Discussion of "On families of distributions with shape parameters", by M.C. Jones

SCOPUS: ar.j

Hydrocarbon field size distributions: a case study in mixed integer nonlinear programming

Of key importance to oil and gas companies is the size distribution of fields in the areas that they are drilling. Recent arguments suggest that there are many more fields yet to be discovered in mature provinces than had previously been thought because the underlying distribution is monotonic not peaked. According to this view the peaked nature of the distribution for discovered fields reflects not the underlying distribution but the effect of economic truncation. This paper contributes to the discussion by analysing up-to-date exploration and discovery data for two mature provinces using the discovery-process model, based on sampling without replacement and implicitly including economic truncation effects. The maximum likelihood estimation involved generates a high-dimensional mixed-integer nonlinear optimization problem. A highly efficient solution strategy is tested, exploiting the separable structure and handling the integer constraints by treating the problem as a masked allocation problem in dynamic programming.

Measuring distributions of jump rates in disordered metal–hydrogen systems by nuclear magnetic relaxation

Monte Carlo calculations of the nuclear magnetic relaxation rate in a disordered metal–hydrogen system having a distribution of jump rates are reported. The calculations deal specifically with the spin-locked rotating-frame relaxation time T1ρ. The results demonstrate that the temperature variation of the rate is only weakly dependent on the distribution and it is therefore unlikely that the jump rate distribution can be extracted from relaxation measurements in which temperature is the main variable. It is shown that the alternative of measuring the relaxation rate over a wide range of spin-locking field strengths at a constant temperature can lead to an evaluation of the distribution.

Birth-death processes with an instantaneous reflection barrier

A new structure with the special property that an instantaneous reflection barrier is imposed on the ordinary birth-death processes is considered. An easy-checking criterion for the existence of such Markov processes is first obtained. The uniqueness criterion is then established. In the nonunique case, all the honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. It is proved that honest processes are always ergodic without necessarily imposing any extra conditions. Equilibrium distributions for all these ergodic processes are established. Several examples are provided to illustrate our results.

Birth-death processes with disaster and instantaneous resurrection

A new structure with the special property that instantaneous resurrection and mass disaster are imposed on an ordinary birth-death process is considered. Under the condition that the underlying birth-death process is exit or bilateral, we are able to give easily checked existence criteria for such Markov processes. A very simple uniqueness criterion is also established. All honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. Surprisingly, it can be proved that all the honest processes are not only recurrent but also ergodic without imposing any extra conditions. Equilibrium distributions are then established. Symmetry and reversibility of such processes are also investigated. Several examples are provided to illustrate our results.

Markovian bulk-arriving queues with state-dependent control at idle time

This paper considers a Markovian bulk-arriving queue modified to allow both mass arrivals when the queue is idle and mass departures which allow for the possibility of removing the entire workload. Properties of queues which terminate when the server becomes idle are developed first, since these play a key role in later developments. Results for the case of mass arrivals, but no mass annihilation, are then constructed with specific attention being paid to recurrence properties, equilibrium queue-size structure, and waiting-time distribution. A closed-form expression for the expected queue size and its Laplace transform are also established. All of these results are then generalised to allow for the removal of the entire workload, with closed-form expressions being developed for the equilibrium size and waiting-time distributions.

Bivariate distribution modeling with tree diameter and height data

The Logit-Logistic (LL), Johnson's SB, and the Beta (GBD) are flexible four-parameter probability distribution models in terms of the (skewness-kurtosis) region covered, and each has been used for modeling tree diameter distributions in forest stands. This article compares bivariate forms of these models in terms of their adequacy in representing empirical diameter-height distributions from 102 sample plots. Four bivariate models are compared: SBB, the natural, well-known, and much-used bivariate generalization of SB; the bivariate distributions with LL, SB, and Beta as marginals, constructed using Plackett's method (LL-2P, etc.). All models are fitted using maximum likelihood, and their goodness-of-fits are compared using minus log-likelihood (equivalent to Akaike's Information Criterion, the AIC). The performance ranking in this case study was SBB, LL-2P, GBD-2P, and SB-2P

