Discrete Probability Mass Functions Research Articles

Discrete q-Exponential Limit Order Cancellation Time Distribution

Modeling financial markets based on empirical data poses challenges in selecting the most appropriate models. Despite the abundance of empirical data available, researchers often face difficulties in identifying the best fitting model. Long-range memory and self-similarity estimators, commonly used for this purpose, can yield inconsistent parameter values, as they are tailored to specific time series models. In our previous work, we explored order disbalance time series from the broader perspective of fractional L’evy stable motion, revealing a stable anti-correlation in the financial market order flow. However, a more detailed analysis of empirical data indicates the need for a more specific order flow model that incorporates the power-law distribution of limit order cancellation times. When considering a series in event time, the limit order cancellation times follow a discrete probability mass function derived from the Tsallis q-exponential distribution. The combination of power-law distributions for limit order volumes and cancellation times introduces a novel approach to modeling order disbalance in the financial markets. Moreover, this proposed model has the potential to serve as an example for modeling opinion dynamics in social systems. By tailoring the model to incorporate the unique statistical properties of financial market data, we can improve the accuracy of our predictions and gain deeper insights into the dynamics of these complex systems.

RECOVERING PROBABILITY FUNCTIONS WITH FOURIER SERIES

The COS method was introduced in Fang & Oosterlee (2008) and then was applied to pricing a variety of stock options for continuous random variables. This paper adapts the Fourier-cosine series (COS) method to recover discrete probability mass functions. We approximate mixture and compound probability distributions with cosine series. Enormous precision and computational speed are the qualities of the function estimates here obtained. We also develop the pricing framework to trade derivatives subject to discrete random variables. We apply the method to calculate, for the first time, the price of an interest rate derivative of recent vintage introduced in the Brazilian financial market. Parameter calibration confirms the quality of the model.

Consistent second-order discrete kernel smoothing using dispersed Conway–Maxwell–Poisson kernels

The histogram estimator of a discrete probability mass function often exhibits undesirable properties related to zero probability estimation both within the observed range of counts and outside into the tails of the distribution. To circumvent this, we formulate a novel second-order discrete kernel smoother based on the recently developed mean-parametrized Conway–Maxwell–Poisson distribution which allows for both over- and under-dispersion. Two automated bandwidth selection approaches, one based on a simple minimization of the Kullback–Leibler divergence and another based on a more computationally demanding cross-validation criterion, are introduced. Both methods exhibit excellent small and large sample performance. Computational results on simulated datasets from a range of target distributions illustrate the flexibility and accuracy of the proposed method compared to existing smoothed and unsmoothed estimators. The method is applied to the modelling of somite counts in earthworms, and the number of development days of insect pests on the Hura tree.

A Discrete Density Approach to Bayesian Quantile and Expectile Regression with Discrete Responses

For decades, regression models beyond the mean for continuous responses have attracted great attention in the literature. These models typically include quantile regression and expectile regression. But there is little research on these regression models for discrete responses, particularly from a Bayesian perspective. By forming the likelihood function based on suitable discrete probability mass functions, this paper introduces a discrete density approach for Bayesian inference of these regression models with discrete responses. Bayesian quantile regression for discrete responses is first developed, and then this method is extended to Bayesian expectile regression for discrete responses. The posterior distribution under this approach is shown not only coherent irrespective of the true distribution of the response, but also proper with regarding to improper priors for the unknown model parameters. The performance of the method is evaluated via extensive Monte Carlo simulation studies and one real data analysis.

Entropy and Information Content of Geostatistical Models

Geostatistical models quantify spatial relations between model parameters and can be used to estimate and simulate properties away from known observations. The underlying statistical model, quantified through a joint probability density, most often consists of both an assumed statistical model and the specific choice of algorithm, including tuning parameters controlling the algorithm. Here, a theory is developed that allows one to compute the entropy of the underlying multivariate probability density when sampled using sequential simulation. The self-information of a single realization can be computed as the sum of the conditional self-information. The entropy is the average of the self-information obtained for many independent realizations. For discrete probability mass functions, a measure of the effective number of free model parameters, implied by a specific choice of probability mass function, is proposed. Through a few examples, the entropy measure is used to quantify the information content related to different choices of simulation algorithms and tuning parameters.

