Tolerance Intervals for Hypergeometric and Negative Hypergeometric Variables

Transcription

1 Sankhyā : The Indian Journal of Statistics 2015, Volume 77-B, Part 1, pp c 2014, Indian Statistical Institute Tolerance Intervals for Hypergeometric and Negative Hypergeometric Variables Derek S. Young University of Kentucky, Lexington, USA Abstract Tolerance intervals for discrete variables are widely used, especially in industrial applications. However, there is no thorough treatment of tolerance intervals when sampling without replacement. This paper proposes methods for constructing one-sided tolerance limits and two-sided tolerance intervals for hypergeometric and negative hypergeometric variables. Equal-tailed tolerance intervals (i.e., tolerance intervals that control the percentages in both tails) are studied followed by a small adjustment to the nominal coverage level to obtain tolerance intervals that control a specified inner percentage of the sampled distribution. The tolerance interval calculations implicitly use confidence bounds for, the unknown number of elements possessing a certain attribute in the finite population of size N. Three different methods for obtaining such confidence bounds are suggested: a large sample approach, an approach with a continuity correction, and an exact method based on nonrandomization. The intervals are examined for desirable coverage probabilities and expected widths. The methods are also illustrated using some examples. AS (2000) subject classification. Primary 62F25; Secondary 62F03. Keywords and phrases. Acceptance sampling, coverage probability, exact confidence bounds, expected width, monotone likelihood ratio property, tolerance package. 1 Introduction A statistical tolerance interval is an interval that is expected to contain at least a certain proportion of the sampled population (P ) with a specified confidence level (1 α). Tolerance intervals are important for applications in quality control, engineering, and the pharmaceutical industry. See the texts Electronic supplementary material The online version of this article (doi: /s ) contains supplementary material, which is available to authorized users.

2 Hypergeometric and negative hypergeometric tolerance intervals 115 by Hahn and eeker (1991) and Krishnamoorthy and athew (2009) for examples. Like confidence and prediction intervals, (approximate) tolerance intervals are available for numerous continuous and discrete distributions, regression models, and some multivariate settings. The literature on constructing tolerance intervals for continuous distributions is extensive and dates back to the seminal works of Wilks (1941, 1942). However, there is considerably less literature on tolerance intervals for discrete distributions, where the majority of such works focus on the binomial, Poisson, and negative binomial distributions. One of the earliest works is Zacks (1970), who developed uniformly most accurate (UA) upper tolerance limits for discrete distributions that possess the monotone likelihood ratio property. While the work of Zacks (1970) is firmly rooted in a theoretical framework, Hahn and Chandra (1981) took a more pragmatic approach to the problem so that practitioners could easily calculate tolerance limits (intervals) for binomial and Poisson random variables. Zaslavsky (2007) provided numerous examples of calculating discrete tolerance limits for clinical trials having dichotomous outcomes. athew and Young (2013) used a fiducial approach to construct tolerance intervals for functions of discrete random variables. Using the framework of Hahn and Chandra (1981), Young (2014) provided an extensive simulation study of different methods for computing negative binomial tolerance intervals. The approaches for constructing tolerance intervals for the above discrete distributions are often conservative. However, methods to improve their coverage probabilities have been investigated. Wang and Tsung (2009) proposed a coverage-adjustment procedure to compute the exact minimum and average coverage probabilities of binomial and Poisson tolerance intervals. Krishnamoorthy et al. (2011) applied this coverage adjustment to obtain two-sided binomial and Poisson tolerance intervals that do not necessarily control the percentages in both tails. Cai and Wang (2009) utilized a probability-matching technique to construct tolerance intervals for any distribution belonging to the natural discrete exponential family having a quadratic variance function. Their approach involves a two-term Edgeworth expansion that re-centers the tolerance interval by causing the first-order smoothing term (i.e., first-order probability matching) or the first-order and second-order smoothing terms (i.e., second-order probability matching) to vanish. This removes systematic bias and results in better coverage probabilities. The above references highlight most of the work that has been done regarding tolerance limits for the binomial, Poisson, and negative binomial distributions. However, there is little work that properly treats the tolerance

