Discrete Weibull Research Articles

Economic optimization of off-line inspection with rework consideration

This paper addresses issues relating to off-line inspection with rework consideration. We develop a mathematical model to determine the first unit inspected and the average number to be inspected in a batch, assuming the involvement of a quality-control department with a standard procedure. Numerical analyses are used to simulate the effect of changes in the various parameters on the optimal solution when the probability of a transition between states follows the discrete Weibull distribution.

Discrete Program-Size Dependent Software Reliability Assessment: Modeling, Estimation, and Goodness-of-Fit Comparisons

In this paper we propose a discrete program-size dependent software reliability growth model flexibly describing the software failure-occurrence phenomenon based on a discrete Weibull distribution. We also conduct model comparisons of our discrete SRGM with existing discrete SRGMs by using actual data sets. The program size is one of the important metrics of software complexity. It is known that flexible discrete software reliability growth modeling is difficult due to the mathematical manipulation under a conventional modeling-framework in which the time-dependent behavior of the cumulative number of detected faults is formulated by a difference equation. Our discrete SRGM is developed under an existing unified modeling-framework based on the concept of general order-statistics, and can incorporate the effect of the program size into software reliability assessment. Further, we discuss the method of parameter estimation, and derive software reliability assessment measures of our discrete SRGM. Finally, we show numerical examples of discrete software reliability analysis based on our discrete SRGM by using actual data.

Modeling Tree Recruitment with Zero-Inflated Models: The Example of Hardwood Stands in Southern Québec, Canada

Abstract In recruitment modeling, the response variable is a count and its distribution is often characterized by an excess number of zeros. As a result, standard distributions of probabilities, such as Poisson, are inappropriate. A common approach in forestry consists of using two-part conditional models. These models have two distinct components aimed to predicting the occurrence and abundance of recruitment, respectively. For such data, zero-inflated models might provide a more adequate framework by combining the two components into a joint distribution of probabilities. In this article, a conditional model is compared with two different zero-inflated models, namely, a zero-inflated Poisson (ZIP) and a zero-inflated discrete Weibull (ZIdiW) model. The three models were calibrated using a data set provided by permanent sample plots located in hardwood stands. Parsimony criteria (the Akaike information criterion and the Bayesian information criterion) and diagnostic plots were used to perform the comparison. The results show that the ZIdiW model has the best fit. The flexibility of the Weibull function and the possibility of obtaining a more parsimonious model are two advantages related to the use of a ZIdiW model in recruitment modeling.

Improved duration-based backtesting of value-at-risk

Backtesting value-at-risk forecasts is an important issue. A duration-based approach has recently been proposed by Christoffersen and Pelletier (2004). Their method is very appealing because it allows one to test for both correct conditional and unconditional coverage against quite general alternatives, thus generalizing earlier approaches. Despite the discrete nature of the problem, the authors used the continuous Weibull distribution in their implementation of the method. In this paper, we employ the discrete counterpart of this model instead. We argue that the discrete approach has two advantages. First, the parameters involved have a clear-cut interpretation in risk management terms. Moreover, and more importantly, simulations indicate that the discrete model has superior power properties to the continuous candidate. As the discrete Weibull distribution is not well known in the risk management literature, its properties relevant to backtesting are also briefly discussed.

Optimal lot sizing for products sold under free-repair warranty

Repairable items sold under a free-repair warranty (FRW) policy are examined. In the literature, the lot-size problem is treated as a quality-control problem and not as an inventory problem for a continuous-type deteriorating production system, in which there is a shift from an in-control state to an out-of-control state and it is assumed that the system has a geometric survival distribution. Consequently, smaller lots result in a reduction in warranty cost per item, but such a reduction is achieved at the expense of an increased manufacturing cost per item. In order to control such a production system economically under an FRW, tradeoffs between manufacturing cost and warranty cost must be analyzed. In this paper, analytical results are extended to a discrete general shift distribution, to provide an optimal lot size so that the long run total cost of the setup, inventory holding, and warranty is minimized. Different conditions for optimality, properties and bounds on the optimal lot size are provided. A numerical example is given to see the adequacy of using the geometric distribution when the actual distribution is discrete Weibull.

Discretization of Continuous Distributions with an Application to Stress-Strength Reliability

A generic approach has been studied for discretization of continuous distributions in the univariate and bivariate situations. In particular, discretization of uniform distribution has been used to develop a bivariate binary model. Characterization results have also been studied to establish a direct link between the discrete Weibull distribution and its continuous counterpart. Attempts have also been made for approximating reliability of complex systems using suitably adjusted discrete concentrations of continuous design variables.

