### Generalized Pivot Quantity Methods

Organizers: Chih-Ming Wang, NIST and Hari Iyer, Colorado StateSession Chair: Chih-Ming Wang, NIST

**Inference for the Lognormal Mean and Quantiles Based on Samples with Non-Detects**

Thomas Mathew

University of Maryland Baltimore County

In the talk, the problems of finding confidence limits for the mean and quantiles of a lognormal distribution are addressed based on a sample that includes values below the detection limit, i.e., a type I singly left censored sample. Generalized inferential procedures which utilize approximate pivotal quantities based on the maximum likelihood estimators, and some likelihood based methods, are proposed. The latter includes methodology based on the signed log-likelihood ratio test (SLRT) statistic and the modified signed log-likelihood ratio test (MSLRT) statistic. The merits of the methods are evaluated using Monte Carlo simulation. For inference concerning the lognormal mean, the SLRT is to be preferred for left-tailed testing, the generalized inference procedure is to be preferred for right-tailed testing, and all three approaches provide nearly the same performance for two-tailed testing. These conclusions hold even when the proportion of sample values below the detection limit is as large as 0.70. For inference concerning quantiles, both the generalized inference approach and the MSLRT provide nearly identical performance, and both are satisfactory. In view of its simplicity as well as ease of understanding and implementation, the recommendation is to use the generalized inference procedure for practical applications that require inference concerning the quantiles. The results are illustrated with an example.

The material presented in this talk is based on joint work with K. Krishnamoorthy and A. Mallick.

**On Fiducial Inference in Linear Mixed Models**

Jan Hannig

University of North Carolina Chapel Hill

R. A. Fisher's fiducial inference has been the subject of many discussions and controversies ever since he introduced the idea during the 1930's. The idea experienced a bumpy ride, to say the least, during its early years and one can safely say that it eventually fell into disfavor among mainstream statisticians. However, it appears to have made a resurgence recently under the label of generalized inference. In this new guise fiducial inference has proved to be a useful tool for deriving statistical procedures for problems where frequentist methods with good properties were previously unavailable. Therefore we believe that the fiducial argument of R.A. Fisher deserves a fresh look from a new angle.

In this talk we first generalize Fisher's fiducial argument and obtain a fiducial recipe applicable in virtually any situation. We demonstrate this fiducial recipe on examples of varying complexity. We also investigate, by simulation and by theoretical considerations, some properties of the statistical procedures derived by the fiducial recipe showing they often posses good repeated sampling, frequentist properties. As an example we will discuss generalized fiducial inference for linear mixed models.

Portions of this talk are based on joint work with Hari Iyer, Thomas C.M. Lee, E. Lidong, and Jessi Cisewski.

**Fiducial Prediction Intervals**

Chih-Ming (Jack) Wang

NIST

Prediction intervals, used in many practical applications, are statistical intervals that contain, with a specific probability, future realizations of a random variable. In this talk, we present a general procedure for constructing prediction intervals for arbitrary distributions. The procedure we propose is based on fiducial inference. We illustrate the procedure with several examples.

This is joint work with Hari K. Iyer and Jan Hannig.