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This paper develops a new family of estimators, the minimum density power divergence estimators (MDPDEs), for the parameters of the one-shot device model as well as a new family of test statistics, Z-type test statistics based on MDPDEs, for testing the corresponding model parameters. The family of MDPDEs contains as a particular case the maximum likelihood estimator (MLE) considered in Balakrishnan and Ling (2012). Through a simulation study, it is shown that some MDPDEs have a better behavior than the MLE in relation to robustness. At the same time, it can be seen that some Z-type tests based on MDPDEs have a better behavior than the classical Z-test statistic also in terms of robustness.

New goodness-of-fit tests for exponentiality based on a particular property of exponential law are constructed. Test statistics are functionals of U-empirical processes. The first of these statistics is of integral type, the second one is a Kolmogorov type statistic. We show that the kernels corresponding to our statistics are nondegenerate. The limiting distributions and large deviations of new statistics under the null hypothesis are described. Their local Bahadur efficiency for various parametric alternatives is calculated and is compared with simulated powers of new tests. Conditions of local optimality of new statistics in Bahadur sense are discussed and examples of "most favorable" alternatives are given. New tests are applied to reject the hypothesis of exponentiality for the length of reigns of Roman emperors which was intensively discussed in recent years.

We study the distribution of the occurrence of rare patterns in sufficiently mixing Gibbs random fields on the lattice $\mathbb{Z}^d$, $d\geq 2$. A typical example is the high temperature Ising model. This distribution is shown to converge to an exponential law as the size of the pattern diverges. Our analysis not only provides this convergence but also establishes a precise estimate of the distance between the exponential law and the distribution of the occurrence of finite patterns. A similar result holds for the repetition of a rare pattern. We apply these results to the fluctuation properties of occurrence and repetition of patterns: We prove a central limit theorem and a large deviation principle.

For $\sigma\in (0,+\infty)$, denote by $B_\sigma^{\infty}$ the Bernstein space (of type $\sigma$) of all entire functions of exponential type $\leq \sigma$ bounded on real axis $\R$. Let $I_d\subset \R$ be a segment of length $d>0$. We announce complete description of non-uniqueness sequences of points for $B_\sigma^\infty$ and criteria of completeness of exponential system in $C(I_d)$ or $L^p(I_d)$ to within one or two exponential functions.

The distribution of the geometric distances of connected neurons is a practical factor underlying neural networks in the brain. It can affect the brain\'s dynamic properties at the ground level. Karbowski derived a power-law decay distribution that has not yet been verified by experiment. In this work, we check its validity using simulations with a phenomenological model. Based on the in vitro two-dimensional development of neural networks in culture vessels by Ito, we match the synapse number saturation time to obtain suitable parameters for the development process, then determine the distribution of distances between connected neurons under such conditions. Our simulations obtain a clear exponential distribution instead of a power-law one, which indicates that Karbowski's conclusion is invalid, at least for the case of in vitro neural network development in two-dimensional culture vessels.

A {\it scenery} is a coloring $\xi$ of the integers. Let $\{S_t\}_{t\geq 0}$ be a recurrent random walk on the integers. Observing the scenery $\xi$ along the path of this random walk, one sees the color $\chi_t:=\xi(S_t)$ at time $t$. The {\it scenery reconstruction problem} is concerned with recovering the scenery $\xi$, given only the sequence of observations $\chi:=(\chi_t)_{t\geq 0}$. The scenery reconstruction methods presented to date require the random walk to have bounded increments. Here, we present a new approach for random walks with unbounded increments which works when the tail of the increment distribution decays exponentially fast enough and the scenery has five colors.

In simulations of some economic gas-like models, the asymptotic regime shows an exponential wealth distribution, independently of the initial wealth distribution given to the system. The appearance of this statistical equilibrium for this type of gas-like models is explained in a rigorous analytical way.

This paper provides a framework for investigations in fluctuation theory for L\'evy processes with matrix-exponential jumps. We present a matrix form of the components of the infinitely divisible factorization. Using this representation we establish generalizations of some results known for compound Poisson processes with exponential jumps in one direction and generally distributed jumps in the other direction.

Dufour gave a conjecture on a characterization of an arbitrary exponential distribution based on a right-censored ordered sample. The conjecture was shown to be true by Leslie and van Eeden (1993) in the case when the number of censored observations is no larger than (1/3)n - 1, and by Xu and Yang (1995) and Rao and Shanbhag (1995) when the number is less than or equal to max?O,n - 5!, where n (greater than or equal to 3) is the sample size. In the present paper we give extended Versions of the results concerning the Dufour conjecture.

