تجارت کردن بین حجم نمونه و دقت: مورد اندازه گیری های تحت عدم قطعیت فاصله

In many practical situations, we are not satisfied with the accuracy of the existing measurements. There are two possible ways to improve the measurement accuracy: • First, instead of a single measurement, we can make repeated measurements; the additional information coming from these additional measurements can improve the accuracy of the result of this series of measurements. • Second, we can replace the current measuring instrument with a more accurate one; correspondingly, we can use a more accurate (and more expensive) measurement procedure provided by a measuring lab – e.g., a procedure that includes the use of a higher quality reagent. In general, we can combine these two ways, and make repeated measurements with a more accurate measuring instrument. What is the appropriate trade-off between sample size and accuracy? This is the general problem that we address in this paper.

In many practical situations, we are not satisfied with the accuracy of the existing measurements. There are two possible ways to improve the measurement accuracy. First, instead of a single measurement, we can make repeated measurements; the additional information coming from these additional measurements can improve the accuracy of the result of this series of measurements. Second, we can replace the current measuring instrument with a more accurate one; correspondingly, we can use a more accurate (and more expensive) measurement procedure provided by a measuring lab – e.g., a procedure that includes the use of a higher quality reagent. In general, we can combine these two ways, and make repeated measurements with a more accurate measuring instrument. What is the appropriate trade-off between sample size and accuracy? Traditional engineering approach to this problem assumes that we know the exact probability distribution of all the measurement uncertainties. In many practical situations, however, we do not know the exact distributions. For example, we often only know the upper bound on the corresponding measurement (or estimation) uncertainty; in this case, after the measurements, we only know the interval of possible values of the quantity of interest. We first show that in such situations, traditional engineering approach can sometimes be misleading, so for interval uncertainty, new techniques are needed. Then, we describe proper techniques for achieving optimal tradeoff between sample size and accuracy under interval uncertainty. In general, if the measurement uncertainty consists of several components, then the optimal trade-off between the accuracy D and the same size n occurs when these components are approximately of the same size. In particular, if we want to achieve the overall accuracy D0, as a first approximation, it is reasonable to take D ¼ D0=2 – and select the sample size for which the resulting overall measurement uncertainty is D0. A more accurate description of optimal selections in different situations is as follows:  For the case when we measure a single well-defined quantity (or the average value of varying quantity), we should take D ¼ 13  D0.  For the case when we are interested in reconstructing all the values xðtÞ of a smooth quantity x depending on d parameters t ¼ ðt1; . . . ; tdÞ, we should take D ¼ 1 dþ1  D0.  For the case when are interested in reconstructing all the values xðtÞ of a non-smooth quantity x depending on d parameters t ¼ ðt1; . . . ; tdÞ, we should take D ¼ b dþb  D0, where b is the exponent of the power law that describes how the difference xðt þ DtÞ  xðtÞ changes with kDtk.  For the case of more accurate measuring instruments, when the cost FðDÞ of a single measurement starts growing as c=D3, we should take D ¼ 35  D0. In general, if FðDÞ ¼ c=Da, we should take D ¼ a aþ2  D0.