Recursive Schemes for Calculating Common Cumulative Distributions

This article gives a method for obtaining accurate (5 decimal places) estimates of nine common cumulative distributions. Starting with a positive series expansion, we use the common ratio of each term to the preceding term and proceed as with a geometric series (the ratio may involve the term number). This avoids calculating terms in the the numerator or denominator which can be large enough to overflow or small enough to underflow the machine. The method is fast because it eliminates the necessity of calculating each term of the series in its entirety.     

Irregularities in X(Y) from Y(X) in linear calibration

Let X be an input measurement and Y the output reading of a calibrated instrument, with Y(X) as the calibration curve. Solving X(Y) projects an instrumental reading back onto the scale of measurements as an object of pivotal interest. The arrays of instrumental readings are projected in this manner in practice, yielding arrays of calibrated measurements, typically subject to errors of calibration. The effects of calibration errors on the properties of calibrated measurements are examined here under linear calibration. Irregularities arise as induced dependencies, inflated variances, non-standard distributions, inconsistent sample means, the underestimation of measurement variance, and other unintended consequences. On the other hand, conventional properties are seen to remain largely in place in the use of selected regression diagnostics and in one-way comparative experiments using calibrated data.     

Approximate MLEs of the parameters of location-scale models under type II censoring

Log-location-scale distributions are widely used parametric models that have fundamental importance in both parametric and semiparametric frameworks. The likelihood equations based on a Type II censored sample from location-scale distributions do not provide explicit solutions for the para-meters. Statistical software is widely available and is based on iterative methods (such as, Newton Raphson Algorithm, EM algorithm etc.), which require starting values near the global maximum. There are also many situations that the specialized software does not handle. This paper provides a method for determining explicit estimators for the location and scale parameters by approximating the likelihood function, where the method does not require any starting values. The performance of the proposed approximate method for the Weibull distribution and Log-Logistic distributions is compared with those based on iterative methods through the use of simulation studies for a wide range of sample size and Type II censoring schemes. Here we also examine the probability coverages of the pivotal quantities based on asymptotic normality. In addition, two examples are given.     

The Lindeberg-Feller theorem for positive definite densities

We consider the Lindeberg-Feller model for independent random variables and focus our attention on the behaviour of the probability densities q_{n} of sums S_{n}, n\geq 1 . We obtain a theorem on the convergence of q_{n} to the standard normal density \varphi which resembles the well known limit theorem for distribution functions--provided that the q_{n} are positive definite. A special case is the following: if q_{n}(0)\rightarrow\varphi(0) as n\rightarrow\infty then the Lindeberg condition guarantees that the convergence of q_{n} to \varphi continues to the real line.     

The existence of waiting time moments for single server queues with mixed renewal and poisson inputs

The following queuing system is considered: Two independent recurrent input streams (streams 1 and 2) arrive at a server. It is assumed that stream 1 is of Poisson type. Three priority disciplines are studied in case that customers of type 1 have priority: head-of-the-line, preemptive-resume, and preemptive-repeat discipline. For all three cases, the limiting distribution function of actual waiting times of low-priority customers is considered, and conditions are given for the existence of moments related to these limiting distributions.     

Heterogeneous data modeling with two-component Weibull–Poisson distribution

The mixture distribution models are more useful than pure distributions in modeling of heterogeneous data sets. The aim of this paper is to propose mixture of Weibull–Poisson (WP) distributions to model heterogeneous data sets for the first time. So, a powerful alternative mixture distribution is created for modeling of the heterogeneous data sets. In the study, many features of the proposed mixture of WP distributions are examined. Also, the expectation maximization (EM) algorithm is used to determine the maximum-likelihood estimates of the parameters, and the simulation study is conducted for evaluating the performance of the proposed EM scheme. Applications for two real heterogeneous data sets are given to show the flexibility and potentiality of the new mixture distribution.     

Reliability estimation in stress–strength models: an MCMC approach

In this paper, we consider the estimation of the stress–strength parameter R=P(Y<X) when X and Y are independent and both are modified Weibull distributions with the common two shape parameters but different scale parameters. The Markov Chain Monte Carlo sampling method is used for posterior inference of the reliability of the stress–strength model. The maximum-likelihood estimator of R and its asymptotic distribution are obtained. Based on the asymptotic distribution, the confidence interval of R can be obtained using the delta method. We also propose a bootstrap confidence interval of R. The Bayesian estimators with balanced loss function, using informative and non-informative priors, are derived. Different methods and the corresponding confidence intervals are compared using Monte Carlo simulations.     

Truncated-exponential skew-symmetric distributions

The family of skew distributions introduced by Azzalini and extended by others has received widespread attention. However, it suffers from complicated inference procedures. In this paper, a new family of skew distributions that overcomes the difficulties is introduced. This new family belongs to the exponential family. Many properties of this family are studied, inference procedures developed and simulation studies performed to assess the procedures. Some particular cases of this family, evidence of its flexibility and a real data application are presented. At least 10 advantages of the new family over Azzalini's distributions are established.     

Discriminating between the Weibull and log-normal distributions for Type-II censored data

Log-normal and Weibull distributions are the two most popular distributions for analysing lifetime data. In this paper, we consider the problem of discriminating between the two distribution functions. It is assumed that the data are coming either from log-normal or Weibull distributions and that they are Type-II censored. We use the difference of the maximized log-likelihood functions, in discriminating between the two distribution functions. We obtain the asymptotic distribution of the discrimination statistic. It is used to determine the probability of correct selection in this discrimination process. We perform some simulation studies to observe how the asymptotic results work for different sample sizes and for different censoring proportions. It is observed that the asymptotic results work quite well even for small sizes if the censoring proportions are not very low. We further suggest a modified discrimination procedure. Two real data sets are analysed for illustrative purposes.