In a previous paper ('A Multivariate Exponential Distribution,' AD- 634 335) the authors have derived a multivariate exponential distribution from points of view designed to indicate the applicability of the distribution. Two of these derivations are based on 'shock models' and one is bases on the requirement that residual life is independent of age. The practical importance of the univariate exponential distribution is partially due to the fact that it governs waiting times in a Poisson process. In this paper, the distribution of joint waiting times in a bivariate Poisson process is investigated. There are several ways to define 'joint waiting time.' Some of these lead to the bivariate exponential distribution previously obtained by the authors, but others lead to a generalization of it. This generalized bivariate exponential distribution is also derived from shock models. The moment generating function and other properties of the distribution are investigated.

This study examined the assumption that the underlying failure distribution for inertial measurement units is exponential. Failure data from three inertial measurement units, the KT-73 unit from the A-7D/E and AC-130H, the FLIP unit from the C-5A, and the LN-15 unit from the B-52G/H, were analyzed by use of the SIMFIT computer program. This program is a curve fitting technique employed for fitting a sample of data to twelve theoretical distributions. The research hypothesis was to show that the 'exponential assumption' of failures for the above units was inappropriate by finding a failure distribution which would better model the failure data. The results indicated that the gamma was, in fact, the best model of the distributions tested and several other models, such as the Weibull and lognormal, were generally found to be better than the exponential model. Other areas of reliability discussed include renewal, data aggregation, and infant mortality. (Author)

Empirical Bayes tests for testing H(0) : theta </= theta(0) against H1 : theta > theta(0) in the distribution family having density f(x\theta) = Ia(x)/A(theta) for 0<x<theta are considered under the linear loss. We construct the empirical Bayes tests delta(n) and Delta(n) and prove that both of them have a good asymptotic property. As a special case, taking a(x) = 1 leads to a result obtained by Karunamuni (1999). But our result gets rid of the condition that the critical point b(G) falls in some interval O, P0, where P0 < theta(0) is a known constant.

Our objective is to develop methodology for analyzing life test data. Initially, there is only data-no mathematical models. Through an exploratory data analysis or an analysis based on the physical processes generating the data, an exponential life distribution model may be judged as appropriate for the analysis of the data. Specifically: where lambda is the unknown constant failure rate. The vertical bar in F(x/lambda) indicates that we are conditioning on the parameter lambda; i.e., for specified lambda the distribution is exponential with failure rate lambda. The corresponding density is F(x/lambda) = lambda exp -LAMBDA X, x or =, lambda 0.

Our objective is to develop methodology for analyzing life test data. Initially, there is only data-no mathematical models. Through an exploratory data analysis or an analysis based on the physical processes generating the data, an exponential life distribution model may be judged as appropriate for the analysis of the data. Specifically: where lambda is the unknown constant failure rate. The vertical bar in F(x/lambda) indicates that we are conditioning on the parameter lambda; i.e., for specified lambda the distribution is exponential with failure rate lambda. The corresponding density is F(x/lambda) = lambda exp -LAMBDA X, x or =, lambda 0.

It is shown that the exponential is the only distribution which satisfies a certain regression equation. This characterization equation involves the conditional expectation (regression function) of a record value given a pair of record values, one previous and one future, as covariates. The underlying distribution is exponential if and only if the above regression equals the expected value of an appropriately defined Beta distributed random variable. In a particular case, the expected value of the Beta variable reduces to a weighted average of the covariates.

This paper indicates how exponential random variables can be effeciently used in a variety of simulation problems. One of the problems is the simulation of order statistics from a normal population. The authors discuss the general problem of simulating order statistics and then consider the normal case. They start by showing how the Von-Neumann rejection approach via the exponential can be modified to become an efficient algorithm for generating a normal and then present a method for generating normal order statistics. They show how to use the exponential to efficiently simulate random permutations with weights. They consider the problem of simulating a 2-dimensional Poissin process both for a homogeneous and nonhomogeneous Poisson process. (Author)

In this note two characterizations of the exponential distribution are discussed, based on a generalization of the lack of memory property. These results were motivated by the notion of 'relevation of distributions' introduced by Krakowski (1973). (Author)

In this note two characterizations of the exponential distribution are discussed, based on a generalization of the lack of memory property. These results were motivated by the notion of 'relevation of distributions' introduced by Krakowski (1973). (Author)

