As seen in Chapter 4, usually the marginal distributions of financial time series are not well fit by normal distributions. Fortunately, there are a number of suitable alternative models, such as t-distributions, generalized error distributions, and skewed versions of t- and generalized error distributions. All of these will be introduced in this chapter.

Let {v_n(θ)} be a sequence of statistics such that whenθ =θ₀,v_n(θ₀) $$\mathop \to \limits^D $$ N_p(0,Σ), whereΣ is of rankp andθ εR^d. Suppose that underθ =θ₀, {Σ_n} is a sequence of consistent estimators ofΣ. Wald (1943) shows thatv_n^T (θ₀)Σ_n⁻¹v_n(θ₀) $$\mathop \to \limits^D $$ x²(p). It often happens thatv_n(θ₀) $$\mathop \to \limits^D $$ N_p(0,Σ) holds butΣ is singular. Moore (1977) states that under certain assumptionsv_n^T (θ₀)Σ_n⁻v_n(θ₀) $$\mathop \to \limits^D $$ x²(k), wherek = rank (Σ) andΣ_n⁻ is a generalized inverse ofΣ_n. However, Moore’s result as stated is incorrect. It needs the additional assumption that rank (Σ_n) =k forn sufficiently large. In this article, we show that Moore’s result (as corrected) holds under somewhat different, but easier to verify, assumptions.

In classical extreme value theory probabilities of extreme events are estimated assuming all the components of a random vector to be in a domain of attraction of an extreme value distribution. In contrast, the conditional extreme value model assumes a domain of attraction condition on a sub-collection of the components of a multivariate random vector. This model has been studied in Heffernan and Tawn (JRSS B 66(3):497–546, 2004), Heffernan and Resnick (Ann Appl Probab 17(2):537–571, 2007), and Das and Resnick (2009). In this paper we propose three statistics which act as tools to detect this model in a bivariate set-up. In addition, the proposed statistics also help to distinguish between two forms of the limit measure that is obtained in the model.

We analyze the computational efficiency of approximate Bayesian computation (ABC), which approximates a likelihood function by drawing pseudo-samples from the associated model. For the rejection sampling version of ABC, it is known that multiple pseudo-samples cannot substantially increase (and can substantially decrease) the efficiency of the algorithm as compared to employing a high-variance estimate based on a single pseudo-sample. We show that this conclusion also holds for a Markov chain Monte Carlo version of ABC, implying that it is unnecessary to tune the number of pseudo-samples used in ABC-MCMC. This conclusion is in contrast to particle MCMC methods, for which increasing the number of particles can provide large gains in computational efficiency.

Summary

For regression problems where observations may be taken at points in a set X which does not coincide with the set Y on which the regression function is of interest, we consider the problem of finding a design (allocation of observations) which minimizes the maximum over Y of the variance function (of estimated regression). Specific examples are calculated for one-dimensional polynomial regression when Y is much smaller than or much larger than X. A related problem of optimum estimation of two regression coefficients is studied. This paper contains proofs of results first announced at the 1962 Minneapolis Meeting of the Institute of Mathematical Statistics. No prior knowledge of design theory is needed to read this paper.

This article considers the convergence to steady states of Markov processes generated by the action of successive i.i.d. monotone maps on a subset S of an Eucledian space. Without requiring irreducibility or Harris recurrence, a “splitting” condition guarantees the existence of a unique invariant probability as well as an exponential rate of convergence to it in an appropriate metric. For a special class of Harris recurrent processes on [0,∞) of interest in economics, environmental studies and queuing theory, criteria are derived for polynomial and exponential rates of convergence to equilibrium in total variation distance. Central limit theorems follow as consequences.

When residual analysis shows that the residuals are correlated, then one of the key assumptions of the linear model does not hold, and tests and confidence intervals based on this assumption are invalid and cannot be trusted. Fortunately, there is a solution to this problem: Replace the assumption of independent noise by the weaker assumption that the noise process is station- ary but possibly correlated. One could, for example, assume that the noise is an ARMA process. This is the strategy we will discuss in this section.