Calculating the Net Activity Uncertainty if the Background Includes a

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Calculating the Net Activity Uncertainty if the Background Includes a Nonstationary Component Alex Ulianov,* Maria Honisch, and Othmar Muntener Institute of Earth Sciences, University of Lausanne, Géopolis, 1015 Lausanne, Switzerland ABSTRACT: Knowing the uncertainty of the net signal activity is essential in order to calculate the concentration uncertainty of an analyte in a sample or to decide whether the analyte activity is sufficient for detection. The net signal activity is usually calculated as a difference between the gross signal activity and the blank activity obtained from a material that is added to the sample during the analysis (e.g., acid to dilute the sample or gas to transport it as an aerosol). It is often assumed a priori that the analyte concentration in the blank is the same for any blank measurement, and the dispersion in the blank activities obtained from the individual measurements is related to the intrinsic imprecision of the measurement process only. This case can be called stationary; the relevant uncertainty calculation methods are widely known and used. However, in real applications where only some of the blank activity sources are well characterized, the blank stationarity needs to be demonstrated. The nonstationarity of the blank has serious consequences: (1) for a “well-known background”, the net signal uncertainty increases, since it includes the variance of the nonstationary background component; (2) for paired measurements, the net signal uncertainty does not change but cannot be estimated from one single blank analysis, thus compromising the utility of the paired measurement approach. Presenting the lower and upper uncertainty limits is appropriate for the evaluation of the uncertainty of a net signal if the assumption of a stationary blank is questionable.

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equal to the uncertainty of the gross content? Below we analyze this question with a special emphasis on cases when the blank is not stationary, i.e., the contamination level varies from analysis to analysis.

he common view of the background correction, rooted in the wet chemistry technique, implies that the gross analyte content in the analyzed sample solution is a sum of two terms, the (net) sample analyte content and the analyte content in the solvent. The analyte content in the solvent is often called “blank” or “background” content; it can be estimated in a blank experiment usually performed at the conditions identical to the sample solution analysis or calculated as an average from a replicate series of a number (n) of such experiments. The former approach is called “paired measurements”, the latter, “well-known background”.1,2 For a unimodal symmetrical distribution, the average content is the most probable among all individual estimates of the true analyte content in the solvent. In the absence of inaccuracies, it is approximately equal to the latter (the uncertainty of the average content can be estimated as the standard deviation of the mean; for large n, it is small and rarely used in calculations2). Thus, the uncertainty of the net analyte content becomes equal to the uncertainty of the gross content. This traditional approach is adequate if the blank is stationary. In the example above, the stationarity of the blank means that the same solvent, with the same analyte contamination level, is used in each analysis and this solvent is the only source of the blank. Let us now consider the situation when vials used to store sample solutions before the analysis are contaminated with the analyte, the contamination level being unique for each of the vials (Figure 1a,b). A series of n blank analyses carried out to constrain the “well-known background” yields an average contamination level of the vials. Increasing n makes the uncertainty of this average level negligible. Is then the uncertainty of the net analyte content © 2013 American Chemical Society

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DEFINING THE NET RESPONSE IN THE MASS (CONCENTRATION) DOMAIN

Basics. The net analyte content, represented in mass units, is equal to the gross content minus the content that is not due to the sample: mnet = mgross − mnot‑due‑to‑the‑sample. The not-dueto-the-sample content (i.e., true contamination level per analysis) is part of the gross content and cannot be measured separately during an analysis of the gross content. It is substituted by an estimate obtained from the individual blank analyses, the precision of this estimate depending on the number of replicates n. Two limiting cases arise. (1) The stationary case: the contamination level mnot‑due‑to‑the‑sample is the same for any individual analysis (e.g., defined by the analyte content in the solvent). If mnot‑due‑to‑the‑sample is estimated precisely (“well-known background”), we obtain Received: September 8, 2012 Accepted: January 18, 2013 Published: January 18, 2013 2589

