of measurement standards on the basis of a mixture of distributions
Since in the MRA there is no clear definition of the reference value and, correspond- ingly, the equivalence of measurement standards, then various interpretations of these concepts which on the whole do not contradict the MRA are possible [95, 96, 105, 117, 118, 546, 547].
The interpretation of the equivalence of measurement standards as the compatibility of measurement results obtained using these standards is the most common. Thus, if a result of the i-th laboratory is incompatible with a KCRV within the limits of the uncertainty claimed, then the standard given is not equivalent to the remaining group. The expansion of measurement uncertainties with the purpose of making key comparison measurement results consistent, as a rule, is not anticipated in evaluating key comparison data.
Section 1.4 Expression of the DE on the basis of a mixture of distributions 13 Within the framework of this subsection an alternative interpretation of the equiv- alence of measurement standards as the reproducibility of measurement results, ob- tained by a group of NMIs participating in key comparisons, is treated. Correspond- ingly, the degree of equivalence is understood as a quantitative measure of this repro- ducibility [95, 96, 105].
This interpretation can be useful for evaluating data of comparisons conducted for the first time, when from the very start there are doubts about taking into account all factors and, above all, systematic ones, in the budget of measurement uncertainty.
Let a measurement result be presented in the form of observation model that corre- sponds to model (1.2):
xi DxCbiC"i,
where x is the measurand value, bi is the bias value of measurement results in a laboratory given, and"iis its error.
The measurement results obtained in laboratories can be interpreted as a sample of a general totality of measurements carried out by a group of laboratories participating in comparisons. It is important to note, that this distribution significantly depends on a membership of the group of participants. It is the group of definite laboratories that
“generates” a new distribution, different from distributions inside each laboratory- participant, i.e., the mixture of the distributions:
F .x/D 1 N
XFi.x/,
whereFi.x/ is the frequency distribution function of the random quantity Xi, the realization of which provides results in thei-th laboratory, with the mathematical ex- pectationEXi DxCbi.
It is suggested to choose the mathematical expectation of mixture distribution as the reference value:
EX DxC 1 N
Xbi.
Then the degree of equivalence of measurement standards can be defined as the difference of the mathematical expectations of two distributions by the formula
di DEXiEXDaCbia NbDbi Nb,
and may be interpreted as the difference between the “reference laboratory value”
and reference value of key comparisons, or as the difference of a systematic bias of measurements in a particular laboratory and systematic bias of the reference value. It is necessary to note that in such an approach the degree of equivalence and the reference value are defined through model parameters.
Let a linear combination of measurement results be considered as an estimate of the reference value:
v_ DX
!ixi, Ev_ DX
!iExi DaCX
!ibi.
From this it follows that an unbiased estimate is obtained at the arithmetical mean:
!i D 1
N, vD 1 N
Xxi, D./D 1 N
Xu2i. Correspondingly, the estimate of degree of equivalence is
_
di Dxi Nx, u2.
_
di/Du2i
1 2 N
C 1
N2 Xu2i.
The distribution mix functionF .x/D N1 P
Fi.x/is the most general form for de- scribing the dispersion of measurements in the group of comparison participants. This function, on its own, its derivative or particular characteristics, for example, a tolerance interval, can serve as the measure ofmutual equivalence of a group of measurement standards.
When introducing the quantitative equivalence measures on this basis of a mixture of probability distributions, it is necessary to distinguish different types of equivalence:
mutual equivalence of a groupof measurement standards,pair-wise equivalencein- side of a group of measurement standards, and lastly,equivalence of a particular mea- surement standard to a groupof measurement standards. The pair-wise equivalence of any pair of comparison participants are described by the difference of two inde- pendent random quantitiesX,Y. Each of these random quantities has the distribution F .x/. The density of difference distribution is the convolution of initial probability distribution densities:p.z/DR
f .zx/f .x/dx.
In the same manner, the concept of equivalenceof each measurement standard to a groupis defined, which is quantitatively expressed by the distribution (or tolerance in- terval) of deviations of results obtained by this laboratory from any result of the group XiX, whereXi 2 Fi.x/,X 2 F .x/. Correspondingly, the density of difference distribution is the convolutionpi.z/DR
fi.zx/f .x/dx.
It should be noted that application of mixture of distribution to key comparison data analysis, which is advocated in this subsection is different from those used in [117, 546].
Section 1.5 Evaluation of regional key comparison data 15