An Investigation into Estimating Type B Degrees of Freedom

Transcription

1 An Investigation into Estimating Type B Degrees of H. Castrp President, Integrated Sciences Grop Jne, 00 Backgrond The degrees of freedom associated with an ncertainty estimate qantifies the amont of information that went into the estimate. As this amont of information grows, the degrees of freedom becomes larger. Conversely, if or information is meager or nclear, the degrees of freedom shrinks. Another way of saying this is that, if the ncertainty or variance in an ncertainty estimate is large, the degrees of freedom is small. An approximate expression of this relationship, as applied to compting the degrees of freedom for a Type B estimate, is provided in Eq. G.3 of the GUM [] ( x. ( [ x ( ] This is an intitively appealing reslt in that the degrees of freedom is inversely proportional to the sqare of the ncertainty in the ncertainty estimate. Hence, a large variance in the ncertainty estimate yields a small degrees of freedom. While the relation is appealing, the GUM offers no advice on how to compte [(x]. Fortnately, a method was developed since the pblication of the GUM, that was reported at the 000 MSC [], and has proved sefl in practice. This method will be referred to herein as the Castrp method. Recent Findings An investigation of has been ndertaken [3] in which sets of normal deviates are generated with known standard deviations for a variety of sample sizes. Following this, standard deviations for each set are calclated sing both Type A and Type B procedres. The Type A estimate is, of corse, while the Type B estimate is given by n s ( x xi x, ( n x i L. (3 + p Φ In this expression, the variable L is an arbitrary limit sed to bond vales of x arond the mean vale and p is the observed fraction of vales fond within ± L of the mean. For example, if x is the ncertainty in the bias of a toleranced parameter, ± L wold most likely be the tolerance limits for a parameter s deviation from nominal and p wold be the observed fraction in-tolerance. In discssing Type B ncertainty estimates, the limits ±L are referred to as containment limits and p is called the containment probability. Preliminary Reslts The reslts to date show that, for most of the simlated trials, as the sample size n grows to 50 or more, the Type B ncertainty estimate actally comes closer to the nderlying poplation standard deviation (which is a known parameter in generating the random normal deviates than does the Type A estimate.

2 However, sing the Castrp method, the estimated Type B degrees of freedom trns ot to be considerably less than the sample degrees of freedom for the Type A estimate. This is somewhat disqieting, since the Type B estimate is at least as good as the Type A estimate. Of corse, obtaining a smaller degrees of freedom for the Type B estimate is a conseqence of the difference in the mathematical forms of Eqs. ( and (3. Investigation The qestion is, why doesn t the Castrp method yield a larger degrees of freedom? It may be that the method sed to determine [(x] is flawed, or the GUM relation is flawed, or both. As a first step in isolating the problem, an attempt was made to independently derive Eq. (. The attempt begins with a consideration of the variance of a sample standard deviation. Let s represent the standard deviation, taken on a sample of size n + of a N(0, variable x. We know that the qantity s / is χ -distribted with degrees of freedom. The χ -distribtion has the pdf Accordingly, we set x For a χ -distribted x, we have Sbstitting in the variance of s /, or ( / s yields as contrasted with Eq. G3 of the GUM x f( x ( / x / / e Γ s x, and compte the variance in ( s var ( s var ( x. ( x var ( s. s., ( GUM. (5 We will now compare these alternative expressions by sing the Castrp method to obtain ( s ( s /. Using Eq. (3 of Ref, we have ( s ( s π L + ϕ p / and e, (6 L ϕ where L is the ncertainty in the containment limit L and p is the ncertainty in the containment probability p. the fnction ϕ is defined as Using the method of Ref, we can show that Comparison of Eq. (6 with Eq. (7 reveals that [ p ] ϕ( p Φ ( + /. ( s π L ϕ p e +. (7 L ϕ ( s s (,

