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Let F(νj)={Qmjνj,mj∈(m−νj,m+νj)}, j=1,2, be two Cauchy–Stieltjes Kernel (CSK) families induced by non-degenerate compactly supported probability measures ν1 and ν2. Introduce the set of measures F=F(ν1)⊞F(ν2)={Qm1ν1⊞Qm2ν2,m1∈(m−ν1,m+ν1)andm2∈(m−ν2,m+ν2)}. We show that if F remains a CSK family, (i.e., F=F(μ) where μ is a non-degenerate compactly supported measure), then the measures μ, ν1 and ν2 are of the Marchenko–Pastur type measure up to affinity. A similar conclusion is obtained if we substitute (in the definition of F) the additive free convolution ⊞ by the additive Boolean convolution ⊎. The cases where the additive free convolution ⊞ is replaced (in the definition of F) by the multiplicative free convolution ⊠ or the multiplicative Boolean convolution ⨃ are also studied.

Several authors are interested in the study of limiting distributions for large symmetrical random matrices. Our approach of studying limiting distributions is different and is related to the new concept of variance functions of Cauchy–Stieltjes Kernel (CSK) families of probabilities. By means of the variance functions machinery, some novel limiting probability measures are provided involving the free multiplicative law of large numbers, using both free and Boolean (additive and multiplicative) convolutions. Several examples of these limiting probability measures are given in the context of free probability.

Let K+(μi)={Qsiμi,si∈(m0μi,m+μi)}, i=1,2, be two CSK families generated by the nondegenerate probability measures μ1 and μ2 with support bounded from above. Define the set of measures L=K+(μ1)•K+(μ2)={Qs1μ1•Qs2μ2,s1∈(m0μ1,m+μ1)ands2∈(m0μ2,m+μ2)}, where Qs1μ1•Qs2μ2 denotes the Fermi convolution of Qs1μ1 and Qs2μ2. We prove that if L is still a CSK family (that is, L=K+(σ) for some nondegenerate probability measure ()σ), then the probability measures σ, μ1 and μ2 are of the free Poisson type and follow the free Poisson law up to affinity. The same result, regarding the free Poisson measure, is obtained if we consider the t-deformed free convolution t replacing the Fermi convolution • in the family of measures L.

In free probability, the theory of Cauchy–Stieltjes Kernel (CSK) families has recently been introduced. This theory is about a set of probability measures defined using the Cauchy kernel similarly to natural exponential families in classical probability that are defined by means of the exponential kernel. Within the context of CSK families, this article presents certain features of the Marchenko–Pastur law based on the Fermi convolution and the t-deformed free convolution. The Marchenko–Pastur law holds significant theoretical and practical implications in various fields, particularly in the analysis of random matrices and their applications in statistics, signal processing, and machine learning. In the specific context of CSK families, our study of the Marchenko–Pastur law is summarized as follows: Let K+(μ)={Qmμ(dx);m∈(m0μ,m+μ)} be the CSK family generated by a non-degenerate probability measure μ with support bounded from above. Denote by Qmμ•s the Fermi convolution power of order s>0 of the measure Qmμ. We prove that if Qmμ•s∈K+(μ), then μ is of the Marchenko–Pastur type law. The same result is obtained if we replace the Fermi convolution • with the t-deformed free convolution t.

In this article, we make mathematical and practical contributions to the Bell-X family of absolutely continuous distributions. As a main member of this family, a special distribution extending the modeling perspectives of the famous Burr XII (BXII) distribution is discussed in detail. It is called the Bell-Burr XII (BBXII) distribution. It stands apart from the other extended BXII distributions because of its flexibility in terms of functional shapes. On the theoretical side, a linear representation of the probability density function and the ordinary and incomplete moments are among the key properties studied in depth. Some commonly used entropy measures, namely Rényi, Havrda and Charvat, Arimoto, and Tsallis entropy, are derived. On the practical (inferential) side, the associated parameters are estimated using seven different frequentist estimation methods, namely the methods of maximum likelihood estimation, percentile estimation, least squares estimation, weighted least squares estimation, Cramér von-Mises estimation, Anderson-Darling estimation, and right-tail Anderson-Darling estimation. A simulation study utilizing all these methods is offered to highlight their effectiveness. Subsequently, the BBXII model is successfully used in comparisons with other comparable models to analyze data on patients with acute bone cancer and arthritis pain. A group acceptance sampling plan for truncated life tests is also proposed when an item's lifetime follows a BBXII distribution. Convincing results are obtained.