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Figure 1. Limiting cases of the background uncertainty (“well-known” background). (a,b) Stationary background and nonstationary background with precise mass estimates, respectively. Even if the range of mass estimates appears the same in both cases, the meaning of these estimates is different (see Basics). (c) Nonstationary background with imprecise mass estimates. The estimated masses show a larger scatter than the true masses (see Generalization for Imprecise Measurements). n

mnot‐due‐to‐the‐sample =

∑i = 1 (mnot‐due‐to‐the‐sample)estimate i n

Var(mnot‐due‐to‐the‐sample) = 0 n → +∝

unimodal symmetrical distribution, this is the most probable of ;

all estimates of the true contamination level mnot‑due‑to‑sample. To get mnet, we thus assume

and

Var(mnet ) = Var(mgross)

mnot‐due‐to‐the‐sample = (mnot‐due‐to‐the‐sample)estimate mean n

If it is estimated from a single analysis (“paired measurements”), then

=

mnot‐due‐to‐the‐sample = (mnot‐due‐to‐the‐sample)estimate ; Var(mnot‐due‐to‐the‐sample)estimate > 0

∑i = 1 (mnot‐due‐to‐the‐sample)estimate i n

Let us imagine that the mass estimates in the equation above

and

are precise, i.e., each of them equals the true contamination level (mnot‑due‑to‑sample)true in the corresponding blank analysis. In

Var(mnet ) = Var(mgross) + Var(mnot‐due‐to‐the‐sample)estimate

our example, the precise mass measurement can be ensured by

Here, (mnot‑due‑to‑sample)estimate is the result of an individnual weighing of the analyte content in the blank, i.e., a weight e s t i m a t e r e l at ed t o t h e b a l a n c e p r e c i s i o n . V a r (mnot‑due‑to‑sample)estimate reflects the uncertainty of this estimate (quantifying this uncertainty from one single weighing does not seem possible). (2) The nonstationary case: the contamination level varies from analysis to analysis. mnot‑due‑to‑sample for an individual sample analysis cannot be obtained. By the analysis of a series of blanks, only the mean value of it can be computed. For a

using a precise balance with very low relative standard deviation for masses to measure or by replicating the weighing for each of the blanks and computing the corresponding mean weights of the contaminant. With a probability depending on the d i s t r i b u t i o n t y p e , t h e t r u e co n t a mi n a t i o n l e v e l (mnot‑due‑to‑sample)true for a particular analysis falls in an interval of ± s(mnot‑due‑to‑sample)true centered at the mean value above: 2590

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individual estimates of this level. For each of the blanks, let us introduce a coefficient k according to the following equation:

s(mnot‐due‐to‐the‐sample)true n

= {[ ∑ ((mnot‐due‐to‐the‐sample)estimate mean

k(mnot‐due‐to‐the‐sample)true = (mnot‐due‐to‐the‐sample)estimate

i=1

The true contamination level in the same blank remains the same, while its estimates randomly vary. The coefficient above is thus a random value; in the absence of inaccuracies, it is centered around unity; its standard deviation is equal to the relative standard deviation of the estimated contamination level for our blank. On the basis of the linearization of the k(mnot‑due‑to‑sample)true product, for homoscedastic k, we obtain

− (mnot‐due‐to‐the‐sample)estimate i )2 ]/(n − 1)}1/2 n

= {[ ∑ ((mnot‐due‐to‐the‐sample)true mean i=1

− (mnot‐due‐to‐the‐sample)true i )2 ]/(n − 1)}1/2

For the both “well-known background” and “paired measurements” approaches, we obtain

Var(k(mnot‐due‐to‐the‐sample)true )

mnet = mgross − mnot‐due‐to‐the‐sample Var(mnot‐due‐to‐the‐sample)true > 0

= Var(mnot‐due‐to‐the‐sample)estimate over all blanks Var(mnot‐due‐to‐the‐sample)true

