Showing papers on "Natural exponential family published in 2018"
TL;DR: This work introduces a truly sequential algorithm that achieves Hannan-consistent log-loss regret performance against true probability distribution without requiring any information about the observation sequence, and can be efficiently used in big data applications.
Abstract: We investigate online probability density estimation (or learning) of nonstationary (and memoryless) sources using exponential family of distributions. To this end, we introduce a truly sequential algorithm that achieves Hannan-consistent log-loss regret performance against true probability distribution without requiring any information about the observation sequence (e.g., the time horizon $T$ and the drift of the underlying distribution $C$ ) to optimize its parameters. Our results are guaranteed to hold in an individual sequence manner. Our log-loss performance with respect to the true probability density has regret bounds of $O(({CT})^{1/2})$ , where $C$ is the total change (drift) in the natural parameters of the underlying distribution. To achieve this, we design a variety of probability density estimators with exponentially quantized learning rates and merge them with a mixture-of-experts notion. Hence, we achieve this square-root regret with computational complexity only logarithmic in the time horizon. Thus, our algorithm can be efficiently used in big data applications. Apart from the regret bounds, through synthetic and real-life experiments, we demonstrate substantial performance gains with respect to the state-of-the-art probability density estimation algorithms in the literature.
20 citations
TL;DR: In this article, the authors studied linear differential equations with exponential polynomial coefficients, where exactly one coefficient is of order greater than all the others, and they showed that a nontrivial exponential Polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient.
17 citations
TL;DR: In this article, a new generalized geometric distribution through a weight function was derived, which can also be viewed as a discrete analog of weighted exponential distribution introduced by Gupta, and can be seen as a deterministic version of the exponential distribution.
Abstract: In this article, we derive a new generalized geometric distribution through a weight function, which can also be viewed as a discrete analog of weighted exponential distribution introduced by Gupta...
9 citations
TL;DR: In this paper, the modified slashed Rayleigh distribution (SRLD) is proposed and studied. But the proposed SRLD is an extension of the ordinary Rayleigh distributions, being more flexible in terms of distributional kurtosis, and it arises as a quotient of two independent random variables.
Abstract: A new family of slash distributions, the modified slashed-Rayleigh distribution, is proposed and studied. This family is an extension of the ordinary Rayleigh distribution, being more flexible in terms of distributional kurtosis. It arises as a quotient of two independent random variables, one being a Rayleigh distribution in the numerator and the other a power of the exponential distribution in denominator. We present properties of the proposed family. In addition, we carry out estimation of the model parameters by moment and maximum likelihood methods. Finally, we conduct a small-scale simulation study to evaluate the performance of the maximum likelihood estimators and apply the results to a real data set, revealing its good performance.
9 citations
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TL;DR: In this paper, the authors consider the problem of computing the tail probabilities of the distribution of a random sum of positive random variables, where the individual variables follow a reproducible natural exponential family (NEF) distribution and the random number has a counting distribution with a cubic variance function.
Abstract: In this paper we consider the problem of computing tail probabilities of the distribution of a random sum of positive random variables. We assume that the individual variables follow a reproducible natural exponential family (NEF) distribution, and that the random number has a NEF counting distribution with a cubic variance function. This specific modelling is supported by data of the aggregated claim distribution of an insurance company. Large tail probabilities are important as they reflect the risk of large losses, however, analytic or numerical expressions are not available. We propose several simulation algorithms which are based on an asymptotic analysis of the distribution of the counting variable and on the reproducibility property of the claim distribution. The aggregated sum is simulated efficiently by importancesampling using an exponential cahnge of measure. We conclude by numerical experiments of these algorithms.
9 citations
TL;DR: Two real data applications regarding the strength data and Proschan's air-conditioner data are used to show that the new distribution is better than the BE2 distribution and some other well-known distributions in modeling lifetime data.
Abstract: The binomial exponential 2 (BE2) distribution was proposed by Bakouch et al. as a distribution of a random sum of independent exponential random variables, when the sample size has a zero truncated...
6 citations
TL;DR: In this paper, a cost model based on generalized exponential distribution as shock model was proposed, which seems to be a better alternative in many cases, based on this new shock model, extended Banerjee and Rahim's model in non-uniform sampling scheme and compared that with uniform design.
