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2,335 results · page 98 of 98
Max-Olivier Hongler, Olivier Gallay, Michael Hülsmann, Philip Cordes · 5 authors
No abstract is available for this record.
Herbert Gintis
No abstract is available for this record.
Torsten Eymann, Stefan Sackmann, Günter Müller, Ingo Pippow
The mobile and increasingly ubiquitous use of information technology leads to more dynamic, constantly self-reconfiguring networks. Their services are available anytime and anywhere; as software agents, they can make local, context-aware decisions. F. A. von Hayek developed a theory for economic coordination based on individual decision making. This paper presents the explanation concepts of economic self-organization as at least one option for the design of decentralized coordination of information systems consisting of autonomous software agents with limited information processing capacity and incomplete information. Experiments using a multi-agent system show that a targeted change of this basic rule set directly influences the behavior of the individual elements and indirectly the behavior of the overall system.
Joel N. Morse
No abstract is available for this record.
Jiarun Hu, Qian Luo, Jiaen Zhang
Security breaches of the cryptocurrency exchanges usually cause the price fluctuation in the market. Approximately one hundred cryptocurrency thefts, including hacks and scams, has occurred since 2012 to 2018, half of which are hacks of Bitcoins. Based on the thirty Bitcoin hacks, this study portrays the general price pattern during the hack. And it illustrates the link between the size of the hack and the subsequent price change of Bitcoin. The tests reveal that the larger the volume of the hack, the stronger the price drop. However, a similar obvious relationship does not exist for the recovery of the price. The study might be the first piece of research focus on the hacks and the price pattern in a short time period.
Grace Wahba
Let $\{X(t), t = \cdots -1, 0, 1, \cdots\}$ be a $P$ dimensional zero mean stationary Gaussian time series, $X(t) = \begin{pmatrix}X_1(t)\\X_2(t)\\\vdots\\X_P(t)\end{pmatrix}$ we let $R(\tau) = EX(t)X' (t + \tau)$, where $R(\tau) = \{R_{ij}(\tau), i,j = 1, 2, \cdots P\}$, and $F(\omega) = (2\pi)^{-1} \sum^\infty_{\tau=-\infty}e^{-i\omega\tau}R(\tau)$. It is assumed that $\sum^P_{i,j=1} \sum^\infty_{\tau=-\infty} |\tau| |R_{ij}(\tau)| < \infty$, and hence $F(\omega)$ exists and the elements possess bounded derivatives. It is further assumed that $F(\omega)$ is strictly positive definite, all $\omega$. Knowledge of $F(\omega)$ serves to specify the process. $F(\omega)$, and $S$, the covariance matrix of $x = \begin{pmatrix}x_1 \\ x_2\ \\ vdots\\x_P\end{pmatrix}$, a Normal $(0, S)$ random vector are known to enjoy many analogous properties. (See [7].) To cite two examples, the hypothesis that $X_i(s)$ is independent of $X_j(t)$ for $i \neq j = 1, 2, \cdots P$, any $s, t$, is equivalent to the hypothesis that $F(\omega)$ is diagonal, all $\omega$, while the hypothesis that $x_i$ is independent of $x_j$, for $i \neq j = 1, 2, \cdots