A Type-1 Imbalanced Bivariate Poisson Distribution Demystifies Patient’s Phobia Visiting Physician Often and Its Implications

In this article, a new bivariate Poisson distribution, called the type-1 imbalanced bivariate Poisson distribution, is introduced for dependence modeling among the random number of visits made by a patient and the number of prescriptions written by a physician. By utilizing the proposed model, the article discovers patterns on the implications of the over-visits made by the patients. This new discrete bivariate Poisson probability mass function is quite interesting and useful to the healthcare researchers. Its statistical properties are derived. The concepts and methodology around its imbalance parameter, $$ \phi $$ , are constructed and illustrated using the Australian Health Survey data for 1997–1998. The data analytic results in the illustration demystify the implications when the patients make more visits to the physician.

Estimating and interpreting migration of Amazonian forests using spatially implicit and semi-explicit neutral models.

With many sophisticated methods available for estimating migration, ecologists face the difficult decision of choosing for their specific line of work. Here we test and compare several methods, performing sanity and robustness tests, applying to large‐scale data and discussing the results and interpretation. Five methods were selected to compare for their ability to estimate migration from spatially implicit and semi‐explicit simulations based on three large‐scale field datasets from South America (Guyana, Suriname, French Guiana and Ecuador). Space was incorporated semi‐explicitly by a discrete probability mass function for local recruitment, migration from adjacent plots or from a metacommunity. Most methods were able to accurately estimate migration from spatially implicit simulations. For spatially semi‐explicit simulations, estimation was shown to be the additive effect of migration from adjacent plots and the metacommunity. It was only accurate when migration from the metacommunity outweighed that of adjacent plots, discrimination, however, proved to be impossible. We show that migration should be considered more an approximation of the resemblance between communities and the summed regional species pool. Application of migration estimates to simulate field datasets did show reasonably good fits and indicated consistent differences between sets in comparison with earlier studies. We conclude that estimates of migration using these methods are more an approximation of the homogenization among local communities over time rather than a direct measurement of migration and hence have a direct relationship with beta diversity. As betadiversity is the result of many (non)‐neutral processes, we have to admit that migration as estimated in a spatial explicit world encompasses not only direct migration but is an ecological aggregate of these processes. The parameter m of neutral models then appears more as an emerging property revealed by neutral theory instead of being an effective mechanistic parameter and spatially implicit models should be rejected as an approximation of forest dynamics.

Effective sample size for importance sampling based on discrepancy measures

The Effective Sample Size (ESS) is an important measure of efficiency of Monte Carlo methods such as Markov Chain Monte Carlo (MCMC) and Importance Sampling (IS) techniques. In the IS context, an approximation ESS^ of the theoretical ESS definition is widely applied, involving the inverse of the sum of the squares of the normalized importance weights. This formula, ESS^, has become an essential piece within Sequential Monte Carlo (SMC) methods, to assess the convenience of a resampling step. From another perspective, the expression ESS^ is related to the Euclidean distance between the probability mass described by the normalized weights and the discrete uniform probability mass function (pmf). In this work, we derive other possible ESS functions based on different discrepancy measures between these two pmfs. Several examples are provided involving, for instance, the geometric mean of the weights, the discrete entropy (including the perplexity measure, already proposed in literature) and the Gini coefficient among others. We list five theoretical requirements which a generic ESS function should satisfy, allowing us to classify different ESS measures. We also compare the most promising ones by means of numerical simulations.

Maximum likelihood estimation of a unimodal probability mass function

We develop an estimation procedure of a discrete probability mass function (pmf) with unknown support. We derive the maximum likelihood estimator of this pmf under the mild and natural shape-constraint of unimodality. Shape-constrained estimation is a powerful and robust technique, which additionally provides smoothing of the empirical distribution yielding thereby gains in eciency. We show that our unimodal estimator is consistent when the model is specied, and that it converges to the best projection of the true pmf on the unimodal class under model misspecication. Furthermore, we derive the limiting distribution of the the estimator, and use the obtained result to build asymptotic condence bands for the unknown pmf when the latter is unimodal. We illustrate our approach using time-to-onset data of the Ebola virus during the 1976 outbreak in the former republic of Zaire.

A general formula for computing maximum proportion correct scores in various psychophysical paradigms with arbitrary probability distributions of stimulus observations.