We analyze the problem of the analytical characterization of the probability distribution of financial returns in the exponential Ornstein-Uhlenbeck model with stochastic volatility. In this model the prices are driven by a Geometric Brownian motion, whose diffusion coefficient is expressed through an exponential function of an hidden variable Y governed by a mean-reverting process. We derive closed-form expressions for the probability distribution and its characteristic function in two limit cases. In the first one the fluctuations of Y are larger than the volatility normal level, while the second one corresponds to the assumption of a small stationary value for the variance of Y. Theoretical results are tested numerically by intensive use of Monte Carlo simulations. The effectiveness of the analytical predictions is checked via a careful analysis of the parameters involved in the numerical implementation of the Euler-Maruyama scheme and is tested on a data set of financial indexes. In particular, we discuss results for the German DAX30 and Dow Jones Euro Stoxx 50, finding a good agreement between the empirical data and the theoretical description.

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We employ a parameter-free distribution estimation framework where estimators are random distributions and utilize the Kullback-Leibler (KL) divergence as a loss function. Wu and Vos [J. Statist. Plann. Inference 142 (2012) 1525-1536] show that when an estimator obtained from an i.i.d. sample is viewed as a random distribution, the KL risk of the estimator decomposes in a fashion parallel to the mean squared error decomposition when the estimator is a real-valued random variable. In this paper, we explore how conditional versions of distribution expectation ($E^{\dagger}$) can be defined so that a distribution version of the Rao-Blackwell theorem holds. We define distributional expectation and variance ($V^{\dagger}$) that also provide a decomposition of KL risk in exponential and mixture families. For exponential families, we show that the maximum likelihood estimator (viewed as a random distribution) is distribution unbiased and is the unique uniformly minimum distribution variance unbiased (UMV$^{\dagger}$U) estimator. Furthermore, we show that the MLE is robust against model specification in that if the true distribution does not belong to the exponential family, the MLE is UMV$^{\dagger}$U for the KL projection of the true distribution onto the exponential families provided these two distribution have the same expectation for the canonical statistic. To allow for estimators taking values outside of the exponential family, we include results for KL projection and define an extended projection to accommodate the non-existence of the MLE for families having discrete sample space. Illustrative examples are provided.

We consider the gravitational potential and the gravitational rotation field generated by an spherical mass distribution with exponential density, when the force between any two mass elements is not the usual Newtonian one, but some general central force. We invert the usual integral relations and obtain the elemental interaction (between two point-like masses) as a function of the macroscopic gravitational field (the one generated by the distribution). Thus, we have a direct way for testing the possibility of finding a correction to the Newtonian law of gravity that can explain the observed dynamics at large scales without the need of dark matter. We show that this method can be used even in the case of spiral galaxies with a good level of confidence.

A generalized so called magnetically controlled ballistic rain-like deposition (MBD) model of granular piles has been numerically investigated in 2D. The grains are taken to be elongated disks whence characterized by a two-state scalar degree of freedom, called ''nip'', their interaction being described through a Hamiltonian. Results are discussed in order to search for the effect of nip flip (or grain rotation from vertical to horizontal and conversely) probability in building a granular pile. The characteristics of creation of + (or $-$) nip's clusters and clusters of holes (missing nips) are analyzed. Two different cluster-mass regimes have been identified, through the cluster-mass distribution function which can be exponential or have a power law form depending on whether the nip flip (or grain rotation) probability is large or small. Analytical forms of the exponent are empirically found in terms of the Hamiltonian parameters.

We consider a class of nonlinear Boltzmann equations describing return to thermal equilibrium in a gas of colliding particles suspended in a thermal medium. We study solutions in the space $L^{1}(\mathbb{R}^{3}\times \mathbb{T}^3).$ Special solutions of these equations, called "Maxwellians," are spatially homogenous static Maxwell velocity distributions at the temperature of the medium. We prove that, for dilute gases, the solutions corresponding to smooth initial conditions in a weighted $L^{1}$-space converge to a Maxwellian in $L^{1},$ exponentially fast in time.

In this paper non-asymptotic exact exponential estimates are derived for the tail of maximum distribution of random field in the terms of majoring measures or, equally, generic chaining.

We characterize the exponential distribution as the only one which satisfies a regression condition. This condition involves the regression function of a fixed record value given two other record values, one of them being previous and the other next to the fixed record value, and none of them are adjacent. In particular, it turns out that the underlying distribution is exponential if and only if given the first and last record values, the expected value of the median in a sample of record values equals the sample midrange.