The effects of the warranty cost on the imperfect EMQ model with general discrete shift distribution

In this paper, we investigate the effect of the warranty cost on optimization of the economic manufacturing quality (EMQ). This is done for a deteriorating process where the production process shifts from the in-control state to the out-of-control state following a general discrete probability distribution. Once the production process goes out of control, the production process produces some defective items. The defective item cost includes reworking and warranty costs. Thus, in order to economically operate a production-inventory system with products sold under warranty, the tradeoffs among the production setup, inventory, and defective item cost, including the reworked cost before sale and the warranty cost after sale, needed to be analysed. This objective in this paper is to determine the production lot size while minimizing the total cost per unit of time per unit of time. Various special cases are presented. Two of them are extensions of results obtained previously in the literature. Finally, a numerical example is given which uses a discrete Weibull probability distribution. Sensitivity analysis of the model with respect to cost and time parameters is also performed.

The Rising Hazards of Party Incumbency: A Discrete Renewal Analysis

Recent interest in voters' anti-incumbent sentiments focuses on generational change as well as public weariness with partisan control of a long duration. Theories on the electoral effects of such behaviors predict partisan cycles that suggest rising hazards of party incumbency. This article provides an analytical framework for examining changes and durations of party control in presidential elections as a discrete point process. We introduce the discrete Weibull distribution for testing contagion in the context of renewal theory and develop the notion of pseudo-periodicity for a binary process. Our findings based on this event history approach confirm the claim that party incumbency engenders rising hazards. The partisan cycles that we identified have a pseudo-period of approximately six to eight presidential terms.

Curtailment planning of catalogue delivery in direct mail: A theoretical study

Catalogues are an essential medium for direct mail businesses to inform customers of the merchandise that they offer. Considerable expense is therefore devoted to designing, publishing and delivering catalogues. Some customers do not respond to the catalogues they receive: they make no purchases even when they receive a series of issues. Further catalogue delivery to such customers should be curtailed, as the cost per catalogue is high. This study discusses conditions whereby catalogue delivery is curtailed if the customer has not responded to the last K successive issues of a catalogue. The value of K is determined by introducing the expected catalogue profit per customer. Two models are proposed: one for customers who have responded only a few times, and one for customers who have responded many times. The conditions for an optimal curtailment time are discussed for the two model types. The optimal curtailment strategy for catalogue delivery is proposed on the basis of these models. A case with a discrete Weibull distribution is also presented with numerical examples.

Approximate MLE's of the parameters of a discrete Weibull distribution with type I censored data

Approximate Maximum Likelihood Estimators of the parameters of a discrete Weibull distribution under censoring is discussed. These approximations are obtained by approximating the likelihood functions at a set of estimators which are easy to obtain. Simulations are carried cut to illustrate the performance of the suggested procedure.

On estimating parameters in a discrete Weibull distribution

Two discrete Weibull distributions are discussed, and a simple method is presented to estimate the parameters for one of them. Simulation results are given to compare this method with the method of moments. The estimates obtained by the two methods appear to have almost similar properties. The discrete Weibull data arise in reliability problems when the observed variable is discrete. The modeling of such a random phenomenon has already been accomplished. Estimation of parameters in these models is considered. Since the usual methods of estimation are not easy to apply, a simple method is suggested to estimate the unknown parameters. The estimates obtained by this method are comparable to those obtained by the method of moments. The method can be applied in most inferential problems. Though the authors have restricted themselves to type I distribution, their method of proportions for the estimation of parameters can be easily applied to the type II distribution as well.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A new discrete Weibull distribution : William E. Stein and Ronald Dattero. IEEE Trans. Reliab.R-33 (2) 196 (June 1984)

A new discrete Weibull distribution : William E. Stein and Ronald Dattero. IEEE Trans. Reliab.R-33 (2) 196 (June 1984)

A New Discrete Weibull Distribution

This paper introduces another discrete Weibull distribution. A connection is shown to the famous Birthday Problem and to the lifetime of a series system of components.

The discrete Weibull distribution : Toshio Nakagawa and Shunji Osaki. IEEE Trans. Reliab.R-24, (5) 300 (Dec. 1975)

The discrete Weibull distribution : Toshio Nakagawa and Shunji Osaki. IEEE Trans. Reliab.R-24, (5) 300 (Dec. 1975)

The Discrete Weibull Distribution

The discrete Weibull distribution is defined to correspond with the Weibull distribution in continuous time. A few properties of the discrete Weibull distribution are discussed.