.

The results show that the income distributions of all low‐income household groups follow the beta general probability distribution. The poverty measures derived using these distributions show that among the different income groups, the estate low‐income group has the highest incidence of poverty, followed by the rural low‐income group.

According to the best of the authors' knowledge, empirical estimation of income distribution functions for South Asia has never been attempted. The results of this study, even though based on Sri Lankan data, will be relevant to most developing countries in South Asia and will be very useful in developing poverty alleviation strategies.

Purpose – In this chapter, copula theory is used to model dependence structure between hedge fund returns series.

Methodology/approach – Goodness-of-fit tests, based on the Kendall's functions, are applied as selection criteria of the “best” copula. After estimating the parametric copula that best fits the used data, we apply previous results to construct the cumulative distribution functions of the equally weighted portfolios.

Findings – The empirical validation shows that copula clearly allows better estimation of portfolio returns including hedge funds. The three studied portfolios reject the assumption of multivariate normality of returns. The chosen structure is often of Student type when only indices are considered. In the case of portfolios composed by only hedge funds, the dependence structure is of Franck type.

Originality/value of the chapter – Introducing goodness-of-fit bootstrap method to validate the choice of the best structure of dependence is relevant for hedge fund portfolios. Copulas would be introduced to provide better estimations of performance measures.

This chapter reviews recent developments in the density discontinuity approach. It is well known that agents having perfect control of the forcing variable will invalidate the popular regression discontinuity designs (RDDs). To detect the manipulation of the forcing variable, McCrary (2008) developed a test based on the discontinuity in the density around the threshold. Recent papers have noted that the sorting patterns around the threshold are often either the researcher’s object of interest or may relate to structural parameters such as tax elasticities through known functions. This, in turn, implies that the behavior of the distribution around the threshold is not only informative of the validity of a standard RDD; it can also be used to recover policy-relevant parameters and perform counterfactual exercises.

This paper provides a theoretical framework for application of Chance-Constrained Programming (CCP) in situations where the coefficient matrix is random and its elements are not normally distributed. Much of the CCP literature proceeds to derive deterministic equivalent in computationally implementable form on the assumption of “normality”. However, in many applications, such as air pollution control, right skewed distributions are more likely to occur. Two types of models are considered in this paper. One assumes an exponential distribution of matrix coefficients, and another one uses an empirical approach. In case of exponential distributions, it is possible to derive exact “deterministic” equivalent to the chance-constrained program. Each row of the coefficient matrix is assumed to consist of independent, exponentially distributed random variables and a simple example illustrates the complexities associated with finding a numerical solution to the associated deterministic equivalent. In our empirical approach, on the other hand, simulated data typically encountered in air pollution control are provided, and the data-driven (empirical) solution to the implicit form of deterministic equivalent is obtained. Post-optimality analyses on model results are performed and risk implications of these decisions are discussed. Conclusions are drawn and directions for future research are indicated.

This chapter proposes a test for a parametric specification of the autoregressive function of a given stationary autoregressive time series. This test is based on the integrated square difference between the empirical distribution function estimate and a convolution-type distribution function estimate of the stationary distribution function obtained from the autoregressive residuals. Some asymptotic properties of the proposed convolution-type distribution function estimate are studied when the model’s innovation density is unknown. These properties are in turn used to derive the asymptotic null distribution of the proposed test statistic. We also discuss some finite sample properties of the test statistic based on the block bootstrap methodology. A simulation study shows that the proposed test competes favorably with some existing tests in terms of the empirical level and power.

Fat‐tailed distributions have been found in many financial and economic variables ranging from forecasting returns on financial assets to modeling recovery distributions in bankruptcies. They have also been found in numerous insurance applications such as catastrophic insurance claims and in value‐at‐risk measures employed by risk managers. Financial applications include:

To discuss subcopula estimation for discrete models.

The convergence of estimators is considered under the weak convergence of distribution functions and its equivalent properties known in prior works.

The domain of the true subcopula associated with discrete random variables is found to be discrete on the interior of the unit hypercube. The construction of an estimator in which their domains have the same form as that of the true subcopula is provided, in case, the marginal distributions are binomial.

To the best of our knowledge, this is the first time such an estimator is defined and proved to be converged to the true subcopula.

Purpose – The purpose of this chapter is to estimate non-Gaussian distributions by means of Johnson distributions. An empirical illustration on hedge fund returns is detailed.

Methodology/approach – To fit non-Gaussian distributions, the chapter introduces the family of Johnson distributions and its general extensions. We use both parametric and non-parametric approaches. In a first step, we analyze the serial correlation of our sample of hedge fund returns and unsmooth the series to correct the correlations. Then, we estimate the distribution by the standard Johnson system of laws. Finally, we search for a more general distribution of Johnson type, using a non-parametric approach.

Findings – We use data from the indexes Credit Suisse/Tremont Hedge Fund (CSFB/Tremont) provided by Credit Suisse. For the parametric approach, we find that the SU Johnson distribution is the most appropriate, except for the Managed Futures. For the non-parametric approach, we determine the best polynomial approximation of the function characterizing the transformation from the initial Gaussian law to the generalized Johnson distribution.

Originality/value of chapter – These findings are novel since we use an extension of the Johnson distributions to better fit non-Gaussian distributions, in particular in the case of hedge fund returns. We illustrate the power of this methodology that can be further developed in the multidimensional case.

In this article we design an econometric test for monotone comparative statics (MCS) often found in models with multiple equilibria. Our test exploits the observable implications of the MCS prediction: that the extreme (high and low) conditiona l quantiles of the dependent variable increase monotonically with the explanatory variable. The main contribution of the article is to derive a likelihood-ratio test, which to the best of our knowledge is the first econometric test of MCS proposed in the literature. The test is an asymptotic “chi-bar squared” test for order restrictions on intermediate conditional quantiles. The key features of our approach are: (1) we do not need to estimate the underlying nonparametric model relating the dependent and explanatory variables to the latent disturbances; (2) we make few assumptions on the cardinality, location, or probabilities over equilibria. In particular, one can implement our test without assuming an equilibrium selection rule.