Exponential Marginal Distributions Research Articles

Modelling and Residual Analysis of Nonlinear Autoregressive Time Series in Exponential Variables

SUMMARY An approach to modelling and residual analysis of nonlinear autoregressive time series in exponential variables is presented; the approach is illustrated by an analysis of a long series of wind velocity data which has first been detrended and then transformed into a stationary series with an exponential marginal distribution. The stationary series is modelled with a newly developed type of second order autoregressive process with random coefficients, called the NEAR(2) model; it has a second order autoregressive correlation structure but is nonlinear because its coefficients are random. The exponential distributional assumptions involved in this model highlight a very broad four parameter structure which combines five exponential random variables into a sixth exponential random variable; other applications of this structure are briefly considered. Dependency in the NEAR(2) process not accounted for by standard autocorrelations is explored by developing a residual analysis for time series having autoregressive correlation structure; this involves defining linear uncorrelated residuals which are dependent, and then assessing this higher order dependence by standard time series computations. The application of this residual analysis to the wind velocity data illustrates both the utility and difficulty of nonlinear time series modelling.

Modelling correlated arrivals with a phase-type process

One considers semi-Markov processes using phase-type distributions, such that the marginal distribution of the interevent times are identically distributed but not independent. By suitably choosing the phase-type distributions, one obtains an exponential marginal distribution.

A Mixed Exponential Time Series Model

The simple model NMEAR (1) is described for a stationary dependent sequence of random variables which have a mixed exponential marginal distribution; the model is a first-order stochastic difference equation with random coefficients and is first-order Markovian. It should be broadly applicable for stochastic modelling in operations analysis. In particular, it provides a model for simulating interarrival times in queuing systems when these random variables are overdispersed relative to an exponential random variable, and moreover are positively correlated. The model also has capability to model a variable which may be zero, but which otherwise is exponentially distributed. Such variables are found as waiting times in queuing models. Because of the (random) linearity of the process, it is easily extended to the modelling of cross-coupled sequences of interarrival and service times. The model can also be extended quite simply to a mixed exponential process with mixed pth order autoregressive and qth order moving average correlation structure, NMEAR (p, q), so that non-Markovian dependence can be handled.

Optimal replacement of parts having observable correlated stages of deterioration

AbstractA single component system is assumed to progress through a finite number of increasingly bad levels of deterioration. The system with level i (0 ≤ i ≤ n) starts in state 0 when new, and is definitely replaced upon reaching the worthless state n. It is assumed that the transition times are directly monitored and the admissible class of strategies allows substitution of a new component only at such transition times. The durations in various deterioration levels are dependent random variables with exponential marginal distributions and a particularly convenient joint distribution. Strategies are chosen to maximize the average rewards per unit time. For some reward functions (with the reward rate depending on the state and the duration in this state) the knowledge of previous state duration provides useful information about the rate of deterioration.

The mixed exponential solution to the first-order autoregressive model

This paper gives the distribution of the independent variable in a first-order autoregressive equation which is necessary for the modelled variable to have a mixed exponential marginal distribution. Restricting attention first to a mixture of two exponentials which is probabilistic, a solution is shown to exist over a restricted range of the first serial correlation; tabulations of the boundary values are given. The independent variable has a distribution which includes a discrete component at zero and a not necessarily probabilistic mixture of three exponentials. The marginal distribution is then considered with a negative mixture weight; this weight is shown to be lower bounded in value, but not to depend on the first serial correlation.

The mixed exponential solution to the first-order autoregressive model

This paper gives the distribution of the independent variable in a first-order autoregressive equation which is necessary for the modelled variable to have a mixed exponential marginal distribution. Restricting attention first to a mixture of two exponentials which is probabilistic, a solution is shown to exist over a restricted range of the first serial correlation; tabulations of the boundary values are given. The independent variable has a distribution which includes a discrete component at zero and a not necessarily probabilistic mixture of three exponentials. The marginal distribution is then considered with a negative mixture weight; this weight is shown to be lower bounded in value, but not to depend on the first serial correlation.

The Exponential Autoregressive-Moving Average Earma(P,Q) Process

Summary A new model for pth-order autoregressive processes with exponential marginal distributions, ear(p), is developed and an earlier model for first-order moving average exponential processes is extended to qth-order, giving an ema(q) process. The correlation structures of both processes are obtained separately. A mixed process, earma(p,q), incorporating aspects of both ear(p) and ema(q) correlation structures is then developed. The earma(p, q) process is an analog of the standard arma(p, q) time series models for Gaussian processes and is generated from a single sequence of independent and identically distributed exponential varables.

Statistical inference for a certain class of bivariate distributions

The paper deals with statistical inference for a certain class of bivariate distributions. The class of marginal distributions is given and is shown to include distributions with only location and scale parameters. A normalizing transformation is applied to the marginal distributions and the parameters are estimated by maximum likelihood. For this class there is a great deal of simplification in the calculations for the asymptotic covariance matrix of the vector of parameter estimators. Statistics for tests of zero correlation are discussed. Also, the analysis is carried out for exponential marginal distributions.

A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1)

A stationary sequence of random variables with exponential marginal distributions and the correlation structure of an ARMA (1, 1) process is defined. The process is formed as a random linear combination of i.i.d. exponential random variables and is very simple to generate on a computer. Moments and joint distributions for the sequence are obtained, as well as limiting properties of sums of the random variables and of the point process whose intervals have the EARMA (1, 1) structure.

An exponential moving-average sequence and point process (EMA1)

A construction is given for a stationary sequence of random variables {Xi} which have exponential marginal distributions and are random linear combinations of order one of an i.i.d. exponential sequence {εi}. The joint and trivariate exponential distributions of Xi−1, Xi and Xi+ 1 are studied, as well as the intensity function, point spectrum and variance time curve for the point process which has the {Xi} sequence for successive times between events. Initial conditions to make the point process count stationary are given, and extensions to higher-order moving averages and Gamma point processes are discussed.

A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1)

A stationary sequence of random variables with exponential marginal distributions and the correlation structure of an ARMA (1, 1) process is defined. The process is formed as a random linear combination of i.i.d. exponential random variables and is very simple to generate on a computer. Moments and joint distributions for the sequence are obtained, as well as limiting properties of sums of the random variables and of the point process whose intervals have the EARMA (1, 1) structure.

An exponential moving-average sequence and point process (EMA1)

A construction is given for a stationary sequence of random variables {Xi } which have exponential marginal distributions and are random linear combinations of order one of an i.i.d. exponential sequence {ε i }. The joint and trivariate exponential distributions of Xi −1, Xi and Xi + 1 are studied, as well as the intensity function, point spectrum and variance time curve for the point process which has the {Xi } sequence for successive times between events. Initial conditions to make the point process count stationary are given, and extensions to higher-order moving averages and Gamma point processes are discussed.

Reliability Applications of a Bivariate Exponential Distribution

This paper examines some two-unit systems in which the lifetimes of the two units in service are not independent but depend upon one another in a particular way. This dependence is characterized by the bivariate exponential distribution of Marshall and Olkin, which has exponential marginal distributions and other physically motivating properties. Two measures of reliability are determined: the first is the distribution and mean of the time to system failure (i.e., when all units are failed) and the second gives steady-state probabilities of the number of working units. Some graphical results are given to illustrate the deviation of these quantities from the values obtained under the classical assumption of independent lifetimes.