We prove that the exponential distribution is the only one which satisfies a regression identity. This identity involves conditional expectation of the sample mean of record values given two record values outside of the sample.

The main aim of this article is to characterize and investigate the three parameter exponentiated exponential Poisson probability distribution ${\rm EEP}(\alpha, \beta, \lambda)$ by giving explicit closed form expressions for its characteristic function $\phi_\xi(t)$ and moment generating function $M_\xi(t)$, and finally, to show that the existing series and integral form expressions for positive integer order moments $\mathbb E\xi^\nu, \nu \in \mathbb N$ are in fact valid for all $\nu>1-\alpha, \alpha>0$.

The gravitational potential and the gravitational rotation field generated by a thin-disk mass distribution with exponential density are considered in the case when the force between any two mass elements is not the usual Newtonian one, but some general central force. We use an approximation such that in the Newtonian case the gravitational field generated by the disk reduces to the familiar expression that results from applying the Gauss' law. In this approximation, we invert the usual integral relations in such a way that the elemental interaction (between two point-like masses) is obtained as a function of the overall gravitational field (the one generated by the distribution). Thus, we have a direct way for testing whether it is possible or not to find a correction to the Newtonian law of gravity that can explain the observed dynamics in spiral galaxies without dark matter.

The gravitational potential and the gravitational rotation field generated by a thin-disk mass distribution with exponential density are considered in the case when the force between any two mass elements is not the usual Newtonian one, but some general central force. We use an approximation such that in the Newtonian case the gravitational field generated by the disk reduces to the familiar expression that results from applying the Gauss' law. In this approximation, we invert the usual integral relations in such a way that the elemental interaction (between two point-like masses) is obtained as a function of the overall gravitational field (the one generated by the distribution). Thus, we have a direct way for testing whether it is possible or not to find a correction to the Newtonian law of gravity that can explain the observed dynamics in spiral galaxies without dark matter.

The gravitational potential and the gravitational rotation field generated by a thin-disk mass distribution with exponential density are considered in the case when the force between any two mass elements is not the usual Newtonian one, but some general central force. We use an approximation such that in the Newtonian case the gravitational field generated by the disk reduces to the familiar expression that results from applying the Gauss' law. In this approximation, we invert the usual integral relations in such a way that the elemental interaction (between two point-like masses) is obtained as a function of the overall gravitational field (the one generated by the distribution). Thus, we have a direct way for testing whether it is possible or not to find a correction to the Newtonian law of gravity that can explain the observed dynamics in spiral galaxies without dark matter.

A new characterization of the exponential distribution is established. It is proven that the well-known Sukhatme-Renyi necessary condition is also sufficient for exponentiality. A method of proof due to Arnold and Villasenor based on the Maclaurin series expansion of the density is utilized.

Consider a complex system consisting of several components in series. The system as a whole is either impossible or too expensive to test. The only data available are the times to failure of the components when tested individually. Based on these data an exact lower confidence bound for the reliability of the system is obtained. Initially, the method is developed for the case when the components are independent. Later the method has been extended to the case when the components are dependent and have multivariate exponential time to failure distribution. The sampling method considered is type II censoring of r out of n with or without replacement. An asymptotically smallest confidence interval has also been developed.

We derive a closed-form (infinite series) representation for the distribution of the ruin time for the Sparre Andersen model with exponentially distributed claims. This extends a recent result of Dickson et al. (2005) for such processes with Erlang inter-claim times. We illustrate our result in the cases of gamma and mixed exponential inter-claim time distributions.

In [1], we introduced a new, matrix algebraic, performance analysis framework for wireless systems with fading channels based on the matrix exponential distribution. The main idea was to use the compact, powerful, and easy-to-use, matrix exponential (ME)-distribution for i) modeling the unprocessed channel signal to noise ratio (SNR), ii) exploiting the closure property of the ME-distribution for SNR processing operations to give the effective channel random variable (r.v.) on ME-distribution form, and then to iii) express the performance measure in a closed-form based on ME-distribution matrix/vector parameters only. In this work, we aim to more clearly present, formalize, refine and develop this unified bottom-up analysis framework, show its versatility to handle important communication cases, performance evaluation levels, and performance metrics. The bivariate ME-distribution is introduced here as yet another useful ME-tool, e.g. to account for dependency among two r.v.s. We propose that the ME-distribution may, in addition to fading, also characterize the pdf of discrete-time signal r.v.s, thus extending the ME-distribution matrix form to new generalized 1D/2D-Gaussian-, and Rayleigh-, distribution-like matrix forms. Our findings here, strengthen the observation from [1], [2], and indicates that the ME-distribution can be a promising tool for wireless system modeling and performance analysis.