P$ is equivalent to the hypothesis that $S$ is diagonal. The conditional expectation of $x_1$, given $x_2, \cdots x_P$ is \begin{equation*}E(x_1\mid x_2, \cdots x_P) = S_{12}S^{-1}_{22}\begin{pmatrix}x_2 \\ \vdots \\ x_P\end{pmatrix}, S = \bigg(\begin{array}{c|c} S_{11} & S_{12} \\ \hline S_{21} & S_{22}\end{array} \bigg)\end{equation*}. The corresponding regression problem for stationary Gaussian time series goes as follows. If \begin{equation*}E\{X_1(t)\mid X_2(s), \cdots X_P(s), s = \cdots -1, 0, 1, \cdots\} = \sum^P_{j=2} \sum^\infty_{s=-\infty} b_j(t - s)X_j(s)\end{equation*} then $B(\omega)$, defined by $B(\omega) = (B_2(\omega), \cdots B_P(\omega)), B_j(\omega) = \sum^\infty_{s=-\infty} b_j(s)e^{i\omega s}$ satisfies \begin{equation*}B(\omega) = F_{12}(\omega)F_{22}^{-1}(\omega), \quad F(\omega) = \bigg(\begin{array}{c|c}f_{11}(\omega) & F_{12}(\omega) \\ \hline F_{21}(\omega) & F_{22}(\omega)\end{array} \bigg).\end{equation*} It is interesting to ask how well these and similar analogies carry over to sampling theory and hypothesis testing. Goodman [3] gave a heuristic argument to support the conclusion that $\hat{F}_X(\omega_k)$, a suitably formed estimate of the spectral density matrix $F(\omega_k)$ has the complex Wishart distribution. The question is met here by the following results. Firstly if $\hat{F}_X(\omega_l), l = 1, 2, \cdots M$ are estimates of the spectral density matrix, each consisting of averages of $(2n + 1)$ periodograms based on a record of length $T$, with the $\omega_l$ equally spaced and $(2n + 1)M \leqq \frac{1}{2} T$, then it is possible to construct, on the same sample space as $X(t), M$ independent complex Wishart matrices $\hat{F}{\bar{\bar{X}}}(\omega_l), l = 1, 2, \cdots M$ such that $\{\hat{F}_X(\omega_l), l = 1, 2, \cdots M\}$ converge simultaneously in mean square to $\{\hat{F}_{\bar{\bar{X}}}(\omega_l), l = 1, 2,\cdots M\}$, as $n, M$ get large. Secondly, it is legitimate to use the natural analogies from multivariate analysis to test hypotheses about time series. One example is presented, as follows. The likelihood ratio test statistic for testing $S$ diagonal is $|\hat{S}|/\mathbf{\prod}^P_{i=1} \hat{s}_{ii}$ where $\hat{S} = \{\hat{s}_{ij}$ is the sample covariance matrix. The analogous statistic $\psi$ for testing $X_i(s), X_j(t)$ independent, $i,j 1 = 2, \cdots P$ from a record of length $T$ is $\psi = \prod^M_{l=1} \lbrack|\hat{F}_X(\omega_l)|/\prod^P_{i=1} \hat{f}_{ii}(\omega_l)\rbrack$ where $\hat{F}_X(\omega_l) = \{\hat{f}_{ij}(\omega_l)\}$ are the sample spectral density matrices as above. Letting ${\bar{dbar{\psi}}} = \prod^M_{l=1} \lbrack|\hat{F}_{\bar{\bar{x}}}(\omega_l)|/\prod^P_{i=1} \hat{h}_{ii}(\omega_l)\rbrack$ where $\hat{F}_{\bar{\bar{x}}}(\omega_l) = \{\hat{h}_{ij}(\omega_l)\}$ are the independent complex Wishart matrices referred to above, we show $EC_{n,M} |\log \psi - \log {\bar{\bar{\psi}}} \rightarrow 0$ for large $n, M$, where $C_{n,M}$ are chosen to make the result non-trival. The method of proof applies to any statistic which is a product over $l$ of sufficiently smooth functions of the entries