%0 Journal Article
%T Acceptance Sampling Plans Based on Truncated Life Tests for Gompertz Distribution
%A Wenhao Gui
%A Shangli Zhang
%J Journal of Industrial Mathematics
%D 2014
%R 10.1155/2014/391728
%X An acceptance sampling plan for Gompertz distribution under a truncated life test is developed. For different acceptance numbers, consumer¡¯s confidence levels and values of the ratio of the experimental time to the specified mean lifetime, the minimum sample sizes required to ensure the specified mean lifetime are obtained. The operating characteristic function values and the associated producer¡¯s risks are also presented. An example is provided to illustrate the acceptance sampling plan. 1. Introduction Gompertz [1] introduced the Gompertz distribution to describe human mortality and establish actuarial tables. It is a well-known lifetime model and has many applications such as biology [2], gerontology [3], and marketing science [4]. Recently, Gompertz distribution has attracted much attention. Some characteristics of this distribution are obtained by Pollard and Valkovics [5], Wu and Lee [6], Read [7], and Kunimura [8]. Sara£¿o£¿lu et al. [9] investigated the statistical inference for reliability and stress strength for Gompertz distribution. Jaheen [10] conducted a Bayesian analysis of record statistics from the Gompertz model. Lenart [11] derived exact formulas for its moment-generating function and central moments using the generalised integroexponential function. The probability density function of the Gompertz distribution is given by where is the shape parameter, is the scale parameter, and is the life time. We denote it by writing . If , the density function is increasing and then decreasing with mode . If , then the density function is decreasing with mode 0. The corresponding cumulative distribution function of is Gompertz distribution has an exponentially increasing failure rate function and it is given by . The mean of the Gompertz distribution is given by where is known as the upper incomplete gamma function. Its mean is positively proportional to the scale parameter when the shape parameter is fixed. The mean acts as quality level for the lifetime distribution under consideration. On the other hand, acceptance sampling plan is an important tool for ensuring quality in the field of statistical quality control. It is widely used when the testing is destructive and the cost of complete and thorough inspection is very high and/or it takes too long. A random sample is selected from the lot and on the basis of information yielded by the sample a decision is made regarding the accepting or rejecting the lot. Since the lifetime of a product is expected to be very high and it might be time consuming to wait until all the products fail, it is usual
%U http://www.hindawi.com/journals/jim/2014/391728/