and

= [Var(mnot‐due‐to‐the‐sample)estimate over all blanks

Var(mnet ) = Var(mgross) + Var(mnot‐due‐to‐the‐sample)true

− (mnot‐due‐to‐the‐sample 2)true mean Var(k)]/k 2

It is worth emphasizing again that mnot‑due‑to‑sample in the above expression for the net mass, calculated as a mean value of all available blank estimates or estimated from one single blank analysis, is only an estimate, the most probable one in the case of “well-known background”, of the true contamination level in a specific sample analysis. The real value of the true contamination level associated with this analysis is unknown. To simplify the example, we can imagine that the sample measurement is precise [ Var(mgross) = 0]. Still, the net intensity mnet receives an uncertainty defined by the variance of the true contamination level Var(mnot‑due‑to‑sample)true. In the “paired measurements”, this uncertainty cannot be obtained from one single weighing compromising the utility of the “paired measurements” approach for nonstationary backgrounds. Generalization for Imprecise Measurements. Let us now consider a more general situation when errors in variables are present, i.e., when the individual mass estimates are imprecise (Figure 1c). The true contamination level (mnot‑due‑to‑sample)true for a given blank cannot be obtained from a single blank analysis, only its imprecise estimate can be obtained from it. Three solutions to calculate the blank and the net uncertainties can be discussed. The first and the most imprecise solution implies that only one imprecise value (mnot‑due‑to‑sample)estimate per blank is available for a series of blank analyses. In the uncertainty calculations, the range of true contamination levels is then substituted by a range of estimated contamination levels:

Since the mean value of k is equal to one, we have Var(mnot‐due‐to‐the‐sample)true = Var(mnot‐due‐to‐the‐sample)estimate − (mnot‐due‐to‐the‐sample 2)true mean Var(k)

Here, Var(mnot‑due‑to‑sample)true is the variance of the true contaminant contents in the blanks, Var(mnot‑due‑to‑sample)estimate is the variance of the (imprecisely estimated) contaminant contents in the blanks obtained from a series of blank analyses, Var(k) is the variance of the estimated-to-true ratios of the contaminant contents in one single blank obtained from a series of analyses of this blank, (mnot‑due‑to‑sample)true mean is the mean contamination level over all blanks. The stationary background and the nonstationary background with precisely measured per-blank contamination levels can be described as special cases of this equation. Indeed, if the background is stationary, then (mnot‑due‑to‑sample)true is constant and Var(mnot‑due‑to‑sample)true= 0: Var(mnot‐due‐to‐the‐sample)true = Var(mnot‐due‐to‐the‐sample)estimate − (mnot‐due‐to‐the‐sample 2)true mean Var(k) = Var(mnot‐due‐to‐the‐sample)estimate

s(mnot‐due‐to‐the‐sample)estimated

− (mnot‐due‐to‐the‐sample 2)true mean (mnot‐due‐to‐the‐sample)estimate × Var (mnot‐due‐to‐the‐sample)true mean

n

= {[ ∑ ((mnot‐due‐to‐the‐sample)estimate mean i=1

− (mnot‐due‐to‐the‐sample)estimate i )2 ]/(n − 1)}1/2

= Var(mnot‐due‐to‐the‐sample)estimate (mnot‐due‐to‐the‐sample 2)true mean − (mnot‐due‐to‐the‐sample 2)true mean

n

≥ {[ ∑ ((mnot‐due‐to‐the‐sample)true mean i=1

− (mnot‐due‐to‐the‐sample)true i )2 ]/(n − 1)}1/2

× Var(mnot‐due‐to‐the‐sample)estimate = 0

The second solution implies that replicate analyses of one blank are additionally available, allowing one to calculate a precise value of the true contamination level for this blank and, most importantly, the variance Var(mnot‑due‑to‑sample)estimate for the

If the background is not stationary, but the contamination levels are measured precisely, then Var(k) = 0. We obtain the same result as discussed for the “precise” nonstationary case in the “Basics”: 2591

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to the optimal selection of the calibration function (ultimately, it is always possible in a Cartesian system to exactly approximate a set of n intensity vs mass pairs with nonrepeating masses by a polynomial with a degree of n − 1); we also do not discuss here, whether the calibration coefficient is obtained using an ordinary least-squares regression or a more complex measurement error model. First, let us discuss the simple classical case of a stationary background with a precisely known calibration coefficient. The value (Inot‑due‑to‑sample)mean yields the true analyte content in the stationary blank at high n and in the absence of inaccuracies and instrumental background: kmnot‑due‑to‑sample = (Inot‑due‑to‑sample)mean, where k is the slope of the calibration line. Thus, the net intensity and its uncertainty are calculated using the mean value Inot‑due‑to‑sample:

Var(mnot‐due‐to‐the‐sample)true = Var(mnot‐due‐to‐the‐sample)estimate − (mnot‐due‐to‐the‐sample 2)true mean × 0 = Var(mnot‐due‐to‐the‐sample)estimate