Abstract: Control chart in statistical process control is an effective way to monitor and improve quality and production cost savings in on-line activities of organizations. Economic design of these charts needs to determine sample size, sampling interval, control limit coefficient and process failure mechanism. Exponential, Gamma, and Weibull distributions are the most famous distributions used in the analysis of lifetime and failure mechanism. However, most of these distributions suffer from some drawbacks that finally affect optimal values of design parameters. Therefore, we here proposed a cost model based on generalized exponential distribution as shock model that seems to be a better alternative in many cases. Based on this new shock model, we extended Banerjee and Rahim’s model in non-uniform sampling scheme and compared that with uniform design. We also used sensitivity analysis to investigate the effect of fitting wrong Weibull, gamma, and exponential distributions instead of generalized exponential. The results showed that inappropriate fitting the wrong distribution leads to misleading values for the average cost and the design parameters.
5 citations
Proceedings Article•
03 Dec 2018TL;DR: It is proved that a contrastive estimator is invariant to model misspecification within the natural exponential family, and moreover, attains the optimal statistical rate of convergence up to a logarithmic factor.
Abstract: Learning from pairwise measurements naturally arises from many applications, such as rank aggregation, ordinal embedding, and crowdsourcing. However, most existing models and algorithms are susceptible to potential model misspecification. In this paper, we study a semiparametric model where the pairwise measurements follow a natural exponential family distribution with an unknown base measure. Such a semiparametric model includes various popular parametric models, such as the Bradley-Terry-Luce model and the paired cardinal model, as special cases. To estimate this semiparametric model without specifying the base measure, we propose a data augmentation technique to create virtual examples, which enables us to define a contrastive estimator. In particular, we prove that such a contrastive estimator is invariant to model misspecification within the natural exponential family, and moreover, attains the optimal statistical rate of convergence up to a logarithmic factor. We provide numerical experiments to corroborate our theory.
4 citations
TL;DR: In this article, a new three-parameter distribution family with a flexible failure rate function arising by mixing the Weibull distribution and power-series distribution is introduced, which includes special cases of some well-used mixing distributions and generalizes the exponential power series distribution.
Abstract: A new three-parameter distribution family with a flexible failure rate function arising by mixing the Weibull distribution and power-series distribution is introduced. This distribution family includes special cases of some well-used mixing distributions and generalizes the exponential power-series distribution. Various properties of the new distribution family are discussed. The maximum likelihood estimation and an EM algorithm are presented for finding the estimates of the distribution family parameters, and expressions for their asymptotic variance and covariance are derived. Intensive simulation studies are implemented and experimental results are illustrated with real data-sets.
4 citations
TL;DR: In this paper, it was shown that the property that t is a median of P t for all t characterizes P as the standard Gaussian law N ( 0, 1 ).
3 citations
TL;DR: The α-power transformation (APT) family of distributions proposed in this journal by Mahdavi and Kundu (2017) is not new, but the same as theexp-G family of distribution as discussed by the authors.
Abstract: Dear Editor,The “α-power transformation” (APT) family of distributions recently proposed in this journal by Mahdavi and Kundu (2017) is not new, but the same as the “exp-G” family of distributions ...
TL;DR: A generalization of the exponential distribution is studied and this new distribution is the natural conjugate prior for the continuous Lindley distribution, which is applied in credibility theory and in other settings.
TL;DR: In this article, the authors provide simple approximations for specific classes of fiducial and confidence distributions and discuss their connections with objective Bayesian posteriors under standard regularity assumptions.
TL;DR: In this article, the authors derived sharp bounds for the Kolmogorov distance between G and the unit exponential distribution, as well as G and an exponential distribution with the same mean as G, and applied the bounds to geometric convolutions and to first passage times.
Abstract: Let F be an NWUE distribution with mean 1 and G be the stationary renewal distribution of F. We would expect G to converge in distribution to the unit exponential distribution as its mean goes to 1. In this paper, we derive sharp bounds for the Kolmogorov distance between G and the unit exponential distribution, as well as between G and an exponential distribution with the same mean as G. We apply the bounds to geometric convolutions and to first passage times.
TL;DR: The statistical curvature properties of this family of skew-normal distributions are studied and the sample size issue is discussed to assess, to what extent the linear and likelihood-based inference of exponential family of distribution can be applicable for the skew- normal family.
