Does the Type of Records Affect the Estimates of the Parameters?

The maximum likelihood estimation of the unknown parameters of inverse Rayleigh and exponential distributions are discussed based on lower and upper records. The aim is to study the effect of the type of records on the behavior of the corresponding estimators. Mean squared errors are calculated through simulation to study the behavior of the estimators. The results shall be of interest to those situations where the data can be obtained in the form of either of the two types of records and the experimenter must decide between these two for estimation of the unknown parameters of the distribution.

Queueing-Inventory with One Essential and m Optional Items with Environment Change Process Forming Correlated Renewal Process (MEP)

We consider a queueing inventory with one essential and m optional items for sale. The system evolves in environments that change randomly. There are n environments that appear in a random fashion governed by a Marked Markovian Environment change process. Customers demand the main item plus none, one, or more of the optional items, but were restricted to at most one unit of each optional item. Service time of the main item is phase type distributed and that of optional items have exponential distributions with parameters that depend on the type of the item, as well as the environment under consideration. If the essential item is not available, service will not be provided. The lead times of optional and main items have exponential distributions having parameters that depend on the type of the item. The condition for stability of the system is analyzed by considering a multi-dimensional continuous time Markov chain that represent the evolution of the system. Under this condition, various performance characteristics of the system are derived. In terms of these, a cost function is constructed and optimal control policies of the different types of commodities are investigated. Numerical results are provided to give a glimpse of the system performance.

Fair Cake Division Under Monotone Likelihood Ratios

This work develops algorithmic results for the classic cake-cutting problem in which a divisible, heterogeneous resource (modeled as a cake) needs to be partitioned among agents with distinct preferences. We focus on a standard formulation of cake cutting wherein each agent must receive a contiguous piece of the cake. Although multiple hardness results exist in this setup for finding fair/efficient cake divisions, we show that, if the value densities of the agents satisfy the monotone likelihood ratio property (MLRP), then strong algorithmic results hold for various notions of fairness and economic efficiency. Addressing cake-cutting instances with MLRP, first we develop an algorithm that finds cake divisions (with connected pieces) that are envy free, up to an arbitrary precision. The time complexity of our algorithm is polynomial in the number of agents and the bit complexity of an underlying Lipschitz constant. We obtain similar positive results for maximizing social, egalitarian, and Nash social welfare. Many distribution families bear MLRP. In particular, this property holds if all the value densities belong to any one of the following families: Gaussian (with the same variance), linear, Poisson, and exponential distributions, linear translations of any log-concave function. Hence, through MLRP, the current work obtains novel cake-cutting algorithms for multiple distribution families.

Kinetic characteristics of transcriptional bursting in a complex gene model with cyclic promoter structure

While transcription occurs often in a bursty manner, various possible regulations can lead to complex promoter patterns such as promoter cycles, giving rise to an important issue: How do promoter kinetics shape transcriptional bursting kinetics? Here we introduce and analyze a general model of the promoter cycle consisting of multi-OFF states and multi-ON states, focusing on the effects of multi-ON mechanisms on transcriptional bursting kinetics. The derived analytical results indicate that bust size follows a mixed geometric distribution rather than a single geometric distribution assumed in previous studies, and ON and OFF times obey their own mixed exponential distributions. In addition, we find that the multi-ON mechanism can lead to bimodal burst-size distribution, antagonistic timing of ON and OFF, and diverse burst frequencies, each further contributing to cell-to-cell variability in the mRNA expression level. These results not only reveal essential features of transcriptional bursting kinetics patterns shaped by multi-state mechanisms but also can be used to the inferences of transcriptional bursting kinetics and promoter structure based on experimental data.

Univariate and bivariate extensions of the generalized exponential distributions

The main aim of this paper is to introduce a new class of continuous generalized exponential distributions, both for the univariate and bivariate cases. This new class of distributions contains some newly developed distributions as special cases, such as the univariate and also bivariate geometric generalized exponential distribution and the exponential-discrete generalized exponential distribution. Several properties of the proposed univariate and bivariate distributions, and their physical interpretations, are investigated. The univariate distribution has four parameters, whereas the bivariate distribution has five parameters. We propose to use an EM algorithm to estimate the unknown parameters. According to extensive simulation studies, we see that the effectiveness of the proposed algorithm, and the performance is quite satisfactory. A bivariate data set is analyzed and it is observed that the proposed models and the EM algorithm work quite well in practice.