Proportion correct (Pc) is a fundamental measure of task performance in psychophysics. The maximum Pc score that can be achieved by an optimal (maximum-likelihood) observer in a given task is of both theoretical and practical importance, because it sets an upper limit on human performance. Within the framework of signal detection theory, analytical solutions for computing the maximum Pc score have been established for several common experimental paradigms under the assumption of Gaussian additive internal noise. However, as the scope of applications of psychophysical signal detection theory expands, the need is growing for psychophysicists to compute maximum Pc scores for situations involving non-Gaussian (internal or stimulus-induced) noise. In this article, we provide a general formula for computing the maximum Pc in various psychophysical experimental paradigms for arbitrary probability distributions of sensory activity. Moreover, easy-to-use MATLAB code implementing the formula is provided. Practical applications of the formula are illustrated, and its accuracy is evaluated, for two paradigms and two types of probability distributions (uniform and Gaussian). The results demonstrate that Pc scores computed using the formula remain accurate even for continuous probability distributions, as long as the conversion from continuous probability density functions to discrete probability mass functions is supported by a sufficiently high sampling resolution. We hope that the exposition in this article, and the freely available MATLAB code, facilitates calculations of maximum performance for a wider range of experimental situations, as well as explorations of the impact of different assumptions concerning internal-noise distributions on maximum performance in psychophysical experiments.

Multiporosity flow in fractured low‐permeability rocks

AbstractA multiporosity extension of classical double and triple‐porosity fractured rock flow models for slightly compressible fluids is presented. The multiporosity model is an adaptation of the multirate solute transport model of Haggerty and Gorelick (1995) to viscous flow in fractured rock reservoirs. It is a generalization of both pseudo steady state and transient interporosity flow double‐porosity models. The model includes a fracture continuum and an overlapping distribution of multiple rock matrix continua, whose fracture‐matrix exchange coefficients are specified through a discrete probability mass function. Semianalytical cylindrically symmetric solutions to the multiporosity mathematical model are developed using the Laplace transform to illustrate its behavior. The multiporosity model presented here is conceptually simple, yet flexible enough to simulate common conceptualizations of double and triple‐porosity flow. This combination of generality and simplicity makes the multiporosity model a good choice for flow modelling in low‐permeability fractured rocks.

A Market Discovery Algorithm to Estimate a General Class of Nonparametric Choice Models

We propose an approach for estimating customer preferences for a set of substitutable products using only sales transactions and product availability data. The underlying demand framework combines a general, nonparametric discrete choice model with a Bernoulli process of arrivals over time. The choice model is defined by a discrete probability mass function (pmf) on a set of possible preference rankings of alternatives, and it is compatible with any random utility model. An arriving customer is assumed to purchase the available option that ranks highest in her preference list. The problem we address is how to jointly estimate the arrival rate and the pmf of the rank-based choice model under a maximum likelihood criterion. Since the potential number of customer types is factorial, we propose a market discovery algorithm that starts with a parsimonious set of types and enlarge it by automatically generating new types that increase the likelihood value. Numerical experiments confirm the potential of our proposal. For a realistic data set in the hospitality industry, our approach improves the root mean square errors between predicted and observed purchases computed under independent demand model estimates by 67% to 93%. This paper was accepted by Serguei Netessine, operations management.

Bayesian inversion and sequential Monte Carlo sampling techniques applied to nearfield acoustic sensor arrays

This paper demonstrates that inverse source reconstruction can be performed using a methodology of particle filters that relies primarily on the Bayesian approach of parameter estimation. The proposed approach is applied in the context of nearfield acoustic holography based on the equivalent source method (ESM). A state-space model is formulated in light of the ESM. The parameters to estimate are amplitudes and locations of the equivalent sources. The parameters constitute the state vector which follows a first-order Markov process with the transition matrix being the identity for every frequency-domain data frame. The implementation of recursive Bayesian filters involves a sequential Monte Carlo sampling procedure that treats the estimates as point masses with a discrete probability mass function (PMF) which evolves with iteration. It is evident from the results that the inclusion of the appropriate prior distribution is crucial in the parameter estimation.

Moment-Based Approximations of Discrete Probability Distributions Using Rational Functions

Provost et al. (2009) introduced an approximation to discrete probability mass functions using a base density function and a polynomial function multiplier. In this article, we examine a new estimator constructed by using a rational function multiplier. The new approximation requires fewer moments to calculate, and generally performs better. Several example cases will be presented.

Global convergence of the log-concave MLE when the true distribution is geometric

Let X1, …, Xn be i.i.d. from a discrete probability mass function (pmf) p. In Balabdaoui et al. [(2013), ‘Asymptotic Distribution of the Discrete Log-Concave mle and Some Applications’, JRSS-B, in press], the pointwise limit distribution of the log-concave maximum-likelihood estimator (MLE) was derived in both the well- and misspecified settings. In the well-specified setting, the geometric distribution was excluded, classified as being degenerate. In this article, we establish the global asymptotic theory of the log-concave MLE of a geometric pmf in all ℓq distances for q∈{1, 2, …}∪{∞}. We also show how these asymptotic results could be used in testing whether a pmf is geometric.