3 which yields, with the aid of Eqs. ( and (5, Discssion From the above, we can write GUM. (8 ( s ( s ( s ( s ( s However, from the properties of the χ distribtion, we have and where If Eq. (9 is valid, we have the relation. (9 ( s, c s, ( c + Γ. Γ + Γ. (0 Γ This relation is a conseqence of Eq. (9, which is, in trn, a conseqence of sing the Castrp method. Also, by Eq. (8, the validity of GUM appears to be dependent on the validity of the Castrp method, which also leads to Eq. (0. I have not fond the eqality in Eq. (0 in any text, handbook or other reference. However, I have tested it by nmerical calclation for vales of ranging from to 000. It trns ot that the eqality is approximate for small vales of (less than 50 or so and improves as increases. For instance, it is exact to for decimal places at 7. The approximate natre of the eqality may be explained in that it springs from the Castrp method, which tilizes an expansion of the error in to first order in L and p, i.e., higher order terms are neglected. The approximate natre of the eqality also indicates that the expression for in Eq. (5 is only approximately eqivalent to the expression for in Eq. (. Utility of the Estimates It is interesting to consider the case where n 00. Sppose and the tolerance limits are ±.96. In this case, we wold observe abot 95% of the observations to be within the limits. This is what occrs when we simlate normal deviates. For example, a typical simlation sing these parameters yielded 93% of observations within the limits ±.96.

4 The sample standard deviation was estimated sing Eqs. ( and (3 and the degrees of freedom were calclated sing Eq. (8. The reslts are shown below. Table A typical simlation with n 00, and L.96. Description Poplation Vale Type A Estimate Type B Estimate Nmber of Observations Nmber Observed within ± L Containment Probability p Deg As has been stated, this example is typical of those for n 00. It is interesting to note that, for sch cases, the Type B standard deviation estimate is at least as good as the Type A estimate. Again, however, we have the reslt that the Type B degrees of freedom is smaller than the Type A vale. There are at least two possible explanations for this information loss that come to mind. One stems from the fact that the variance ( s is evalated sing a first-order expansion of the error in s, with higher-order terms neglected. Another possible explanation is that some loss of information may be de to sing the binomial variance p( p / n to calclate. If there is no ncertainty in L, this leads to the expression p ϕ e n, π p( p which will always be smaller than n. Perhaps a different approach to estimating p is in order. Of corse, it might be arged that a binomial statistic p, based on n independent Bernolli trials, in which a vale of is assigned to a reslt within ±L and a vale of 0 is assigned to a reslt external to ±L, does not really contain as mch information as a sample of data of size n. For instance, if we consider cases where p is nearly or zero, i.e., cases where we may not obtain any vales within ±L or external to ±L, then we have little or no information from which to compte a Type B standard deviation. Given this argment, the smaller Type B degrees of freedom associated with compting with Eq. (3 is an entirely satisfactory otcome. The Castrp method prodces estimates that are consistent with this observation. For example, ten simlations were carried ot with n 300, L.96 and The small, relative to L, prodced high vales of p, reslting in poor estimates for in some of the trials. The Type B degrees of freedom ranged from 5 to 8. The reslts are shown in the table below. Note that, althogh different Type A ncertainty estimates were obtained for each simlation, the Type B ncertainty estimates tended to settle into grops. This is becase the details of the sample (i.e., the sampled vales are open for the Type A estimates and masked or hidden for the Type B estimates. This spports the notion that a lesser amont of information is available for Type B estimates than for Type A estimates de to the binomial character of the former. Again, the smaller degrees of freedom for Type B estimates is to be expected. ϕ Experimenting with larger sample sizes and poplation standard deviations yield reslts that are consistent with those shown in the table.

5 Table The reslts of ten simlations with n 300, 0.75 and L.96. Trial Type A Type B Containment Probability p Deg References [] ANSI/NCSL, Z , U.S. Gide to the Expression of Uncertainty in Measrement, October 9, 997. [] Castrp, H., Estimating Category B Degrees of, Proc. Measrement Science Conference, Anaheim, Janary 000. [3] Work in-progress, Integrated Sciences Grop, Bakersfield, CA