A generalized distribution interpolated between the exponential and power law distributions and applied to the walking data of the pill bug (Armadillidium vulgare)

To determine whether the walking pattern of an organism is a Lévy walk or a Brownian walk, it has been compared whether the frequency distribution of linear step lengths follows a power law distribution or an exponential distribution. However, there are many cases where actual data cannot be classified into either of these categories. In this paper, we propose a general distribution that includes the power law and exponential distributions as special cases. This distribution has two parameters: One represents the exponent, similar to the power law and exponential distributions, and the other is a shape parameter representing the shape of the distribution. By introducing this distribution, an intermediate distribution model can be interpolated between the power law and exponential distributions. In this study, the proposed distribution was fitted to the frequency distribution of the step length calculated from the walking data of pill bugs. The autocorrelation coefficients were also calculated from the time-series data of the step length, and the relationship between the shape parameter and time dependency was investigated. The results showed that individuals whose step-length frequency distributions were closer to the power law distribution had stronger time dependence.

An Alternate Generalized Odd Generalized Exponential Family with Applications to Premium Data

In this article, we use Lehmann alternative-II to extend the odd generalized exponential family. The uniqueness of this family lies in the fact that this transformation has resulted in a multitude of inverted distribution families with important applications in actuarial field. We can characterize the density of the new family as a linear combination of generalised exponential distributions, which is useful for studying some of the family’s properties. Among the structural characteristics of this family that are being identified are explicit expressions for numerous types of moments, the quantile function, stress-strength reliability, generating function, Rényi entropy, stochastic ordering, and order statistics. The maximum likelihood methodology is often used to compute the new family’s parameters. To confirm that our results are converging with reduced mean square error and biases, we perform a simulation analysis of one of the special model, namely OGE2-Fréchet. Furthermore, its application using two actuarial data sets is achieved, favoring its superiority over other competitive models, especially in risk theory.

Shape measures for the generalized beta exponential distribution

This paper contains parametric and informational shape measures for various families of the generalized beta exponential distribution since it is important to determination of the distribution shape for analysing an experimental data set. A logistic parameter is used to select independent types of beta exponential distributions, that it allows to combine the distributions of different subfamilies. In this paper the use of parametric shape measures to pre-define distribution shape is discusses. In particular, the initial and standard central moments for the main types of generalized beta exponential distribution are given. In the paper it is proposes to use the entropy coefficient of unshifted distribution as an independent information measure of the shape of unshifted generalized beta exponential distributions. In order to increase the reliability of the preliminary determination of the shape of the model, expressions for the entropy coefficient of shifted families both the generalized beta exponential distributions of the first and second types, and the generalized gamma exponential distribution were obtained. For practical applied the entropy coefficients of unshifted distributions for various subfamilies of generalized beta exponential distributions can be useful.

Measures of shape for the generalized beta exponential distribution

This paper contains parametric and informational measures of shape for various families of the generalized beta exponential distribution since it is important to determination of the distribution shape for analysing an experimental data set. A logistic parameter is used to select independent types of beta exponential distributions, that it allows to combine the distributions of different subfamilies. In this paper the use of parametric shape measures to predefine distribution shape is discusses. In particular, the initial and standard central moments for the main types of generalized beta exponential distribution are given. In the paper it is proposes to use the entropy coefficient of unshifted distribution as an independent information measure of the shape of unshifted generalized beta exponential distributions. In order to increase the reliability of the preliminary determination of the shape of the model, expressions for the entropy coefficient of shifted families both the generalized beta exponential distributions of the first and second types, and the generalized gamma exponential distribution were obtained. For practical applied the entropy coefficients of unshifted distributions for various subfamilies of generalized beta exponential distributions can be useful.