Length-biased data appear when sampling lifetimes by cross-section. This article presents a nonparametric kernel estimators of entropy function for the length-biased sample under type I censoring. We have shown that the proposed estimator is consistent and asymptotically normal under suitable regularity conditions. We have conducted simulation studies to assess the performance of the proposed estimators.

In this paper, a refined form of the differential quadrature method is proposed to compute the numerical solution of one and two-dimensional convection-diffusion Fisher’s equation. The cubic trigonometric B-spline basis functions are applied in the differential quadrature method in a modified form to obtain the weighting coefficients. The application of the method reduces nonlinear Fisher’s partial differential equation into a system of ordinary differential equations which can be solved by applying the Runge-Kutta method. Six numerical test problems of Fisher’s equation are analyzed numerically to establish the efficiency of the proposed method. The stability of the method is also discussed using the matrix method.

The problem of estimation after selection can be seen in numerous statistical applications. Let be a random sample drawn from the population where Π i follows Pareto distribution with an unknown scale parameter θi and common known shape parameter β. This article is concerned with the problem of estimating θL (or θS ), the scale parameter of the selected Pareto population under the generalized Stein loss function. The uniformly minimum risk unbiased (UMRU) estimators of θL and θS , scale parameters of the largest and the smallest population respectively, are determined. For k = 2, we have obtained a sufficient condition of minimaxity of θS and showed that the generalized Bayes estimator of θS is a minimax estimator for k = 2. Also, a class of linear admissible estimators of the form of θL and θS is found, and a sufficient condition for inadmissibility is provided. Further, we demonstrate that the UMRU estimator of θS is inadmissible. A comparison between the proposed estimators is conducted using MATLAB software and a real data set is analyzed for illustrative purposes. Finally, conclusions and discussion are reported.

The survival function method is a well-known alternative procedure for calculating the expectation of a nonnegative random variable with a continuous probability distribution. This paper proves a generalization of this method that can be used to calculate the expectation of a continuously differentiable function of a nonnegative random variable with a continuous, discrete or mixed probability distribution. A similar result is used in actuarial probability manuals, but proofs for the general case along with required assumptions are often not provided. Moreover, generalizations of the survival function method are typically not mentioned in probability textbooks. One of the main contributions of this paper is that it explores some technical conditions that guarantee that the method can be applied. This paper also provides an alternate proof of the generalized survival function method along with a different set of conditions under which the method can be proved to hold. Finally, examples of how the method can be applied to calculate expectations and moment-generating functions and to derive an integral identity are given.

In this article, we combine the adaptive progressive Type-II censoring model with the general progressive model, to obtain the estimates for the parameters of Gompertz distribution, and the Bayesian prediction intervals. Estimation is executed using the maximum likelihood method (MLE) and the Bayesian method. Bayesian estimates are constructed depending on four types of loss functions. The credible intervals and the asymptotic confidence intervals are determined for the parameters of Gompertz distribution based on the Bayesian estimates and the MLEs, respectively. Finally, a real data example and the simulation study are discussed to compare the proposed methods.

Estimation under equality restrictions is an age old problem and has been considered by several researchers in the past due to practical applications and theoretical challenges involved in it. Particularly, the problem has been extensively studied from classical as well as decision theoretic point of view when the underlying distribution is normal. In this paper, we consider the problem when the underlying distribution is non-normal, say, logistic. Specifically, estimation of the common scale parameter of two logistic populations has been considered when the location parameters are unknown. It is observed that closed forms of the maximum likelihood estimators (MLEs) for the associated parameters do not exist. Using certain numerical techniques the MLEs have been derived. The asymptotic confidence intervals have been derived numerically too, as these also depend on the MLEs. Approximate Bayes estimators are proposed using non-informative as well as conjugate priors with respect to the squared error (SE) and the LINEX loss functions. A simulation study has been conducted to evaluate the proposed estimators and compare their performances through mean squared error (MSE) and bias. Finally, two real life examples have been considered in order to show the potential applications of the proposed model and illustrate the method of estimation.

The traveling purchaser problem (TPP) is a notable generalization of the traveling salesman problem (TSP) which involves selecting a subset of markets at a minimum traveling cost such that the demand for each product is satisfied. A solid green traveling purchaser problem (SGTPP) is a TPP in which, at each market, some conveyances are available to travel to another market with minimum cost considering the environmental impact caused by carbon emission. In this paper, we formulate an SGTPP with travel cost between each pair of markets and purchase price of the products as uncertain variables. Using uncertainty theory, an expected value model is formulated and then transformed into the corresponding deterministic form. Finally, a noble genetic algorithm (nGA) is designed to solve the proposed model. The algorithm is called noble because it adopts a crossover of the combination of a probabilistic selection of three parents, according to real-life In Vitro Fertilization (IVF) techniques. Computational results reveal that our proposal is favorably compared to previous algorithms in the existing literature.

Containerization has been said to be one of the most significant innovations of port management in the last century. Some ports usually tend to be more export driven while some others are import driven. This imbalance in the number of inbound and outbound containers necessitates movement of containers from locations or surplus to those of deficit. A port needs to cater for not only the level of containers required, but also the variety. Many factors affect the cost efficiency of container movement, and hence, that of port operations, and these need to be managed in an integrated manner to optimize cost in a port. We studied the management of a multi-type container system in a port where there can be savings from joint movement of containers, both for empty return and for procurement of replacement containers, and where there is limited storage capacity. We developed a model to determine the optimal integrated return and purchase cycle times and quantities. We illustrated the solution approach with a numerical example and performed sensitivity analysis. We believe this problem is rife and the model can guide the management of container return and replenishment in ports operations.

SYNOPTIC ABSTRACTIn this paper, a new class of dependent Bernoulli random variables is defined. Here the probability of success at a given trial is a function of the number of successes and probabilities of successes in the previous trials. The moment structure for this model is derived. Further, the strong law of large numbers, the central limit theorem and the law of iterated logarithm are established under a condition that the success probabilities be monotone. Simulations are carried out to demonstrate the law of large numbers and the central limit theorem.

SYNOPTIC ABSTRACTThe current study examines a queuing system with two incongruent arrivals and two services. In this regard, two types of customers enter the system by a Poisson process and the service times are assumed to have exponential distributions. After the first service is completed, the system may provide feedback to repeat the first service, leave the system, or continue to give the second service. The same policy is utilized for the other kind of customers. The whole stochastic processes involved in the system are considered as independent random variables. A probability generating function is derived for each type and for the system that yield the performance measures. We examine the validity of the results through numerical approaches.