of $\hat{F}_X(\omega_l)$. Applications to estimation and testing in the regression problem will appear elsewhere [8]. The distribution theory of functions of complex Wishart matrices has been well investigated by a number of authors [3] [5] [6], and hence can be easily applied here to statistics like ${\bar{\bar{\psi}}}$. The results above are shown for $P = 2$, it is clear that the proofs extended to any (fixed) finite $P$. The proofs proceed as follows, via a theorem which has somewhat more general application. For each $T$, let $X$ be the $2 \times T$ random matrix $X = \binom{X_1}{X_2} = \begin{pmatrix}X_1(1), \cdots, X_1(T)\\X_2(1), \cdots, X_2(T)\end{pmatrix}$ and let the $2T \times 2T$ covariance matrix $\Sigma$ be given by $\Sigma = \begin{pmatrix}\sum_{11} \sum_{12} \\ \sum_{21} \sum_{22}\end{pmatrix}$ where $\Sigma_{ij} = EX_i'X_j. \{\hat{F}_X(\omega_l)\}$, the sample spectral density matrices described above based on a record of length $T$, are each of the form $\hat{F}_X(\omega_l) = T^{-1}XQX'$ where $Q$ is a $T \times T$ circulant matrix with largest eigenvalue $ = T(2n + 1)^{-1} \leqq \frac{1}{2}M < <T$. We define circulant matrices $\bar{\Sigma}_{ij}$ which approximate $\Sigma_{ij}$, and a random matrix $\bar{X}$ on the sample space of $X$, $\bar{X} = \binom{\bar{X}_1}{\bar{X}_2} = \begin{pmatrix}\bar{X}_1(1), \cdots, \bar{X}_1(T)\\\bar{X}_2(1), \cdots, \bar{X}_2(T)\end{pmatrix}$ with $E\bar{X}_i'\bar{X}_j = \bar\Sigma_{ij}$. The $2T$ eigenvalues of the block circulant matrix $\bar\Sigma = \begin{pmatrix}\bar\Sigma_{11} \bar\Sigma_{12} \\ \bar\Sigma_{21} \bar\Sigma_{22}\end{pmatrix}$ will be the $2T$ eigenvalues of the $T$ matrices $\{F(2\pi j/T),j = 1, 2, \cdots T\}$. The distribution of random matrices of the form $T^{-1}\bar{X}Q\bar{X}'$ where $Q$ is any circulant matrix are relatively simple to investigate due to the fact that all circulant matrices commute, and their eigenvalues may be exhibited as simple functions of the elements. Circulant quadratic forms in random vectors with circulant covariance matrices are well known in the literature, (See [1] and references cited there). Let $\hat{F}_{X,Q} = T^{-1}XQX'$ and $\hat{F}_{\bar{X},Q} = T^{-1}\bar{X}Q\bar{X}'$ where $Q$ is now any $T \times T$ (real or complex) quadratic form with largest absolute eigenvalue $\leqq q$. The main Theorem allows the replacement of $X$ by $\bar{X}$ in the analysis, and is, that under the assumptions on $F(\omega)$ and $R(\tau)$, for any $T$, \begin{equation*}\tag{1.1} E \operatorname{tr} (\hat{F}_{X,Q} - \hat{F}_{\bar{X},Q})(\hat{F}_{X,Q} - \hat{F}_{\bar{X}, Q})^{\ast'} \leqq cq^2/T^2\end{equation*} where $c$ is a constant depending only on $F(\omega)$ and $R(\tau)$. A lemma, essentially allowing the replacement of $F(\omega)$ by a suitably chosen step-function, together with the application of (1.1) gives the results concerning the $\{\hat{F}_X(\omega_l)\}$ an $\lambda$. Since $\hat{R}(\tau)$, the sample (circularized) autocorrelation function is also of the form $T^{-1}XQX'$ with $Q$ circulant we obtain an easy corollary on the distribution of $\{\hat{R}(\tau)\}$.
