Natural exponential family

About: Natural exponential family is a research topic. Over the lifetime, 1973 publications have been published within this topic receiving 60189 citations. The topic is also known as: NEF.

Online Density Estimation of Nonstationary Sources Using Exponential Family of Distributions

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.

Inference in step-stress models based on failure rates

Abstract: In step-stress modeling with the cumulative exposure model it is well known that, for underlying exponential distributions, explicit expressions can be obtained for maximum likelihood estimators of parameters as well as for their conditional density functions or conditional moment generation functions, given their existence. Applying a failure rate based approach instead, similar results can be also obtained for underlying lifetime distributions out of a general scale family of distributions, which allows for a flexible modeling. Exemplarily, respective results are presented for Type-I and Type-II censored experiments.

Sample Sequences of Maxima

Abstract: Let $X_1, X_2, \cdots, X_n, \cdots$ be a sequence of independent, identically distributed random variables with common distribution function $F$. Let $Z_n = \max \{X_1, X_2, \cdots X_n\}$. Conditions for the stability and relative stability of such sequences with the various modes of convergence have been given by Geffroy [3], and Barndorff-Nielsen [1]. The principal result of this paper is Theorem 2.1, which is an analogue for maxima of the law of the iterated logarithm for sums (Loeve [6] pages 260-1). In Section 3, it is indicated that the theorem is satisfied by a wide class of distributions, and specific forms are given for the normal and exponential distributions.

A New Class of Generalized Inverse Weibull Distribution with Applications

Abstract: The gamma-inverse Weibull (GIW) distribution which includes in- verse Weibull, inverse exponential, gamma-inverse exponential, gamma- inverse Rayleigh, inverse Rayleigh, gamma-Frechet and Frechet distri- butions as special cases is proposed and studied. This new distribu- tion might be useful for failure time data analysis. Some mathematical properties of the new distribution including moments, mean deviations, Bonferroni and Lorenz curves, Shannon and Renyi entropies are pre- sented. Maximum likelihood estimation technique is used to estimate the parameters and applications to real data sets are given to illustrate the usefulness of this new class of distributions.