COVID‐19 as a Transformative Opportunity for Families and Therapists: Harnessing the Possibilities That Constraints Offer Us

This article demonstrates several ways in which the constraints that the COVID‐19 pandemic has posed for families can provide transformative opportunities to strengthen family relationships, become more resilient, and resolve conflicts and longstanding difficulties. Since most families worldwide have been greatly impacted on many fronts by the pandemic, they are looking to us for hope, guidance, and solutions for coping with high levels of anxiety about living in a world where life as we knew it has been turned on its head and we are struggling to cope with uncertainty about what the future will bring. However, in spite of all the limitations and health consequences COVID‐19 has posed for our client families, the majority have been quite resourceful and creative in coming up with their own novel self‐generated coping, problem‐solving, and health‐enhancing strategies well before we see them for the first time. Finally, the article discusses how the constraints of COVID‐19 afford family therapists with many opportunities to further hone their therapeutic alliance‐building skills, gain an intimate insider’s view of their clients’ daily lives that are not accessible in our offices, and co‐design with them therapeutic experiments that can produce high‐quality solutions.     

Statistical Analysis for the Inverse Trinomial Distribution

This article considers parameter estimation, goodness of fit, likelihood ratio and score tests, and model selection by Akaike information criterion for the inverse trinomial (IT) distribution, a classical one-dimensional random walk distribution. The IT distribution has a cubic variance function of the mean and is a generalization of the negative binomial distribution. Basic distributional properties and expressions for the probability mass function, recurrence formula, moments, and score functions are also presented.     

Distributions of Runs and Scans on Higher-Order Markov Trees

In this article, we consider the distributions of the number of success runs of specified length and scans on a higher-order Markov tree under three different enumeration schemes (the “non overlapping”, the “at least”, and the “overlapping” scheme). Recursive formulae for the evaluation of their probability generating functions are established. We provide a proper framework for extending the exact distribution theory of runs and scans from based on sequences to based on directed trees. Some numerical results for the run and scan statistics are given in order to illustrate the computational aspects and the feasibility of our theoretical results. Finally, two special reliability systems are considered, which are closely related to our general results.     

Bimatrix Variate Beta Type IV Distribution: Relation to Wilks's Statistic and Bimatrix Variate Kummer-Beta Type IV Distribution

In this article, the bimatrix variate beta Type IV distribution is derived from independent Wishart distributed matrix variables. We explore specific properties of this distribution which is then used to derive the exact expressions of the densities of the product and ratio of two dependent Wilks's statistics and to define the bimatrix Kummer-beta Type IV distribution.     

Linear Combinations,Products and Ratios of Simplicial or Spherical Variates

The density level sets of the two types of measures under consideration are l _{2, p} -circles with p = 1 and p = 2, respectively. The intersection-percentage function (ipf) of such a measure reflects the percentages which the level set corresponding to the p-radius r shares for each r > 0 with a set to be measured. The geometric measure representation formulae in Richter (2009 Richter , W.-D. (2009). Continuous l _{n, p} -symmetric distributions. Lithuanian Mathemat. J. 49:93–108.[Crossref], [Web of Science ®] , [Google Scholar]) is based upon these ipf's and will be used here for evaluating exact cdf's and pdf's for the linear combination, the product, and the ratio of the components of two-dimensional simplicial or spherically distributed random vectors.     

Reliability design for a MANET with cluster-head gateway routing protocol

ABSTRACT

A MANET is a self-configuring network connected by wireless links. In order to facilitate communication within the network, a routing protocol is used to control the traffic. One of the prominent routing protocols for MANET is CGSR. Network reliability related problems have resulted in the construction of many algorithms. The UGFT has an enterprise approach to analyze the reliability of the MANET. The emphasis in this paper has been on executing the MANET reliability using UGFT. An efficient algorithm has been proposed to guarantee the reliability of MANET using UGFT with CGSR and is validated with a case study in a military test bed.     

Goodness of fit for the poisson distribution based on the probability generating function

For testing the fit of a discrete distribution, use of the probability generating function and its empirical counterpart has been suggested in Koeherlakota and Kocherlakota (1986). In the present paper, a particular functional of the corresponding empirical probability generating function process is proposed as a measure to test the discrepancy between the evidence and the hypothesis. The asymptotic behavior of the empirical probability generating function when a parameter is estimated is obtained, The study is exemplified for the Poisson case only but the procedure can be extended to other discrete distributions.     

Weighted Integral Test Statistics and Components of Smooth Tests of Fit

This paper considers families of statistics for testing the goodness-of-fit of various parametric models such as the normal, exponential or Poisson. Each family consists of weighted integrals over the squared modulus of some measure of deviation from the parametric model, expressed by means of an empirical transform of the data. Letting the rate of decay of the weight function tend to infinity, each test statistic, after a suitable rescaling, approaches a limit that is closely connected to the first non-zero component of Neyman's smooth test for the parametric model.