We introduce the beta generalized exponential distribution that includes the beta exponential and generalized exponential distributions as special cases. We provide a comprehensive mathematical treatment of this distribution. We derive the moment generating function and the $r$th moment thus generalizing some results in the literature. Expressions for the density, moment generating function and $r$th moment of the order statistics also are obtained. We discuss estimation of the parameters by maximum likelihood and provide the information matrix. We observe in one application to real data set that this model is quite flexible and can be used quite effectively in analyzing positive data in place of the beta exponential and generalized exponential distributions.

A finite-support constraint on the parameter space is used to derive a lower bound on the error of an estimator of the correlation coefficient in the bivariate exponential distribution. The bound is then exploited to examine optimality of three estimators, each being a nonlinear function of moments of exponential or Rayleigh observables. The estimator based on a measure of cosine similarity is shown to be highly efficient for values of the correlation coefficient greater than 0.35; for smaller values, however, it is the transformed Pearson correlation coefficient that exhibits errors closer to the derived bound.

In this work we have studied the research activity for countries of Europe, Latin America and Africa for all sciences between 1945 and November 2008. All the data are captured from the Web of Science database during this period. The analysis of the experimental data shows that, within a nonextensive thermostatistical formalism, the Tsallis \emph{q}-exponential distribution $N(c)$ satisfactorily describes Institute of Scientific Information citations. The data which are examined in the present survey can be fitted successfully as a first approach by applying a {\it single} curve (namely, $N(c) \propto 1/[1+(q-1) c/T]^{\frac{1}{q-1}}$ with $q\simeq 4/3$ for {\it all} the available citations $c$, $T$ being an "effective temperature". The present analysis ultimately suggests that the phenomenon might essentially be {\it one and the same} along the {\it entire} range of the citation number. Finally, this manuscript provides a new ranking index, via the "effective temperature" $T$, for the impact level of the research activity in these countries, taking into account the number of the publications and their citations.

For a skip-free Markov process on non-negative integers with generator matrix Q, we evaluate the joint Laplace transform of the occupation times before hitting the state n (starting at 0). This Laplace transform has a very straightforward and familiar expression. We investigate the properties of this Laplace transform, especially the conditions under which the occupation times form a Markov chain.

A hybrid censoring scheme is a mixture of Type-I and Type-II censoring schemes. We study the estimation of parameters of weighted exponential distribution based on Type-II hybrid censored data. By applying EM algorithm, maximum likelihood estimators are evaluated. Also using Fisher infirmation matrix asymptotic confidence intervals are provided. By applying Markov Chain Monte Carlo techniques Bayes estimators, and corresponding highest posterior density confidence intervals of parameters are obtained. Monte Carlo simulations to compare the performances of the different methods is performed and one data set is analyzed for illustrative purposes.

In situations where the sampling units in a study can be more easily ranked based on the measurement of an auxiliary variable, ranked set sampling provide unbiased estimators for the mean of a population that they are more efficient than unbiased estimator based on simple random sample. In this paper, we consider the Morgenstern type bivariate generalized exponential distribution (MTBGED) and obtain several unbiased estimators for a parameter mean of the marginal distribution of MTBGED based on different ranked set sampling schemes. The efficiency of all considered estimators are evaluate and has also been demonstrated with numerical illustrations.

We investigate how significant the spiral structure is on calculations concerning radiative transfer in dusty spiral galaxies seen edge-on. The widely adopted exponential disk model (i.e. both the stars and the dust are distributed exponentially in the radial direction and also perpendicular to the plane of the disk) is now subject to a detailed comparison with a realistic model that includes spiral structure for the stars and the dust in the disk. In particular, model images of galaxies with logarithmic spiral arms are constructed, such that the azimuthally averaged disk is exponential in radius and in height, as the observations suggest. Then, pure exponential disk models (i.e. with no spiral structure) are used to fit the edge-on appearance of the model images. As a result, the parameters derived after the fit are compared to the real values used to create the spiral-structured images. It turns out that the plain exponential disk model is able to give a very good description of the galactic disk with its parameters varying only by a few percent from their true values.