Bayesian approach of nearfield acoustic reconstruction with particle filters

This paper demonstrates that inverse source reconstruction can be performed using a methodology of particle filters that relies primarily on the Bayesian approach of parameter estimation. In particular, the proposed approach is applied in the context of nearfield acoustic holography based on the equivalent source method (ESM). A state-space model is formulated in light of the ESM. The parameters to estimate are amplitudes and locations of the equivalent sources. The parameters constitute the state vector which follows a first-order Markov process with the transition matrix being the identity for every frequency-domain data frame. Filtered estimates of the state vector obtained are assigned weights adaptively. The implementation of recursive Bayesian filters involves a sequential Monte Carlo sampling procedure that treats the estimates as point masses with a discrete probability mass function (PMF) which evolves with iteration. The weight update equation governs the evolution of this PMF and depends primarily on the likelihood function and the prior distribution. It is evident from the simulation results that the inclusion of the appropriate prior distribution is crucial in the parameter estimation.

Option pricing under a normal mixture distribution derived from the Markov tree model

We examine a Markov tree (MT) model for option pricing in which the dynamics of the underlying asset are modeled by a non-IID process. We show that the discrete probability mass function of log returns generated by the tree is closely approximated by a continuous mixture of two normal distributions. Using this normal mixture distribution and risk-neutral pricing, we derive a closed-form expression for European call option prices. We also suggest a regression tree-based method for estimating three volatility parameters σ, σ+, and σ− required to apply the MT model. We apply the MT model to price call options on 89 non-dividend paying stocks from the S&P 500 index. For each stock symbol on a given day, we use the same parameters to price options across all strikes and expires. Comparing against the Black–Scholes model, we find that the MT model’s prices are closer to market prices.

Semiparametric estimation for count data through weighted distributions

This paper is concerned with semiparametric discrete kernel estimators when the unknown count distribution can be considered to have a general weighted Poisson form. The estimator is constructed by multiplying the Poisson estimate with a nonparametric discrete kernel-type estimate of the Poisson weight function. Comparisons are then carried out with the ordinary discrete kernel probability mass function estimators. The Poisson weight function is thus a local multiplicative correction factor, and is considered as the uniform measure to detect departures from the equidispersed Poisson distribution. In this way, the effects of dispersion and zero-proportion with respect to the standard Poisson distribution are also minimized. This method of estimation is also applied to the weighted binomial form for the count distribution having a finite support. The proposed estimators, in addition to being simple, easy-to-implement and effective, also outperform the competing nonparametric and parametric estimators in finite-sample situations. Two examples illustrate this new semiparametric estimation.

A multirate DSP model for estimation of discrete probability density functions

The problem of estimating a probability density function (PDF) from measurements has been widely studied by many researchers. Even though much work has been done in the area of PDF estimation, most of it was focused on the continuous case. We propose a new model-based approach for modeling and estimating discrete probability density functions or probability mass functions. This approach is based on multirate signal processing theory, and it has several advantages over the conventional histogram method. We illustrate the PDF estimation procedure and analyze the statistical properties of the PDF estimates. Based on this model, a novel scheme is introduced that can be used for estimating the PDF in the presence of noise. Furthermore, the proposed ideas are extended to the more general case of estimating multivariate PDFs. Finally, we also consider practical issues such as optimizing the coefficients of a digital filter, which is an integral part of the model. This allows us to apply the proposed model to solve real-world problems. Simulation results are given where appropriate in order to demonstrate the ideas.

Probabilistic analysis of uncertainties in seismic hazard assessment

Sources and current methods of analysis of uncertainty from randomness, fuzziness and ignorance or incomplete knowledge in seismic hazard assessment problem are briefly discussed at beginning; understandings of the authors are then presented in the following order. All three types of uncertainty come from incomplete knowledge. Probabilistic method can be applied to all of them, objective probability for random factors and subjective probability for the other two types of uncertain factors. Discrete subjective probability mass functions for incomplete and fuzzy factors can be obtained from logic-tree and membership functions respectively. Fractile curves may be used to show the scattering of any uncertainty factor, but a unified probabilistic treatment may be applied to any combination of all three types of uncertainty.