3 116 D.S. Young interval problem when sampling without replacement. Eichenberger et al. (2011) presented some work relevant to this topic for sample size determination for social surveys, but it relied on a normal approximation and the notions of size resolution and difference resolution. We provide a rigorous development of tolerance intervals for hypergeometric and negative hypergeometric variables and highlight the corresponding sampling schemes to which they are applicable. For example, the hypergeometric distribution is often used when constructing attribute acceptance sampling plans (see Chapter 15 of ontgomery (2013)). eanwhile, a negative hypergeometric distribution is often used when constructing an attribute inverse sampling plan to determine the total number of samples to draw (without replacement) in order to observe a specified number of the attribute of interest. Acceptance limits for either plan can be framed as tolerance limits for the respective distribution. We note that the negative hypergeometric distribution is, in general, less studied in the literature compared to the hypergeometric distribution, especially regarding the computation of statistical intervals. See Zhang and Johnson (2011) for a discussion of approximate negative hypergeometric confidence intervals. In the literature, the negative hypergeometric distribution is also referred to as the inverse hypergeometric distribution (Guenther, 1975) and the hypergeometric waiting-time distribution (Johnson et al. 2005). It is helpful to briefly comment on some of the notation used throughout this paper. Specifically, we will use Y and Z to denote hypergeometric and negative hypergeometric random variables, respectively. X will denote a general random variable that will often be treated as a discrete random variable. oreover, these random variables will be superscripted with an asterisk (e.g., X ) to indicate when another random variable is drawn independently from the same distributional family, but perhaps with different fixed parameter values. This paper is organized as follows. In Section 2, we describe the general set-up for constructing one-sided tolerance limits and equal-tailed tolerance intervals for discrete distributions following Zacks (1970) and Hahn and Chandra (1981). The method implicitly uses confidence intervals for the parameter of interest. In Section 3, we outline large sample and exact methods for constructing confidence intervals and tolerance intervals for both the hypergeometric and negative hypergeometric distributions. We utilize the theory presented in Lehmann and Romano (2005) and Wright (1997) regarding exact randomized one-sided confidence bounds for hypergeometric distributions, which are known to be uniformly most accurate (UA). We further leverage the monotone likelihood ratio property of the hypergeometric and negative hypergeometric distributions combined with the theory presented

4 Hypergeometric and negative hypergeometric tolerance intervals 117 in Zacks (1970) to develop exact tolerance limits for these two distributions. All two-sided tolerance intervals that we present are conservative; however, we apply a coverage adjustment using a criterion suggested in Krishnamoorthy et al. (2011) for the binomial and Poisson settings, which is based on the methodology of Wang and Tsung (2009). In Section 4, we compare the performance of these different intervals. In Section 5, we present examples of hypergeometric and negative hypergeometric tolerance intervals as well as the corresponding functions that are available in the tolerance package (Young, 2010) for the R programming language (R Development Core Team, 2013). Finally, we end with a brief discussion in Section 6. 2 Tolerance Intervals for Discrete Distributions Let X be a discrete random variable with cumulative distribution F ( ; θ, n), where θ is the parameter of interest and n is a known size parameter (e.g., the number of draws from a hypergeometric distribution or the target number of successes to draw from a negative hypergeometric distribution). Let X follow the same distribution independent of X, but with size m,which may or may not be equal to n. A(1 α, P ) tolerance interval [L(X),U(X)] is constructed so that Pr X{Pr X {L(X) X U(X) X} P } =Pr X{(Pr X {X U(X) X} Pr X {X L(X) 1 X}) P } =Pr X{F (U(X); θ, m) F (L(X) 1; θ, m) P } 1 α. (1) Analogously, a lower (1 α, P ) tolerance bound requires finding the largest integer L 1 (X) such that Pr X {1 F (L 1 (X) 1; θ, m) P } 1 α, (2) while an upper (1 α, P ) tolerance bound requires finding the smallest integer U 1 (X) such that Pr X {F (U 1 (X); θ, m) P } 1 α. (3) In the above, (1 α) is the confidence level and P (called the content) is the proportion of the sampled population that we wish to capture. Note that because of the discrete nature of the problem, the coverage probability requirements above are at least (1 α), whereas in the continuous setting they would be equalities. oreover, we must carefully

