Statistical models parameters are estimated in an effort to have knowledge about unknown quantities. In many situations, however, statisticians provide the estimation of the parameters by using not only information based on the sample, but other information as well. This information may be regarded as nonsample information(NSI) or uncertain prior information (UPI) about the parameter of interest. It is advantageous to utilize the NSI in the estimation procedure, especially when the information based on the sample may be rather limited or even the data quality is questionable. But in some experimental cases, it is not certain whether or not this information holds. Consider the data arising from tumor measurements in mice, for example, at various times following injection of carcinogens. Such data should be thought as coming from an in vivo experiment. Biologists are interested in estimating growth rate parameter θ when it suspected a priori that θ = θ 0. Such θ 0 can be obtained directly from in vitro experiments in which cell behavior may or may not be different because of different environmental conditions. Thus, biologists may have reason to suspect that θ 0 is the true value of the growth parameter for the in vivo experiment, but are not sure.