The third solution, precise and unconstrained but timeintensive, implies that replicate analyses are performed for all blanks available, yielding the mean contamination level for each individual blank. Increasing the number of replicates makes the uncertainty of the mean per-blank contaminant contents negligible. This solution is again reduced to the “precise” nonstationary case described in the “Basics”. For the net intensity, we obtain Var(mnet ) = Var(mgross) + Var(mnot‐due‐to‐the‐sample)true , where

Var(mnot‐due‐to‐the‐sample)true > 0

mnet = mgross − mnot‐due‐to‐the‐sample ⇔ kmnet = kmgross − k(mnot‐due‐to‐the‐sample)mean

It is possible in the examples above that the both solvent and vials are impure, with the solvent having the same analyte content in all analyses and the vials showing a scatter in the analyte contents they are contaminated with. This is a case of the partial stationarity. If the blank is “well-known”, we obtain

⇔ Inet = Igross − (Inot‐due‐to‐the‐sample)mean ⇒ Var(Inet) = Var(Igross)

The presence of an instrumental background, corresponding to a constant − zero − mass and originating, for example, from the detector noise, is not discussed here. The mean intensity of the instrumental background can be considered constant in the calculations, not influencing other uncertainties. If Inot‑due‑to‑sample is estimated from a single measurement (“paired measurements”), then

mnot‐due‐to‐the‐sample = [mnot‐due‐to‐the‐sample]solvent + [mnot‐due‐to‐the‐sample]vial ; Var[mnot‐due‐to‐the‐sample]solvent = 0; Var[(mnot‐due‐to‐the‐sample)true ]vial > 0;

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Var(mnet ) = Var(mgross) + Var[(mnot‐due‐to‐the‐sample)true ]vial

mnot‐due‐to‐the‐sample = (mnot‐due‐to‐the‐sample)estimate = k(Inot‐due‐to‐the‐sample)estimate ;

DEFINING THE NET RESPONSE FOR AN ACTIVITY STATISTICALLY DEPENDENT ON THE CONCENTRATION Estimating the analyte content directly in mass units is easier to understand but is rare in modern analytical techniques. Intensities of various kind related to the presence of an analyte, represented in counts per time unit are often employed instead and converted to mass units with the help of a calibration. For the net intensity, we obtain Inet = Igross − Inot‑due‑to‑sample. The discussion below has much in common with the discussion in the mass domain. The same analyte content in the blank generates a series of somewhat different intensity values during replicate measurements. For a unimodal symmetrical distribution, the mean intensity value is the most probable analyte intensity estimate in the blank; it is precise at high n:

Inet = Igross − (Inot‐due‐to‐the‐sample)estimate ; Var(Inot‐due‐to‐the‐sample)estimate > 0; Var(Inet) = Var(Igross) + Var(Inot‐due‐to‐the‐sample)estimate

Var(Inot‑due‑to‑sample)estimate in the formulas above can be estimated as variance of the mean (time-resolved data or Poisson distributed counts, e.g., in the inductively coupled plasma (ICP) mass spectrometry or radioactivity measurements). This calculus is well-known from the literature3 and is widely used in analytical spectrochemistry without proving the stationarity. Generalization: The Nonstationary Case with Calibration and Intensity Uncertainties. Here, the blank intensity varies from analysis to analysis, reflecting the different contents of the analyte in the blanks and not only the intrinsic imprecision of the measurement process. Modifying the equation k (mnot‑due‑to‑sample)true = (mnot‑due‑to‑sample)estimate from Generalization for Imprecise Measurements, we obtain

n

(Inot‐due‐to‐the‐sample)mean =

∑i = 1 (Iblank )estimate i n

;

Var(Inot‐due‐to‐the‐sample)mean = 0 n → +∝

Converting this intensity to mass (concentration) invokes two additional parameters: the calibration function and the uncertainty associated with the calibration. Our present discussion will be limited by the simple but most practically important case, when intensity linearly depends on concentration and a calibration uncertainty is present. This case also includes (generally) nonlinear intensity vs mass relationships, where the dependence is approximately linear in a range of concentrations sufficient for a given analytical task. We thus leave beyond the scope of this text nontrivial questions related

kcalk int(mnot‐due‐to‐the‐sample)true = (Inot‐due‐to‐the‐sample)estimate

This equation can be paralleled with the equation from Generalization for Imprecise Measurements, provided the coefficient k describes not only the mass, or intensity, measurement uncertainty (component kint) but also the variation range of the calibration coefficient (component kcal); in the general case, the mean value of the product kcalkint is not equal to unity anymore. Using the above equations allows one 2592

