## 1 Introduction

Problems of finding confidence intervals for functions of Poisson means arise naturally in a variety of contexts. Here, are five such problems:

Problem 1. Multiple comparisons procedures for Poisson data with application to comparing defects at an electronics shop over different days (Scheaffer 1980), and to investigating the impact on cancer development of several treatments for Hodgkin's disease (Suissa & Salmi 1989).

Problem 2. In Azerbaijan, about 300 structures that could be oil fields are known onshore, and 66 structures are already recognized in the offshore region. Bagirov & Lerche (1998) wanted to know the fraction of these structures that can be expected to yield horizons with commercial value. To find an answer to this problem, they conducted a statistical analysis of data covering the last 100 years of oil production in Azerbaijan. The number of producing horizons per field was described by a linear combination of Poisson random variables.

Problem 3. A demerit rating system is used to simultaneously monitor counts of several different types of defects in a complex product. The demerit statistic is a linear combination of the counts of these different types of defects. The traditional recommendation is to plot the demerit statistic on a control chart with symmetric 3-sigma control limits. Jones, Woodall & Conerly (1999) proposed an alternative method for determining control limits for the demerit control chart, based on the exact distribution of linear combinations of independent Poisson random variables.

Problem 4. A standard method of estimation with applications to chemistry, the study of geothermal bores and other areas involves adding a radioactive isotope and measuring the number of counts before and after addition. We observe independent Poisson counts in seconds with means , where is the th decay rate, is background noise and is background plus signal. A confidence interval is required for , the signal decay rate.

Problem 5. An increase is observed in the per capita rate of first admissions to psychiatric care. Is the increase significant? Let be the (known) population at time . Let be the number first admitted between times and . Assume for small

where is the (unknown) rate at time . Assume that numbers first admitted in different periods are independent. Then is a Poisson process with mean . The problem of significance can be answered if we have a confidence interval for , where , and are the periods being compared and is an appropriate weight function satisfying . Set , so that . If, in fact, is only available at times (for example, annually) for some then is only estimable if can be expressed as the union of one or more intervals ) and is chosen to be constant over each such interval.

The first four examples and the constrained form of the fifth example are special cases of finding a confidence interval for

where is the number of Poisson variables or Poisson means involved (assumed known), are the weights which are also assumed known and are the unknown Poisson means. We assume that we have observations on independent Poisson variables with means , . The parameters of (1) are: .

These examples are special cases of the following problem: given a known weight function and an observed Poisson process with unknown mean find a confidence interval for

The problem of finding a confidence interval for

where are observed independent Poisson processes with unknown mean functions and known weight functions , is reducible to (2) since we may combine into a single process.

To the best of our knowledge, there are only two papers, Stamey & Hamilton (2006) and Krishnamoorthy & Lee (2010), giving parametric confidence intervals for (1937) and there are none giving parametric confidence intervals for (1999) and (2010). The confidence intervals in Stamey & Hamilton (2006) are variations of approximations based on the Central Limit Theorem (CLT). The confidence intervals in Krishnamoorthy & Lee (2010) are based on normal and chi-square approximations. These intervals may not be as accurate as those proposed here because they are based on the CLT whereas we use tools based on higher order approximations. We have empirical evidence that our intervals provide improved accuracy.

The aim of this paper is to provide accurate parametric confidence intervals for (1937), (1999) and (2010). Section 'Confidence intervals' contains the derivation of these confidence intervals. Section 'Numerical comparisons' assesses the performance of the derived intervals in terms of their widths and coverage probabilities. A part of this assessment is based on simulation. Section 'Data example' demonstrates the importance of the derived intervals using a real data set. Some conclusions are noted in Section 'Conclusions'. Some technical details required for the results in Section 'Confidence intervals' are provided in Appendix I.