5 118 D.S. Young handle complementary probability statements when finding the bounds; i.e., Pr(X x 0 )=Pr(X x 0 1) for integer-valued x 0. Suppose x is an observed value of our random variable X and that F ( ; θ, n) is monotonic in θ. The method presented in Hahn and Chandra (1981) (henceforth referred to as the Hahn-Chandra method ) for constructing equal-tailed tolerance intervals for discrete distributions can be summarized in two steps: 1.Basedontheobservedvaluex, construct a two-sided 100(1 α)% confidence interval for θ, (θ L;α (x, n),θ U;α (x, n)). 2. For a future sample size m, find the maximum integer L(x) andthe minimum integer U(x) such that 1 F (L(x) 1; θ L;α (x, n),m) 1+P 2 (4) and respectively. F (U(x); θ U;α (x, n),m) 1+P 2, (5) The above can easily be modified to obtain one-sided tolerance limits. Specifically, we would replace the confidence interval in Step 1 by the necessary one-sided 100(1 α)% confidence bound for θ, sayθ L1 ;α(x, n) orθ U1 ;α(x, n), andthenreplace(1+p )/2 instep2byp. The Hahn-Chandra method as outlined is a two-sided setting that results in equal-tailed tolerance intervals because we are controlling the percentages in both tails. However, one can control some inner percentage of the distribution by using a slightly different criterion for Step 2. Namely, 2a. For a future sample size m, find integers U(x) >L(x) such that F (U(x); θ U;α (x, n),m) F (L(x) 1; θ L;α (x, n),m) P. (6) Obviously different criteria are employed depending on if the objective is to control the percentages in both tails or to control for some inner (possibly central) percentage. This also impacts how the coverage probabilities of the resulting tolerance intervals are calculated. These differences will be discussed in the following subsection.

6 Hypergeometric and negative hypergeometric tolerance intervals 119 Clearly the method used to construct the confidence interval in Step 1 above will impact the performance of the resulting tolerance interval. As Krishnamoorthy et al. (2011) point out, coverage properties of one-sided tolerance limits are typically similar to those of the confidence intervals that were used to construct them. However, coverage properties of two-sided tolerance intervals often warrant further investigation to gauge their degree of conservatism. Regardless, we will investigate coverage properties for both one-sided tolerance limits and two-sided tolerance intervals for hypergeometric and negative hypergeometric variables. Besides coverage probabilities, expected widths are also important in evaluating the performance of tolerance intervals and statistical intervals in general. See Brown et al. (2001) for such comparisons regarding confidence interval estimation of binomial proportions. We now provide the general formulas for coverage probabilities and expected widths in the discrete set-up Performance easures. In this section, we will define the coverage probabilities and expected widths of discrete two-sided tolerance intervals. It is straightforward to alter the formulas for assessing the performance of onesided upper and one-sided lower tolerance limits. The coverage probability of a discrete tolerance interval is the probability that the calculated tolerance interval captures at least a proportion P of the sampled population. Letting p X ( ; θ, n) denote the probability mass function of X, the coverage probability of a (1 α, P ) tolerance interval [L(X; θ L;α (X, n)),u(x; θ U;α (X, n))] controlling for some inner P 100 % of the sampled population is given by Pr X {Pr X {L(X; θ L;α (X, n)) X U(X; θ U;α (X, n)) P X}} { n U(t;θU;α (t,n)) } = p X (t; θ, n)i p X (x ; θ, m) P, t=0 x =L(t;θ L;α (t,n)) (7) where I{ } is the indicator function. oreover, the expected width of this interval is given by n (U(t; θ U;α (t, n)) L(t; θ L;α (t, n)))p X (t; θ, n). (8) t=0 Due to the more stringent requirement for tolerance intervals when controlling for the tails (Step 2) compared to controlling for some inner percentage (Step 2a), the coverage probability of an equal-tailed tolerance interval must be larger than the true coverage probability. Hence, the coverage