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to compute the blank intensity variance describing the range of true mnot‑due‑to‑sample values in the blanks:

The calculus above requires that both Var(Inot‑due‑to‑sample)estimate and Var(kcalkint) are determined. If only the first of these parameters is available, the calculation of the blank intensity uncertainty characterizing the true range of varying analyte masses in the blanks becomes impossible. The intensity range characterizing the true range of the not-due-to-the-sample masses in the blanks

Var(kcalk int(mnot‐due‐to‐the‐sample)true ) = Var(Inot‐due‐to‐the‐sample)estimate Var(mnot‐due‐to‐the‐sample)true = [Var(Inot‐due‐to‐the‐sample)estimate

Var(Inot‐due‐to‐the‐sample)characterizing the range of true blank masses

− (mnot‐due‐to‐the‐sample 2)true mean Var(kcalk int)]

can then be substituted by the range of intensities characterizing the estimated spread of these masses

/[(kcalk int)mean 2 ]

Var(Inot‐due‐to‐the‐sample)estimate

Var(Inot‐due‐to‐the‐sample)characterizing the range of true blank masses 2

= kcal mean × Var(mnot‐due‐to‐the‐sample)true

This substitution should be mentioned as such in the uncertainty calculations, since it replaces a true parameter with its less precise estimate. The Var(Inot‑due‑to‑sample)estimate can be computed from replicates, if the blank is “well-known”. The situation is more difficult in the “paired measurements”. Contrary to the stationary case, Var(Inot‑due‑to‑sample) cannot be estimated from one single analysis as variance of the mean, even if the data structure allows one to do this. A single blank analysis can be precise, and it can yield a small standard deviation of the mean intensity; however, there can be a large difference between the intensity values obtained from two different blank analyses (e.g., if the corresponding vials are differently contaminated). The partial stationarity is reduced to the combination of the stationary and nonstationary cases. If the blank is “well-known”, we obtain

= kcal mean 2{[Var(Inot‐due‐to‐the‐sample)estimate − (mnot‐due‐to‐the‐sample 2)true mean × Var(kcalk int)]/[(kcalk int)mean 2 ]} = [kcal mean 2 Var(Inot‐due‐to‐the‐sample)estimate − (Inot‐due‐to‐the‐sample 2)true mean × Var(kcalk int)]/[(kcalk int)mean 2 ]

Some special cases can now be considered. If kcal is constant (precise calibration) but the per-blank intensities are measured imprecisely, we obtain Var(Inot‐due‐to‐the‐sample)characterizing the range of true blank masses = [kcal mean 2 Var(Inot‐due‐to‐the‐sample)estimate

mnot‐due‐to‐the‐sample = [mnot‐due‐to‐the‐sample]solvent

− (Inot‐due‐to‐the‐sample 2)true mean Var(kcalk int)]

+ [mnot‐due‐to‐the‐sample]vial ;

2

/[(kcal)mean × 1]

Var[(Inot‐due‐to‐the‐sample)mean ]solvent = 0; Var[(Inot‐due‐to‐the‐sample)characterizing the range of true blank masses ]vial > 0;

= Var(Inot‐due‐to‐the‐sample)estimate 2

− (Inot‐due‐to‐the‐sample )true mean Var(k int)

Var(Inet) = Var(Igross)

If kcal is constant and equal to unity (identity between intensity and mass), we easily derive the mass variance equation from Generalization for Imprecise Measurements:

+ Var[(Inot‐due‐to‐the‐sample)charcacterizing the range of true blank masses ]vial

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RECOGNIZING THE NONSTATIONARY BEHAVIOR OF THE ANALYTE ACTIVITY IN THE BLANK The challenge here is to show whether the dispersion of the individual measurement intensities reflects the imprecision of the measurement process only or it also reflects the difference in the analyte content in the blanks. The general solution is to analyze the blank with a high precision allowing one to distinguish the different analyte contents in the individual blank analyses. In the “well-known background” case, this can be done by increasing the number of analyses of the same blank (e.g., repetitively analyzing a blank solution contained in the same vial), which allows calculating the mean intensity precisely. This can also be done by increasing the counting time per analysis. Some analytical techniques, such as laser ablation ICP mass spectrometry, traditionally use the paired measurements approach combined with short counting times, which often yields very few counts collected per isotope in a blank analysis. If two blank intensity estimates are available and the counting process during an individual blank acquisition is Poissonian, the question is to test if these intensity estimates correspond to the same mean intensity. This can be done using the binomial method of Przyborowski and Wilenski4 or more

