A Two-Parameter Quasi Lindley Distribution in Acceptance Sampling Plans from Truncated Life Tests

In this paper, acceptance sampling plans are developed when the life test is truncated at a pre-assigned time. For different acceptance numbers, confidence levels and values of the ratio of the fixed experiment time to the specified average life time, the minimum sample sizes required to ensure the specified average life are calculate assuming that the life time variate of the test units follows a two-parameter Quasi Lindley distribution ( QLD (2)). The operating characteristic function values of the new sampling plans and the corresponding producer's risk are presented.


Introduction
During the Second World War the acceptance sampling plan is used in the US Military for testing the bullets. Lifetime is a substantial quality variable of a product. The acceptance sampling plans are used to locate the acceptability of a product unit, where the consumer can accept or reject the lot based on a random sample selected from the lot. However, the problem is to find the minimum sample size needful to assert a certain average life when the life test is terminated at a pre-assigned time, t, and when the observed number of failures overtake a given acceptance number, c. Therefore, the decision is to accept a lot if the given life can be established with a pre-determined high probability * , P which support the consumer. To solve and compute the acceptance sampling parameters it is assumed the life time follow a specific model or distribution. Numerous parametric distributions are used in the analysis of lifetime data (Kantam et  In this article we suggest of using a two parameters quasi Lindley distribution QLD(2) (Shanker and Mishra, 2013)  which is positively skewed distribution and the corresponding cumulative distribution function is given by The qth moment about origin of the QLD(2) is defined as It is of interest to note here that when   = , the moments about origin of the QLD(2) will be the respective moments of the QLD(1) (Lindley, 1958;Ghitany et al., 2008). The mode of the QLD (2) The method of moment estimate of  is 2 1 1 X For more details about the QLD(2) see Shanker and Mishra (2013). The rest of the paper is organized as follows. In Section 2, the suggested sampling plans, along with the operating characteristic function are presented. The results are explained and discussed by an example in Section 3. Our conclusions are summarized in Section 4. We assume that the lifetime t of the product follows a QLD (2). The single sampling plan is consists of the following: (1) the number of units n, on test; (2) an acceptance number c, where if c or less failures happen during the test time, the lot is accepted; (3)the maximum test duration time, t; (4) a ratio The producer's risk is the probability of rejecting the lot when 0    (is fixed not to exceed * 1 P − , i.e., the one for which the true average life is below the specified life 0  ) and the producer's risk is known as the probability of rejecting a good lot.
Consider that the lot size is sufficiently large to obtainthe probability of accepting a lot using the binomial distribution. Here, the problem is to determine the smallest sample size n required to satisfy the inequality up to c for given values of * P is the probability of a failure observed during the time t which depends only on the ratio / o t  ; where the population of the QLD(2) distribution is given by: In Table 1,  If the number of observed failures before the time t is less than or equal to the acceptance number c, then based on (12) we may have The probability ( Table 2 for 1  = .
The producer's risk is defined as the probability of rejecting the lot when  Table 3 when 1  = .
For example, for 1  = assume that the experimenter wants to establish the true unknown average life to be at least 1000 hours with confidence * 0.95 P = and it is desired to stop the experiment at 1571 t = hours when the acceptance number 2 = c . Then the required sample size n from Table 1 is 6 n = . Now, the 6 units have to be put on test. If within 1000 hours no more than 2 failures out of 6 units are noticed, then the experimenter can assert with confidence of 0.95 that the average life is at least 1000 hours.

Conclusion
In this paper, acceptance sampling plans based on truncated lifetime tests are developed when the lifetime tests follow a two parameter Quasi Lindley distribution. The minimum sample size needed to assert a certain mean life of the test units, the OC values of the sampling plan, along with the minimum ratio to the specified mean life for accepting a lot with assured producer's risk are calculated for the Quasi Lindley distribution QLD(2) when 1  = .The results obtained in this paper encourage the practitioners to use the suggested sampling plans. However, the suggested sampling plan discussed in this paper can be extended by considering different types of the sampling plans such as the group acceptance sampling plan (Rao, 2009). Other extension can be made by considering repetitive sampling plan (Sherman, 1965).