In this article, we present a concise review of developments on discrete multivariate distributions. We first present some basic definitions and notations. Then, we present several important discrete multivariate distributions and list their significant properties and characteristics.


Keywords:

generating function;
moments;
stirling numbers;
regression;
inflated distributions;
truncated forms;
compound distributions;
multinomial;
negative multinomial;
multivariate poisson;
multivariate hypergeometric;
multivariate Polya–Eggenberger;
multivariate discrete exponential;
multivariate power series;
multivariate hermite;
multivariate occupancy;
multivariate weighted;
dirichlet;
multivariate run-related distributions

Discrete Multivariate Distributions

In Part I of this article, the definitions, prevalence, and clinical presentation of chronic total occlusions (CTOs) were reviewed, the histopathology of CTOs was examined, efforts to replicate human CTOs with experimental models were appraised, and the clinical relevance and rationale for CTO revascularization were evaluated.1 In Part II, we summarize the technical approach to and outcomes after percutaneous coronary intervention (PCI) of occluded coronary arteries, describe the novel devices and drugs approved and undergoing investigation for CTO recanalization, and conclude with practical perspectives on managing the patient with 1 or more chronic coronary occlusions.

### Patient Selection and Revascularization Strategies

PCI of CTOs constitutes as many as 20% of all angioplasty procedures at selected centers,2 although a rate of &10% is more typical,3–6 suggesting that CTO angioplasty is attempted in 50 000 to 100 000 patients per year in the United States. Many more CTOs are present for which PCI is never attempted, representing one of the most common causes for referral to bypass surgery rather than PCI.6–8 Furthermore, a large proportion of patients with CTOs are managed medically, the prognosis of whom may vary depending on the extent of viable myocardium and ischemia, concomitant atherosclerosis in other coronary and noncoronary vascular territories, and other comorbid conditions. The decision to attempt PCI of a CTO (versus continued medical therapy or surgical revascularization) requires an individualized risk/benefit analysis, encompassing clinical, angiographic, and technical considerations. Clinically, the patient’s age, symptom severity, associated comorbidities (eg, diabetes mellitus and chronic renal insufficiency), and overall functional status are major determinants of treatment strategy. Angiographically, the extent and complexity of coronary artery disease (eg, single-vessel versus multivessel disease, single versus multiple total occlusions, likelihood for complete revascularization), left ventricular function, and the presence and degree of valvular heart disease should be considered. The technical probability of achieving …

Percutaneous recanalization of chronically occluded coronary arteries: a consensus document: part II.

Bivariate discrete distributions

A new lifetime distribution

On some lifetime distributions with decreasing failure rate

A lifetime distribution with decreasing failure rate

On an extension of the exponential-geometric distribution

A three-parameter lifetime distribution with increasing, decreasing, bathtub, and upside down bathtub shaped failure rates is introduced. The new model includes the Weibull distribution as a special case. A motivation is given using a competing risks interpretation when restricting its parametric space. Various statistical properties, and reliability aspects are explored; and the estimation of parameters is studied using the standard maximum likelihood procedures. Applications of the model to real data are also included

A Lifetime Distribution With an Upside-Down Bathtub-Shaped Hazard Function

A four-parameter family of Weibull distributions is introduced, as an example of a more general class created along the lines of Marshall and Olkin, 1997. Various properties of the distribution are explored and its usefulness in modelling real data is demonstrated using maximum likelihood estimates.

/pdf/a-family-of-lifetime-distributions-2bl7lw77f5.pdf

S. Loukas

Papers

A lifetime distribution with decreasing failure rate

On an extension of the exponential-geometric distribution

A Lifetime Distribution With an Upside-Down Bathtub-Shaped Hazard Function

A Family of Lifetime Distributions

Factors related to the presence of autoantibodies in patients with chronic mental disorders.