Var(mnot‐due‐to‐the‐sample)true = Var(mnot‐due‐to‐the‐sample)estimate − (mnot‐due‐to‐the‐sample 2)true mean Var(k)

Finally, if the background is stationary [Var (mnot‑due‑to‑sample)true = 0], we have Var(Inot‐due‐to‐the‐sample)characterizing the range of true blank masses = [kcal mean 2 Var(Inot‐due‐to‐the‐sample)estimate − (Inot‐due‐to‐the‐sample 2)true mean (Inot‐due‐to‐the‐sample)estimate × Var ]/[(kcalk int)mean 2 ] (mnot‐due‐to‐the‐sample)true mean = [kcal mean 2 Var(Inot‐due‐to‐the‐sample)estimate −

(Inot‐due‐to‐the‐sample 2)true mean (mnot‐due‐to‐the‐sample 2)true mean

× Var(Inot‐due‐to‐the‐sample)estimate ]/[(kcalk int)mean 2 ] = 0

The net intensity in the case of both “well-known background” and “paired measurements” approaches is calculated as follows: Var(mnet ) = Var(Igross) + Var(Inot‐due‐to‐the‐sample)characterizing the range of true blank masses 2593

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(6) Ng, H. K. T.; Gu, K.; Tang, M. L. Comput. Stat. Data Anal. 2007, 51, 3085−3099. (7) Kaiser, H. Spectrochim. Acta 1947, 3, 40−67.

recent methods5,6 developed with the aim to test the equality of two Poisson means.

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CONCLUDING REMARKS The nonstationarity of the blank response has important consequences for the calculation of the net signal uncertainty. If the background is “well-known”, the net signal uncertainty calculated for a nonstationary blank is higher compared to the stationary case, since it includes the uncertainty of the nonstationary component of the blank activity propagated in squares to get the net signal uncertainty. In the “paired measurements”, the net signal uncertainty does not change; still, calculating this uncertainty based on an estimate derived from an individual blank analysis is impossible in the general case. The calculation of the critical level for detection decision is also influenced by the nonstationarity of the blank. For (quasi-) continuously distributed activities approximated by the normal distribution, the critical level is defined as kσ(Inet),2,7 where the net intensity is obtained from a sample devoid of analyte, while coefficient k corresponds to a desired confidence level. The net intensity uncertainty, σ(Inet), in the general case, includes a nonstationary component. The consequences are a higher critical level compared to the case of a stationary blank, the impossibility to indefinitely decrease the critical level by increasing the precision of individual analyses, and to specify this level based on a single intensity estimate in the case of “paired measurements”. Before calculating the net uncertainty, it is essential to demonstrate that the analyzed blank is stationary. Otherwise, two options arise: (1) to assume that there is an unknown nonstationary component in the total blank activity; the net intensity uncertainty can then be given as an interval, of which the lower and upper limits correspond to the net activity uncertainty in the cases of a fully stationary and a fully nonstationary behavior, respectively; (2) to quantify the uncertainty of the nonstationary component and to provide a specific value for the net uncertainty. In both cases, using the “well-known background” approach (or replicating paired measurements) is appropriate.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.U. thanks Olga Ulianova for her help during the preparation of the manuscript. Constructive reviews by two anonymous colleagues are gratefully acknowledged. This contribution was supported by Grants 200021_138004/1 and 206021-117405 (R’EQUIP) from the Swiss National Science Foundation.

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REFERENCES

(1) Altshuler, B.; Pasternack, B. Health Phys. 1963, 9, 293−298. (2) Currie, L. Anal. Chem. 1968, 40, 586−593. (3) MARLAP. Multi-Agency Radiological Laboratory Analytical Protocol Manual. 2004, Vol. 19−20. (4) Przyborowski, J.; Wilenski, H. Biometrika 1940, 31, 313−323. (5) Krishnamoorthy, K.; Thomson, J. J. Stat. Plann. Inference 2004, 119, 23−35. 2594

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