Abstract: With special reference to the family of skew-normal distributions, we consider geometric curvature of a probability density function as a means to define and identify rare or catastrophic events—a phenomenon common in studying the financial instruments Further, we study the statistical curvature properties of this family of distributions and discuss the sample size issue, to assess, to what extent the linear and likelihood-based inference of exponential family of distribution can be applicable for the skew-normal family
TL;DR: The authors generalize the exponential family of distributions into a wider family which includes important distributions such as the normal, log-normal, Student-t, Cauchy, logistic and Birnbaum-Saunders distributions.
Abstract: In this paper we generalize the exponential family (EF) of distributions into a wider family which includes important distributions such as the normal, log-normal, Student-t, Cauchy, logistic and Birnbaum–Saunders distributions. Furthermore, we derive several characteristics of the proposed family. The importance of such family is also discussed.
TL;DR: In this article, a new three-parameter distribution with decreasing, increasing, and bathtub-shaped hazard rates obtained by compounding geometric, power series, and exponential distributions is introduced.
Abstract: A new three-parameter distribution with decreasing, increasing, and bathtub-shaped hazard rates obtained by compounding geometric, power series, and exponential distributions is introduced. It includes some well-known distributions as particular cases. Various mathematical properties of the new distribution as well as details of the maximum likelihood estimation and a sensitivity analysis for its parameters are presented. Finally, two real data applications are presented.
TL;DR: A new characterization of the exponential distribution is obtained based on an equation involving randomly shifted (translated) order statistics and the Maclaurin series expansion of the probability density of the parent variable.
Abstract: A new characterization of the exponential distribution is obtained. It is based on an equation involving randomly shifted (translated) order statistics. No specific distribution is assumed for the shift random variables. The proof uses a recently developed technique including the Maclaurin series expansion of the probability density of the parent variable.
TL;DR: In this paper, the Conway-Maxwell-Multinomial distribution is studied and its properties are demonstrated, including sufficient statistics and conditions for the propriety of posterior distributions derived from it.
Abstract: The Conway–Maxwell-Multinomial distribution is studied in this paper. Its properties are demonstrated, including sufficient statistics and conditions for the propriety of posterior distributions derived from it. An application is given using data from Mendel’s ground-breaking genetic studies.
TL;DR: In this paper, it was shown that when a random variable has a parametric distribution as a member of an infinitely divisible natural exponential family whose induced measure is absolutely continuous with respect to its basis measure, there exists a deterministic function, referred to as reduction function, such that the random variable transformed by this function is an unbiased estimator of the variance of the variable.
TL;DR: In this paper, a new non-ruin measure associated with the aggregate logarithm of the claim-over-profit ratios was defined and obtained on Pareto-type distributions.
Abstract: In the general risk model (or the Sparre-Andersen model), it is well-known that the following assertion holds: if the claim size is exponentially distributed then the non-ruin probability distribution is a mixture of exponential distributions. In this paper, under some general conditions, we prove that the converse statement of the previous assertion is also true. Besides, we define a new non-ruin measure associated with the aggregate logarithms of the claim-over-profit ratios and obtain a result on Pareto-type distributions.
TL;DR: In this paper, the comparison between the Fisher information on parameters of the weighted distributions and the parent distributions is done, and the most common family of distributions, location-scores, are compared.
Abstract: In this article, the comparison between the Fisher information on parameters of the weighted distributions and the parent distributions is done. The most common family of distributions, location–sc...
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TL;DR: In this article, it was shown that the cross-ratio property for continuous distributions is satisfied if and only if the families are either Gaussian or Gamma distributions, and the proof relies on the fact that only Moebius functions preserve the cross ratio.
Abstract: Suppose that the distribution of $X_a$ belongs to a natural exponential family concentrated on the nonegative integers and is such that $\E(z^{X_a})=f(az)/f(a)$ Assume that $\Pr(X_a\leq k)$ has the form $c_k\int_a ^{\infty}u^k\mu(du)$ for some number $c_k$ and some positive measure $\mu,$ both independent of $a$ We show that this asumption implies that the exponential family is either a binomial, or the Poisson, or a negative binomial family Next, we study an analogous property for continuous distributions and we find that it is satisfied if and only the families are either Gaussian or Gamma Ultimately, the proofs rely on the fact that only Moebius functions preserve the cross ratio,
\textsc{Keywords:} Binomial, Poisson and negative binomial distributions Gaussian and Gamma distributions Moebius transforms Cross ratio