7 120 D.S. Young probability of a (1 α, P ) equal-tailed tolerance interval [L e (X; θ L;α (X, n)), U e (X; θ U;α (X, n))] is given by n t=0 { } p X (t; θ, n)i{ L e (t; θ L;α (t, n)) Q((1 P )/2; θ, m) { U e (t; θ U;α (t, n)) Q((1 + P )/2; θ, m)} }, (9) where Q( ; θ, m) is the quantile function for the distribution characterized by F ( ; θ, m). Note that the above expression is the probability with respect to F ( ; θ, n) that the equal-tailed tolerance interval includes the (1 P )/2 th and (1 + P )/2 th quantiles of the distribution characterized by F ( ; θ, m). Analogous to the setting when controlling for an inner percentage, the expected width of the equal-tailed tolerance interval is given by n (U e (t; θ U;α (t, n)) L e (t; θ L;α (t, n)))p X (t; θ, n). (10) t=0 3 Hypergeometric and Negative Hypergeometric Tolerance Intervals Consider a finite universe of N elements with an unknown number having a particular attribute of interest, where 0 N. When sampling n items without replacement from this universe, we let the random variable Y denote the number of elements in the sample that possess the attribute of interest. Thus, the random variable Y follows a hypergeometric distribution with probability mass function ( )( N ) y n y ( p Y (y;,n) = N, for max{n N +,0} y min{,n} n) 0, otherwise, (11) which we write as Y Hyp(n, N, ). Suppose that we again sample without replacement, but until a specified number of elements k, 1 k, having the attribute of interest is observed. Letting the random variable Z denote the number of elements drawn

8 Hypergeometric and negative hypergeometric tolerance intervals 121 until k successes are observed. We then say that the random variable Z follows a negative hypergeometric distribution with probability mass function ( z 1 )( N z ) k 1 k ( p Z (z;,k) = N, for 0 <k z min{n + k, } ) 0, otherwise, (12) which we write as Z NegHyp(k, N, ). Note that the relationship between the negative hypergeometric and the hypergeometric is similar to that between the negative binomial and the binomial; see iller and Fridell (2007) for further discussion. Also, since N and have the same meaning for each distribution, we use the same notation to avoid introducing additional notation. For each distribution, the value is the unknown parameter of interest. Thus, confidence intervals for are necessary to employ the Hahn-Chandra method. For the hypergeometric distribution, let [ L;α (y; n), U;α (y; n)] denote a two-sided 100(1 α)% confidence interval for. For a future sample size m, we find the appropriate integers L(y) andu(y) that satisfy the requirements in Step 2 or Step 2a, which are the limits for a two-sided (1 α, P ) hypergeometric tolerance interval. Similarly, for the negative hypergeometric distribution, let [ L;α (z; k), U;α (z; k)] denote a two-sided 100(1 α)% confidence interval for. For a future number of target successes l, we also find the appropriate integers L(z) andu(z), which are the limits for a two-sided (1 α, P ) negative hypergeometric tolerance interval. Without loss of generality, we will simply assume that m = n and l = k for the remainder of our discussion as this does not affect the overall results from the comparative study Large Sample Intervals. Confidence intervals based on large sample theory can be constructed for both the hypergeometric and negative hypergeometric distributions. To do this, we utilize the fact that the binomial and negative binomial distributions can be used as large sample approximations for the hypergeometric and negative hypergeometric distributions, respectively; see iller and Fridell (2007) for discussion. oreover, it is easier to first re-parameterize so that we can apply standard Wald-type confidence intervals. When using the binomial approximation to the hypergeometric distribution, ˆp = y/n is the maximum likelihood estimate for the proportion p of elements possessing the attribute of interest. oreover, Var(ˆp) =p(1 p)/n.

