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Considering a representative agent in the market, we study the long-term optimal investment problem in a discrete-time financial market, introducing a set of restrictions in the admissible strategies. The drawdown constraints limit the size of possible losses of the portfolio and impose a floor-based performance measure. The optimal growth rate is characterized, and under suitable hypotheses it is proved that an optimal strategy exists. The approach to solving this problem is based on dynamic programming techniques and a fixed point argument adapted from the theory of Markov decision processes.
We consider the random splitting and aggregating of Hawkes processes. We present the random splitting schemes using the direct approach for counting processes, as well as the immigration–birth branching representations of Hawkes processes. From the second scheme, it is shown that random split Hawkes processes are again Hawkes. We discuss functional central limit theorems (FCLTs) for the scaled split processes from the different schemes. On the other hand, aggregating multivariate Hawkes processes may not necessarily be Hawkes. We identify a necessary and sufficient condition for the aggregated process to be Hawkes. We prove an FCLT for a multivariate Hawkes process under a random splitting and then aggregating scheme (under certain conditions, transforming into a Hawkes process of a different dimension).
Given a graphon $W$ and a finite simple graph $H$, with vertex set $V(H)$, denote by $X_n(H, W)$ the number of copies of $H$ in a $W$-random graph on $n$ vertices. The asymptotic distribution of $X_n(H, W)$ was recently obtained by Hladký, Pelekis, and Šileikis [17] in the case where $H$ is a clique. In this paper, we extend this result to any fixed graph $H$. Towards this we introduce a notion of $H$-regularity of graphons and show that if the graphon $W$ is not $H$-regular, then $X_n(H, W)$ has Gaussian fluctuations with scaling $n^{|V(H)|-\frac{1}{2}}$. On the other hand, if $W$ is $H$-regular, then the fluctuations are of order $n^{|V(H)|-1}$ and the limiting distribution of $X_n(H, W)$ can have both Gaussian and non-Gaussian components, where the non-Gaussian component is a (possibly) infinite weighted sum of centred chi-squared random variables with the weights determined by the spectral properties of a graphon derived from $W$. Our proofs use the asymptotic theory of generalised $U$-statistics developed by Janson and Nowicki [22]. We also investigate the structure of $H$-regular graphons for which either the Gaussian or the non-Gaussian component of the limiting distribution (but not both) is degenerate. Interestingly, there are also $H$-regular graphons $W$ for which both the Gaussian or the non-Gaussian components are degenerate, that is, $X_n(H, W)$ has a degenerate limit even under the scaling $n^{|V(H)|-1}$. We give an example of this degeneracy with $H=K_{1, 3}$ (the 3-star) and also establish non-degeneracy in a few examples. This naturally leads to interesting open questions on higher order degeneracies.
The difficult issue of getting beyond averages when it comes to describing the effects of treatment is addressed. Are women essentially different from men in the way that treatment affects them? If so, how should trials be run to address this? The field of bioequivalence trials is described. These are used to show that generic drugs may safely be used instead of brand name innovator formulations. A claim that bioequivalance is different for women than for men is shown to be false.
An account is made of some early and more modern pioneers in probability and statistics. The purpose of this is not only to provide a historical account of the subject but also to breathe life into important statistical concepts that will appear throughout the book.
Medical statistics as it applies to money, in particular insured sums, is the topic of this chapter which covers the history of annuities and life insurance. The way that this topic has been adapted by medical statistics, in particular as a result of a landmark paper in 1972 by David Cox, is addressed.
Various statistical problems of assessing legal evidence are covered. Poisson's attempts to model the probability of a jury coming to the correct decision are considered. Various versions of the famous Island Problem and possible Bayesian solutions are covered in some detail.
The controversial field of observational studies is covered, taking medicines and their possible side-effects and also lifestyle choices as an example.
A link is made between epistemology – that is to say, the philosophy of knowledge – and statistics. Hume's criticism of induction is covered, as is Popper's. Various philosophies of statistics are described.
This is a new (to the second edition) chapter illustrating many aspects of medical statistics using the COVID-19 pandemic. Topics covered include, reporting cases, case fatality as a function of age, developing vaccines, testing for infection and modelling the spread of infection.
The topic of clinical trials is introduced using the example of the MRC trial in streptomycin in TB. The role of randomization, the subject of design of experiments and ethical problems in conducting trials in patients are covered.