# Lot Acceptance and Change-Point Testing for the Process Mean and Quality Level under Dependent Batch and Panel Based Sampling Designs

Sommer, Andreas; Steland, Ansgar (Thesis advisor); Kössler, Wolfgang (Thesis advisor)

*Aachen (2017, 2018)* [Dissertation / PhD Thesis]

*Page(s): 1 Online Ressource (x, 178 Seiten) : Diagramme*

Abstract

In this thesis we deal with the problem of how to verify if an incoming lot or shipment meets certain quality criteria. Since a full inspection of the lot is inefficient, the decision should better be based on a sample of the delivered items. We propose two statistical procedures for this purpose while always having in mind an application in the area of photovoltaics, which is our main motivation. For instance, high quality photovoltaic modules should produce a certain amount of current output and regular measurements of a possible small sample should ensure this. We establish an acceptance-rejection procedure in order to decide whether to accept or reject a lot. Since the quality of the delivered items can change over time, they should be reinspected at later points in time. Thus, we develop the procedure for an arbitrary number of inspection times. This procedure is based on a possible time-dependent, nonparametric model, thus a fixed panel of the items can be measured at all inspection times. With such a panel model, the costs of measurement can be reduced significantly in many areas of application like photovoltaics, because the measuring instruments can remain on the photovoltaic modules. Moreover, we study here also batch-dependent sampling designs, where the items may be sampled in spatial batches, which is again helpful in photovoltaics where the modules are often arranged in a grid, thus the cabling can be installed efficiently. We calculate the required sample sizes as well as the critical values while controlling the producer's as well as the consumer's risk in terms of probabilities of misspecification. With the help of limiting distributions of the associated test statistics we derive explicit formulas for the sample sizes and critical values. In a first step, we restrict ourselves here to a known underlying distribution before we carry the results over to the more important case of an unknown distribution. In the latter case, suitable estimators for the variance and quantile function are proposed. We examine the finite-sample properties of the procedure in a simulation study and analyse the accuracy of the formulas as well as the stopping time of the procedure and the average outgoing quality. In a further approach, we consider a change-point model in order to test for a possible change in mean within a finite time horizon. In particular, our model allows dependence over time and batch-dependence as well as batch-individual change-points resulting in a batch-dependent panel model. We formulate the change-point problem as a hypothesis test and derive the limiting distribution of the corresponding test statistic under the null hypothesis and show consistency of the test under the alternative. Since the limiting distribution under the null hypothesis involves the usually unknown covariance structure, we also present a bootstrap procedure in order to overcome an estimation of the covariance structure. However, the bootstrap procedure requires a consistent estimation of the change point, and therefore we propose such an estimator, which is also interesting in its own. In particular, this estimator is consistent in the case of no change. We analyse the behaviour of the change-point estimator in a simulation study, where we also consider the bootstrap procedure and calculate empirical rejection rates as well as the power of the test.

### Identifier

- REPORT NUMBER: RWTH-2017-10637