This article demonstrates how Bayesian methods for making inferences about the number, or proportion, of nonconforming units in a quality control setting can be implemented using simulation techniques. Random draws from the posterior distribution of the target quantities of interest are used to construct the needed inferences. The same random draws from the posterior distribution are also used to perform model checking and sensitivity analysis. The methods are illustrated using two examples. The methods can be generalized to more complicated situations which are illustrated by discussing the extensions of both the examples.

Keywords: acceptance sampling, Bayesian methods, posterior distribution, quality control, sensitivity analysis, variable sampling

*by* **T. E. Raghunathan, University of Michigan, Ann
Arbor, MI 48106-1248**

*INTRODUCTION*

A Bayesian data analysis essentially
involves: (1) setting up a probability model that posits a
distribution for observables (measurements or attributes)
conditional on unobservables (parameters) where functions
of some or all of the unobservables and observables (target
quantities) is the object of inference, (2) a prior distribution
that summarizes a priori uncertainty about the likely values
of the parameters, and (3) computing and interpreting the
posterior distribution of the target quantities of interest.
Specifically, suppose that (*x*
| )
is the density function of the measurements, *x*, conditional
on the parameter, ,
and ()
denotes the prior density function of ,
then the posterior density function of
is given by

The above equation is the essential ingredient for making
inferences about the target quantity of interest, (),
a known function of .
For instance, suppose that the objective is to compute the
probability that a scalar function, (),
belongs to a subset *A* in light of the information provided
by the measurements *x*. Here one would simply compute
the integral

In many practical situations,
computation of the above integral can be diffcult. However,
if we had draws from the posterior distribution of ,
then the target quantities (or some functions of them) of
interest can be easily approximated. For instance, if we could
draw a random sample, ,
*i* = 1, 2, . . . , *M* from the posterior distribution
of
given in Equation (1), then the above integral can be approximated
as

where *I*_{[C]} is an indicator function
which takes the value 1 if the condition, *C*, is satisfied
and zero otherwise. These same draws can also be used to draw
inferences about some other function of .
The recent increase in the availability of computational resources
and the development of computational techniques have led to
great advances in the application of Bayesian methods to complicated
problems in various disciplines (Gilks, Richardson, and Spiegelhalter
(1996)).

In solving practical problems,
the choice of a model for *x* can be made based on the
combination of subject matter information and empirical information
provided by the measurements themselves. However, the prior
distribution for
needs to be formulated based on the prior knowledge. This
is usually a diffcult task because such prior knowledge may
not be readily available. In such situations, usually a "noninformative"
prior distribution is used. The basic idea behind formulating
such a prior distribution is that it should be flat so that
the likelihood (the density, (*x*
| ),
evaluated at the observed value of *x*) plays a dominant
role in the construction of the posterior density given in
Equation (1). Jeffreys (1961) formulated such prior distributions
based on certain invariance arguments; often these are not
proper density functions as they are not integrable functions
of .
Although this does not create problems in many situations,
caution must be used in using these priors (DuMouchel and
Waternaux (1992)). When data from prior studies are available
then one can formulate a proper prior distribution. Examples
of these types of situations in a quality control setting
are considered, for example, in Hahn and Raghunathan (1988).
Nevertheless, it is possible that both (*x*
| )
and ()
could be misspecified. Hence, an important component of any
Bayesian analysis is model checking and exploring the extent
to which the inferences about ()
are sensitive to the modest departure from the assumed distributions
(*x*
| )
and ().
The use of simulation techniques also facilitates performing
model checking and the sensitivity analyses. We will demonstrate
how one can perform model checking and sensitivity analyses
while using the simulation methods to draw inferences under
a specific model.

To illustrate the basic ideas, we use two common problems in quality control. The first example, discussed in Hahn and Meeker (1990), involves estimating the proportion of conforming units in a manufacturing process when the specification involves two variables. We show how a Bayesian method implemented through simulations leads to non-asymptotic point and interval estimates for the proportion of nonconforming units, assuming a bivariate normal distribution for these two variables. We also explore sensitivity to model assumptions by drawing inferences under robust models. The details of the analysis are given in the second section.

The second example, discussed in the third section, is concerned with drawing inferences about the number of nonconforming units in the accepted lots after acceptance sampling. Specifically, we obtain point and interval estimates of the mean number of nonconforming units among the units that have not been inspected in the accepted lots. We use a hierarchical model for the data given in Hahn (1986). We also explore sensitivity to the assumed model for the random effects in the hierarchical model. Finally, the fourth section concludes the paper with a discussion and possible extensions for both problems.

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