In this paper, we propose a novel family of distributions called the odd Muth-G distributions by using Transformed-Transformer methodology and study their essential properties. The distinctive feature of the proposed family is that it can provide numerous special models with significant applications in reliability analysis. The density of the new model is expressible in terms of linear combinations of generalized exponentials, a useful feature to extract most properties of the proposed family. Some of the structural properties are derived in the form of explicit expressions such as quantile function, moments, probability weighted moments and entropy. The model parameters are estimated following the method of maximum likelihood principle. Weibull is selected as a baseline to propose an odd Muth-Weibull distribution with some useful properties. In order to confirm that our results converge with minimized mean squared error and biases, a simulation study has been performed. Additionally, a plan acceptance sampling design is proposed in which the lifetime of an item follows an odd Muth-Weibull model by taking median lifetime as a quality parameter. Two real-life data applications are presented to establish practical usefulness of the proposed model with conclusive evidence that the model has enough flexibility to fit a wide panel of lifetime data sets.

In this article, we provide some new limiting laws related to the free multiplicative law of large numbers and involving free and Boolean additive convolutions. Some examples of these limiting laws are presented within the framework of non-commutative probability theory.

A notion of generalized (two-parameterized) t-transformation of free convolution, also called (t=(a,b))-deformed free convolution, is introduced for a∈R and b>0. In this article, some results of t-deformed free convolution are given within the theory of Cauchy-Stieltjes Kernel (CSK) families. The variance function is a fundamental concept in CSK families. An expression is provided for the variance function under t-deformed free convolution power. In addition, through the use of the variance function, an approximation is provided for members of the t-deformed free Gaussian CSK family and members of the t-deformed free Poisson CSK family respectively. Furthermore, by involving the free multiplicative convolution, a new limit theorem is provided with respect to t-deformed free convolution.

In this paper, a new modified Kumaraswamy distribution is proposed, and some of its basic properties are presented, such as the mathematical expressions for the moments, probability weighted moments, order statistics, quantile function, reliability, and entropy measures. The parameter estimation is done via the maximum likelihood estimation method. In order to show the usefulness of the proposed model, some well-established actuarial measures such as value-at-risk, expected-shortfall, tail-value-at-risk, tail-variance, and tail-variance-premium are obtained. A simulation study is carried out to assess the performance of maximum likelihood estimates. The empirical analysis is carried out to show that our proposed model is better in performance as compared to other competitive models related to the extended Kumaraswamy model. Thus, insurance claim data and engineering related real-life data sets are considered to prove this claim.

One of the most used techniques for determining animal abundance is distance sampling. The distance sampling framework depends on the idea of a detection function, and a number of options have been suggested. In this paper, we provide a new flexible parametric model based on the Burr XII distribution. To be more specific, we use the survival function of the Burr XII distribution for novel purposes in this context. The proposed model is appealing because it meets all of the requirements for a reliable detection function model, such as being monotonically decreasing and having a shoulder at the origin. It also has the features of having various asymmetric properties and a heavy right tail, which are rare properties in this setting. In the first part, we provide its key characteristics, such as shapes and moments. Then, the inferential aspect of the model is investigated. The maximum likelihood estimation method is used to estimate the parameters in a data-fitting scenario. The estimates of population abundance are derived and compared with some existing parametric estimates. A simulation is run to assess how well the resulting estimates perform in comparison to other widely applied estimates from the literature. The model is then tested using two real-world data sets. Based on the famous goodness-of-fit statistics, we show that it is preferable to some of the well-established models.

This paper analyzes the time to event data in the presence of collinearity. To address collinearity, the ridge regression estimator was applied in multiple and logistic regression as an alternative to the maximum likelihood estimator (MLE), among others. It has a smaller mean square error (MSE) and is therefore more precise. This paper generalizes the approach to address collinearity in the frailty model, which is a random effect model for the time variable. A simulation study is conducted to evaluate its performance. Furthermore, the proposed method is applied on real life data taken from the largest sample survey of India, i.e., national family health survey (2005–2006 ) data to evaluate the association of different determinants on infant mortality in India.

In this article, we make mathematical and practical contributions to the Bell-X family of absolutely continuous distributions. As a main member of this family, a special distribution extending the modeling perspectives of the famous Burr XII (BXII) distribution is discussed in detail. It is called the Bell-Burr XII (BBXII) distribution. It stands apart from the other extended BXII distributions because of its flexibility in terms of functional shapes. On the theoretical side, a linear representation of the probability density function and the ordinary and incomplete moments are among the key properties studied in depth. Some commonly used entropy measures, namely R'enyi, Havrda and Charvat, Arimoto, and Tsallis entropy, are derived. On the practical (inferential) side, the associated parameters are estimated using seven different frequentist estimation methods, namely the methods of maximum likelihood estimation, percentile estimation, least squares estimation, weighted least squares estimation, Cram'er von-Mises estimation, Anderson-Darling estimation, and right-tail Anderson-Darling estimation. A simulation study utilizing all these methods is offered to highlight their effectiveness. Subsequently, the BBXII model is successfully used in comparisons with other comparable models to analyze data on patients with acute bone cancer and arthritis pain. A group acceptance sampling plan for truncated life tests is also proposed when an item's lifetime follows a BBXII distribution. Convincing results are obtained. [ABSTRACT FROM AUTHOR]