Some properties of extreme stable laws and related infinitely divisible random variables

In the course of studying the moment sequence {nⁿ:n=0,1,…}$\{n^n:n=0,1,\dots \}$, Eaton et al. [1971. On extreme stable laws and some applications. J. Appl. Probab. 8, 794–801] have shown that this sequence, which is, indeed, the moment sequence of a log-extreme stable law with characteristic exponent γ=1$\gamma =1$, corresponds to a scale mixture of exponential distributions and hence to a distribution with decreasing failure rate. Following essentially the approach of Shanbhag et al. [1977. Some further results in infinite divisibility. Math. Proc. Cambridge Philos. Soc. 82, 289–295] we show that, under certain conditions, log-extreme stable laws with characteristic exponent γ∈[1,2)$\gamma \in [1,2)$ are scale mixtures of exponential distributions and hence are infinitely divisible and have decreasing failure rates. In addition, we study the moment problem associated with the log-extreme stable laws with characteristic exponent γ∈(0,2]$\gamma \in (0,2]$ and throw further light on the existing literature on the subject. As a by-product, we show that generalized Poisson and generalized negative binomial distributions are mixed Poisson distributions. Finally, we address some relevant questions on structural aspects of infinitely divisible distributions, and make new observations, including in particular that certain results appearing in Steutel and van Harn [2004. Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker, New York] have links with the Wiener–Hopf factorization met in the theory of random walk.     

Asymptotic properties of the MAMSE adaptive likelihood weights

The weighted likelihood is a generalization of the likelihood designed to borrow strength from similar populations while making minimal assumptions. If the weights are properly chosen, the maximum weighted likelihood estimate may perform better than the maximum likelihood estimate (MLE). In a previous article, the minimum averaged mean squared error (MAMSE) weights are proposed and simulations show that they allow to outperform the MLE in many cases. In this paper, we study the asymptotic properties of the MAMSE weights. In particular, we prove that the MAMSE-weighted mixture of empirical distribution functions converges uniformly to the target distribution and that the maximum weighted likelihood estimate is strongly consistent. A short simulation illustrates the use of bootstrap in this context.     

Robust permutation tests for two samples

We consider robust permutation tests for a location shift in the two sample case based on estimating equations, comparing the test statistics based on a score function and an M-estimate. First we obtain a form for both tests so that the exact tests may be carried out using the same algorithms as used for permutation tests based on the mean. Then we obtain the Bahadur slopes of the tests in these two statistics, giving numerical results for two cases equivalent to a test based on Huber scores and a particular case of this related to a median test. We show that they have different Bahadur slopes with neither exceeding the other over the whole range. Finally, we give some numerical results illustrating the robustness properties of the tests and confirming the theoretical results on Bahadur slopes.     

CRITICAL PERIODS DURING CHILDHOOD AND ADOLESCENCE

We identify the ages that constitute sensitive (or critical) periods in children's development towards their adult health status, skills, and human capital. For this, we use data on families migrating into Sweden from countries that are poorer, with less healthy conditions. Late‐life health is proxied by adult height and other adult outcomes. The relation between siblings’ ages at migration and their adult outcomes allows us to estimate the causal effect of conditions at specific childhood ages. We effectively exploit that, for siblings, the migration occurs simultaneously in calendar time but at different developmental stages (ages). We find evidence that the period just before the puberty growth spurt constitutes a critical period for adult height and we find related critical periods for adult cognition, mental health, and education.     

A WORLD OF CITIES: THE CAUSES AND CONSEQUENCES OF URBANIZATION IN POORER COUNTRIES

Historically, urban growth required enough development to grow and transport significant agricultural surpluses or a government effective enough to build an empire. But there has been an explosion of poor mega‐cities over the last 30 years. A simple urban model illustrates that in closed economies, agricultural prosperity leads to more urbanization, but that in an open economy, urbanization increases with agricultural desperation. The challenge of developing world mega‐cities is that poverty and weak governance reduce the ability to address the negative externalities that come with density. This paper models the connection between urban size and institutional failure, and shows that urban anonymity causes institutions to break down. For large cities with weak governments, draconian policies may be the only way to curb negative externalities, suggesting a painful trade‐off between dictatorship and disorder. A simple model suggests that private provision of infrastructure to reduce negative externalities is less costly when city populations are low or institutions are strong, but that public provision can cost less in bigger cities.