The Richit-Richards family of distributions and its use in forestry

Johnson's SB and the logit-logistic are four-parameter distribution models that may be obtained from the standard normal and logistic distributions by a four-parameter transformation. For relatively small data sets, such as diameter at breast height measurements obtained from typical sample plots, distribution models with four or less parameters have been found to be empirically adequate. However, in situations in which the distributions are complex, for example in mixed stands or when the stand has been thinned or when working with aggregated data, then distribution models with more shape parameters may prove to be necessary. By replacing the symmetric standard logistic distribution of the logit-logistic with a one-parameter “standard Richards” distribution and transforming by a five-parameter Richards function, we obtain a new six-parameter distribution model, the “Richit-Richards”. The Richit-Richards includes the “logit-Richards”, the “Richit-logistic”, and the logit-logistic as submodels. Maximum likelihood estimation is used to fit the model, and some problems in the maximum likelihood estimation of bounding parameters are discussed. An empirical case study of the Richit-Richards and its submodels is conducted on pooled diameter at breast height data from 107 sample plots of Chinese fir (Cunninghamia lanceolata (Lamb.) Hook.). It is found that the new models provide significantly better fits than the four-parameter logit-logistic for large data sets.

Variations in the distributions of the Centropages chierchiae and Temora stylifera (Copepoda: Calanoida) in the north-eastern Atlantic Ocean and western european shelf waters.

Centropages chierchiae and Temora stylifera occurred rarely in the Continuous Plankton Recorder (CPR) survey in the Bay of Biscay, Celtic Sea, and English Channel before 1988. By 2000 they were found frequently and in abundance. The seasonal cycles of abundance of these species differ, C. chierchiae occurring mainly in the summer while T. stylifera was found most frequently in late autumn or winter towards the northern limits of its distribution. The increase in abundance of both species is related to temperature. However, in the years when it was found in the samples, the frequency of occurrence of C. chierchiae was correlated positively with the strength of the shelf edge current and negatively with the North Atlantic Oscillation (NAO) while the reverse was true for T. stylifera.

Nutrient Distributions In An Estuary - Evidence Of Chemical Precipitation Of Dissolved Silicate And Phosphate

Continuous autoanalytical recordings of the axial distributions of dissolved nitrate, silicate and phosphate in the influent freshwater and saline waters of the Tamar Estuary, south-west England have been obtained. Short-term variability in the distributions was assessed by repetitive profiling at approximately 3-h intervals on a single day and seasonal comparisons were obtained from ten surveys carried out between June 1977 and August 1978. Whereas nitrate is always essentially conserved throughout the upper estuary, the silicate- and phosphate-salinity relationships consistently indicate a non-biological removal of these nutrients within the low (0–10%) salinity range. Attempts to quantify precisely the degree of removal and to correlate this with changes in environmental properties (pH, turbidity, chlorophyll fluorescence, salinity, freshwater composition) were mainly inconclusive due to short-term fluctuations in the riverine concentrations of silicate and phosphate advected into the reactive region and to the rapid changes in turbidity brought about by tidally-induced resuspension and deposition of bottom sediment.

Vertical Distributions Of Calanus-Finmarchicus And Calanus-Helgolandicus (Crustacea Copepoda)

The vertical distributions of the spring populations of Calanus finmarchicus (Gunnerus) and C. helgolandicus Claus are described and compared. The differences we observed between the two species have probably confused the understanding of the vertical distribution and development of the populations of Calanus spp. in the shelf seas around the United Kingdom where the species occur together. The results imply that these two congeneric species have different behaviour patterns which minimise interspecific competition where the species have sympatric distributions. C. finmarchicus has its younger development stages overlying the older stages in the water column. In C. helgolandicus the converse is true; i. e., the majority of the populations of Stage I and II copepodites of the first spring generations are found below the thermocline. It is also suggested that the different behaviour patterns lead to different feeding regimes and strategies.