This thesis used three criteria to test for equality of p populations with underlying two parameter exponential distributions (theta = location parameter, sigma = scale parameter). The criers use n random samples drawn for each of the p populations. The three criteria are based on three hypotheses. The asymptotic expansion of the distributions are found based on the Neyman-Pearson likelihood ratio. The asymptotic expansions are computed using Bernoulli polynomials and a recursive relationship developed by Kalinin and Shalaevskii. Nine tables of percentage points are computed for each test statistic from the expansions where p = 2(1)10, n = 10(1)20(5)50(10)100, and alpha = .100, 050, .025, .010, .005. These tables along with a practical illustration give the analyst a good technique that can be applied to many exponentially related situations. Originator-supplied keywords include: Asymptotic series, Exponential Distributions, Distribution functions, Hypotheses, Percentage point tables, and Likelihood ratio.

The two-parameter distribution known as exponential-Poisson (EP) distribution, which has decreasing failure rate, was introduced by Kus (2007). In this paper we generalize the EP distribution and show that the failure rate of the new distribution can be decreasing or increasing. The failure rate can also be upside-down bathtub shaped. A comprehensive mathematical treatment of the new distribution is provided. We provide closed-form expressions for the density, cumulative distribution, survival and failure rate functions; we also obtain the density of the $i$th order statistic. We derive the $r$th raw moment of the new distribution and also the moments of order statistics. Moreover, we discuss estimation by maximum likelihood and obtain an expression for Fisher's information matrix. Furthermore, expressions for the R\'enyi and Shannon entropies are given and estimation of the stress-strength parameter is discussed. Applications using two real data sets are presented.

The two-parameter distribution known as exponential-Poisson (EP) distribution, which has decreasing failure rate, was introduced by Kus (2007). In this paper we generalize the EP distribution and show that the failure rate of the new distribution can be decreasing or increasing. The failure rate can also be upside-down bathtub shaped. A comprehensive mathematical treatment of the new distribution is provided. We provide closed-form expressions for the density, cumulative distribution, survival and failure rate functions; we also obtain the density of the $i$th order statistic. We derive the $r$th raw moment of the new distribution and also the moments of order statistics. Moreover, we discuss estimation by maximum likelihood and obtain an expression for Fisher's information matrix. Furthermore, expressions for the R\'enyi and Shannon entropies are given and estimation of the stress-strength parameter is discussed. Applications using two real data sets are presented.

In this paper three new characterizing theorems of exponential distribution are presented. They are based on equidistribution of some functions of order statistics. All of them include the median of sample of size three.

We introduce the concept of weakly Markov dynamical systems. We prove for them the asymptotic convergence, along the radii of relative Lebesgue measure converging very fast to one, to the exponential law of the distribution of the the first return times to almost every shrinking ball. We prove that weakly Markov dynamical systems include such large classes of smooth dynamical systems as expanding repellers, holomorphic endomorphisms of complex projective spaces, and Axiom A diffeomorphisms (none of them are assumed to be conformal) and conformal ones such as conformal iterated function systems, conformal graph directed Markov systems, conformal expanding repellers, rational functions of the Riemann sphere, and transcendental meromorphic functions. For the conformal systems we in fact prove much more, namely that the convergence to the exponential law is along all radii. This is achieved by proving the thin annuli property.

We analyze waiting times for price changes in a foreign currency exchange rate. Recent empirical studies of high frequency financial data support that trades in financial markets do not follow a Poisson process and the waiting times between trades are not exponentially distributed. Here we show that our data is well approximated by a Weibull distribution rather than an exponential distribution in a non-asymptotic regime. Moreover, we quantitatively evaluate how much an empirical data is far from an exponential distribution using a Weibull fit. Finally, we discuss a phase transition between a Weibull-law and a power-law in the asymptotic long waiting time regime.

We study holonomic gradient decent for maximum likelihood estimation of exponential-polynomial distribution, whose density is the exponential function of a polynomial in the random variable. We first consider the case that the support of the distribution is the set of positive reals. We show that the maximum likelihood estimate (MLE) can be easily computed by the holonomic gradient descent, even though the normalizing constant of this family does not have a closed-form expression and discuss determination of the degree of the polynomial based on the score test statistic. Then we present extensions to the whole real line and to the bivariate distribution on the positive orthant.

A statistical model of quantization based on an exponential distribution of infinitesimal action is proposed. Trajectory which does not extremize the action along an infinitesimal short segment of path is allowed to occur with a very small probability following an exponential law. Planck constant is argued to give the average deviation from the infinitesimal stationary action.