Optimal confidence intervals for the geometric parameter

Abstract

This article discusses optimal confidence estimation for the geometric parameter and shows how different criteria can be used for evaluating confidence sets within the framework of tail functions theory. The confidence interval obtained using a particular tail function is studied and shown to outperform others, in the sense of having smaller width or expected width under a specified weight function. It is also shown that it may not be possible to find the most powerful test regarding the parameter using the Neyman-Pearson lemma. The theory is illustrated by application to a fecundability study.     

Control charts for high-quality processes: MAX or CUMAX?

For attribute data with (very) small failure rates control charts were introduced which are based on subsequent groups of r   failure times, for some r≥1$r\ge 1$. Within this family, it was shown to be attractive to stop once the maximum of such a group is sufficiently small, because this choice allows a very satisfactory nonparametric adaptation. The question we address here is whether a cumulative approach offers even further improvement. Thus instead of fixed groups, we shall use the first sequence of r consecutive sufficiently small failure times to produce a signal. A further reason for considering this type of chart is the fact that it forms the nonparametric counterpart of the well-known sets method.     

Weighted geometric distribution with new characterizations of geometric distribution

In this article, we derive a new generalized geometric distribution through a weight function, which can also be viewed as a discrete analog of weighted exponential distribution introduced by Gupta and Kundu (2009 Gupta, R. D., and D. Kundu. 2009. A new class of weighted exponential distributions. Statistics 43:621–34.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). We derive some distributional properties like moments, generating functions, hazard function, and infinite divisibility followed by different estimation methods to estimate the parameters. New characterizations of the geometric distribution are presented using the proposed generalized geometric distribution. The superiority of the proposed distribution to other competing models is demonstrated with the help of two real count datasets.     

Discrete Weibull geometric distribution and its properties

In this article, the discrete analog of Weibull geometric distribution is introduced. Discrete Weibull, discrete Rayleigh, and geometric distributions are submodels of this distribution. Some basic distributional properties, hazard function, random number generation, moments, and order statistics of this new discrete distribution are studied. Estimation of the parameters are done using maximum likelihood method. The applications of the distribution is established using two datasets.     

STRATEGY, NONTRANSITIVE DOMINANCE AND THE EXPONENTIAL DISTRIBUTION

An easily programmed recursive formula for the evaluation of the distribution function of ratios of linear combinations of independent exponential random variables is developed. This formula is shown to yield the probability that one team beats another in a contest we call the special gladiator game. This game generates tournaments which exhibit nontransitive dominance and have some surprising consequences. Similar results are obtained for a recursive formula based on the geometric distribution.     

Computing the Distributions of Economic Models via Simulation

We study a Monte Carlo algorithm for computing marginal and stationary densities of stochastic models with the Markov property, establishing global asymptotic normality and O_P(n^–1/2) convergence. Asymptotic normality is used to derive error bounds in terms of the distribution of the norm deviation.     

The optimal choice of negative binomial charts for monitoring high-quality processes

Good control charts for high quality processes are often based on the number of successes between failures. Geometric charts are simplest in this respect, but slow in recognizing moderately increased failure rates p. Improvement can be achieved by waiting until r>1 failures have occurred, i.e. by using negative binomial charts. In this paper we analyze such charts in some detail. On the basis of a fair comparison, we demonstrate how the optimal r is related to the degree of increase of p. As in practice p will usually be unknown, we also analyze the estimated version of the charts. In particular, simple corrections are derived to control the nonnegligible effects of this estimation step.     

Non‐Parametric Estimation of the Conditional Distribution of the Interjumping Times for Piecewise‐Deterministic Markov Processes

This paper presents a non‐parametric method for estimating the conditional density associated to the jump rate of a piecewise‐deterministic Markov process. In our framework, the estimation needs only one observation of the process within a long time interval. Our method relies on a generalization of Aalen's multiplicative intensity model. We prove the uniform consistency of our estimator, under some reasonable assumptions related to the primitive characteristics of the process. A simulation study illustrates the behaviour of our estimator.     

从A－A相到Berry几何相

研究核磁共振系统产生A－A相的条件,计算出核磁共振系统的A－A相,并在绝热极限下从A－A相得到Berry几何相。     

A complementary generalized linear failure rate-geometric distribution

In this article, we shall attempt to introduce a new class of lifetime distributions, which enfolds several known distributions such as the generalized linear failure rate distribution and covers both positive as well as negative skewed data. This new four-parameter distribution allows for flexible hazard rate behavior. Indeed, the hazard rate function here can be increasing, decreasing, bathtub-shaped, or upside-down bathtub-shaped. We shall first study some basic distributional properties of the new model such as the cumulative distribution function, the density of the order statistics, their moments, and Rényi entropy. Estimation of the stress-strength parameter as an important reliability property is also studied. The maximum likelihood estimation procedure for complete and censored data and Bayesian method are used for estimating the parameters involved. Finally, application of the new model to three real datasets is illustrated to show the flexibility and potential of the new model compared to rival models.