John Van Ryzin
This paper is concerned with repetitive sequential play in finite statistical games (decision problems) from the statistician's point of view. We shall assume that the statistician's move at stage $k$ may depend on the previous $k - 1$ moves of Nature as well as the random variable $\mathbf{X}_k = (X_1, \cdots, X_k)$, where the $X_i$ are independent observations (r.v.'s) (possibly vector-valued) from the sequence of statistical games, $k = 1, 2, \cdots$. The play is repetitive in the sense that each component game is identical in structure, with only the moves of the statistician and Nature changing. Furthermore, we impose no assumptions regarding the behavior of the parameter sequence of Nature's moves. The statistician does have the added disadvantage that the finite class of distributions in the component game is not fully specified. However, he does know that class in question has: either (i) all members with discrete distributions or (ii) all members with $q$-dimensional a.e. continuous Lebesgue densities. This same problem when the distributions are fully known has been treated in [6] for statistical as well as more general games in which Nature's space is finite. In the case where the distributions are completely specified but the history of the past moves is unknown to the statistician, see [20], [22], [27], and [28]. The development in this paper is closely connected to and motivated by these results, particularly those of the preceding paper [27]. If for fixed $N$, the empirical distribution $p_N$ of Nature's moves is known, then the statistician could use as a rule for each of the $N$ component games a strategy Bayes against $p_N$ having risk $\phi(p_N)$. In all the papers cited in the previous paragraph, the aim was to construct for the statistician, when $p_N$ is unknown and $N$ not specified, a sequence of randomized decision functions whose $N$th average loss minus $\phi(p_N)$ approaches zero (or has an upper bound approaching zero) in a suitable sense as the number of repetitions of play, $N$, increases. However, in the case of statistical games, all of the above results require that the finite class of distributions be fully specified. In this paper we remove that assumption by estimating the distributions sequentially based on past moves and observations. Then in the present play of the component game the statistician substitutes these estimators into a procedure which is Bayes against the empirical distribution of Nature's previous moves. The resulting sequence of procedures is shown to be "asymptotically good" in the sense that the average loss over the $N$ games $W_N$ minus the Bayes risk $\phi(p_N)$ approaches zero (in an appropriate sense) as $N$, the number of games played, increases. In Section 2 we introduce notation and preliminaries. Section 3 discusses play in repetitive games and defines the proposed sequential procedures $\mathbf{t} = \{\mathbf{t}_k\}$. In Section 4 we prove preliminary results upon which all proofs are founded. Section 5 considers the discrete case giving uniform (in sequences of Nature's moves) convergence theorems (as $N \rightarrow \infty$) for the quantity $W_N - \phi(p_N)$. Theorem 5.1 is a uniform convergence theorem of $O(N^{-\frac{1}{2}})$ of the expected value of $W_N - \phi(p_N)$ for finite discrete classes, each member of which is non-degenerate and satisfies a certain tail probability condition. Under the same conditions, Theorem 5.2 gives uniform convergence to zero in probability for the quantity $N^{\frac{1}{2}} (\log N)^{-1} \{W_N - \phi(p_N)\} \text{as} N \rightarrow \infty$. Uniform convergence of $W_N - \phi(p_N) \rightarrow 0$ in probability for general non-degenerate finite discrete class is presented in Theorem 5.3. Section 6 treats the estimation problem for densities needed to form the randomized strategy sequences $\mathbf{t}$ in the continuous case. The results stated are based on a paper by Cacoullos [3] generalizing the univariate results of Parzen [15]. In Section 7, we present results for the continuous case. Theorem 7.1 and its corollary give uniform convergence of $W_N - \phi(p_N)$ to zero in probability and of its expectation to zero, respectively. The finite continuous classes of Theorem 7.1 are very general in the sense that each member is a continuous a.e. density. Finally, in Section 8 we draw certain conclusions and relate our results to similar results obtained elsewhere. The novelty of the paper rests in the fact that through the past history of Nature's moves and the observations connected with past play, one can construct a sequential strategy, $\mathbf{t} = \{\mathbf{t}_k\}$, with very little knowledge about the finite class of distributions, which approaches asymptotic "optimal" play. The lack of knowledge on the finite class of distributions distinguishes this work from the related "repetitive type" problems in games and/or decision theory treated in [1], [2], [4], [6], [7], [8], [9], [10], [12], [17], [18], [19], [20], [21], [22], [24], [25], [26], [27], [28], and [29]. For possible applications of this work see Neyman [14], especially his Example 3 and his discussion relating to the work of Blackwell [2].