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Register a new userCharacterizations of probability distributions via bivariate regression of record values
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Bairamov et al. (Aust N Z J Stat 47:543-547, 2005) characterize the exponential distribution in terms of the regression of a function of a record value with its adjacent record values as covariates. We extend these results to the case of non-adjacent covariates. We also consider a more general setting involving monotone transformations. As special cases, we present characterizations involving weighted arithmetic, geometric, and harmonic means.
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We prove that if $pgeq 1$ and $-1leq rleq p-1$ then the binomial sequence $binom{np+r}{n}$, $n=0,1,...$, is positive definite and is the moment sequence of a probability measure $ u(p,r)$, whose support is contained in $left[0,p^p(p-1)^{1-p}right]$. If $p>1$ is a rational number and $-1<rleq p-1$ then $ u(p,r)$ is absolutely continuous and its density function $V_{p,r}$ can be expressed in terms of the Meijer $G$-function. In particular cases $V_{p,r}$ is an elementary function. We show that for $p>1$ the measures $ u(p,-1)$ and $ u(p,0)$ are certain free convolution powers of the Bernoulli distribution. Finally we prove that the binomial sequence $binom{np+r}{n}$ is positive definite if and only if either $pgeq 1$, $-1leq rleq p-1$ or $pleq 0$, $p-1leq r leq 0$. The measures corresponding to the latter case are reflections of the former ones.
Many authors have studied the phenomenon of typically Gaussian marginals of high-dimensional random vectors; e.g., for a probability measure on $R^d$, under mild conditions, most one-dimensional marginals are approximately Gaussian if $d$ is large. In earlier work, the author used entropy techniques and Steins method to show that this phenomenon persists in the bounded-Lipschitz distance for $k$-dimensional marginals of $d$-dimensional distributions, if $k=o(sqrt{log(d)})$. In this paper, a somewhat different approach is used to show that the phenomenon persists if $k<frac{2log(d)}{log(log(d))}$, and that this estimate is best possible.
Asymptotic laws of records values have usually been investigated as limits in type. In this paper, we use functional representations of the tail of cumulative distribution functions in the extreme value domain of attraction to directly establish asymptotic laws of records value, not necessarily as limits in type. Results beyond the extreme value value domain are provided. Explicit asymptotic laws concerning very usual laws are listed as well. Some of these laws are expected to be used in fitting distribution
We investigate the confinement-deconfinement transition at finite temperature in terms of the probability distribution of Polyakov-loop complex-phase via the Jensen-Shannon divergence. The Jensen-Shannon divergence quantifies the difference of two probability distributions, namely the target and reference probability distributions. We adopt the complex-phase distributions of the spatially averaged Polyakov loop at $mu/T=0$ and $mu/T=ipi/3$ as the target and reference distributions, respectively. It is shown that the Jensen-Shannon divergence has the inflection point when the target system approaches the Roberge-Weiss endpoint temperature even in the finite-volume system. This means that we can detect the confinement-deconfinement transition from the structural change of probability distributions when we suitably set the reference probability distribution. It is also shown that we can pick up the information of the confinement-deconfinement transition from the quark number density by using the Fourier decomposition; Fourier coefficients have a long tail at around the transition temperature and show a divergent series in calculating the normalized kurtosis.
The Gaussian correlation inequality for multivariate zero-mean normal probabilities of symmetrical n-rectangles can be considered as an inequality for multivariate gamma distributions (in the sense of Krishnamoorthy and Parthasarathy [5]) with one degree of freedom. Its generalization to all integer degrees of freedom and sufficiently large non-integer degrees of freedom was recently proved in [10]. Here, this inequality is partly extended to smaller non-integer degrees of freedom and in particular - in a weaker form - to all infinitely divisible multivariate gamma distributions. A further monotonicity property - sometimes called more PLOD (positively lower orthant dependent) - for increasing correlations is proved for multivariate gamma distributions with integer or sufficiently large degrees of freedom.
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