9 122 D.S. Young Then an estimate for is [N ˆp], where [ ] is the nearest integer function, and a standard large sample 100(1 α)% confidence interval for p is ˆp ± q 1 α/2 ˆp(1 ˆp)/n, (13) where q 1 α/2 is the (1 α/2) th quantile of the standard normal distribution. While there are many other confidence intervals for a binomial proportion (see, for example, Newcombe (1998) and Brown et al. (2001)), our large sample approach will be based on (13). When incorporating a finite population correction factor, we get the following large sample 100(1 α)% confidence interval for : [ ( N ˆp q 1 α/2 ˆp(1 ˆp) n ) ( N n ˆp(1 ˆp), N ˆp + q 1 α/2 N 1 n ) ] N n, N 1 (14) where and are the floor and ceiling functions, respectively. oreover, we can modify the above large sample interval by employing a continuity correction, which results in the following 100(1 α)% confidence interval for : [ ( N ˆp(1 ˆp) N n ˆp q 1 α/2 n N 1 1 ) ( ˆp(1 ˆp) N n, N ˆp+q 1 α/2 2n n N n (15) When using the negative binomial approximation to the negative hypergeometric distribution, ˆν = k/z is the maximum likelihood estimate for the proportion ν of elements possessing the attribute of interest. oreover, Var(ˆν) =ν 2 (1 ν)/z. Then an estimate for is [N ˆν] and a large sample 100(1 α)% confidence interval for ν is ) ] ˆν ± q 1 α/2 ˆν 2 (1 ˆν)/z. (16) Again, many other confidence intervals are available for a negative binomial proportion (see, for example, Tian et al. (2009) and Young (2014)), but our large sample approach will be based on (16). Like the hypergeometric setting, we can incorporate a finite population correction factor to get the following large sample 100(1 α)% confidence interval for : [ ( N ˆν q 1 α/2 ˆν2 (1 ˆν) z ) ( N z ˆν2 (1 ˆν), N ˆν+q 1 α/2 N 1 z N z N 1 ) ]. (17).

10 Hypergeometric and negative hypergeometric tolerance intervals 123 oreover, the 100(1 α)% large sample confidence interval for with a continuity correction is [L 1 (X; θ L1 ;α/2(x + R, n)),u 1 (X; θ U1 ;α/2(x + R, n))]. (18) [ ˆν N( ˆν q 2 (1 ˆν) 1 α/2 z ( ˆν N ˆν + q 2 (1 ˆν) 1 α/2 z N z N 1 1 ), 2z N z N ) ]. (19) 2z Using any of the above set-ups, we can then plug the respective confidence interval into Step 2 of the Hahn-Chandra method to obtain the appropriate (1 α, P ) equal-tailed tolerance interval. oreover, it is easy to obtain onesided confidence limits for and construct one-sided (1 α, P ) tolerance limits similarly Exact Intervals. In this section, we show how to obtain UA (or exact) one-sided upper (1 α, P ) tolerance limits. The construction of onesided lower tolerance limits is completely analogous and so it is enough to consider the one-sided upper setting. We will then use these limits to compute an exact-based two-sided tolerance interval. Consider again the discrete random variable X and the corresponding distribution function F (X; θ, n). Suppose that we are interested in the hypothesis test H 0 : θ = θ 0 (20) H A : θ<θ 0. For size α tests involving discrete distributions, it is usually not possible to choose a critical region consisting of realizations that yield test statistics of size exactly α. However, the theory in Chapter 3 of Lehmann and Romano (2005) shows how any such randomized test based on X has the representation of a nonrandomized test based on X and an independent standard uniform random variable U. Specifically, the statistic T = X + U is equivalent to the pair (X, U), since with probability 1, X = T and U = T T. Thus, the distribution of T is continuous and confidence bounds can be based on this statistic. Wright (1997) applied this approach when studying both randomized and nonrandomized 100(1 α)% confidence bounds when attributes are rare in finite universes. Now, define P as the family of distribution functions F (x; θ, n), where θ Θ; i.e., P = {F (x; θ, n) : θ Θ}. Assume that P is a monotone likelihood ratio family, which means for each θ<θ, p(x; θ, n)/p(x; θ,n)is

11 124 D.S. Young non-decreasing in x. We now state a definition and theorem, both due to Zacks (1970): Definition 1. An upper (1 α, P ) tolerance limit is UA if, subject to Equation (3), it has the optimum property at all θ Θ, that Pr X {U 1 (X) F 1 (P ; θ,m)} is at a minimum for all θ such that F 1 (P ; θ,m) >F 1 (P ; θ, m). Note in Zacks (1970) that m is simply taken to be n and suppressed from the formulas. Using the representation of T defined above, we can state the following theorem: Theorem 1. If P = {F (x; θ, n) :θ Θ} is a monotone likelihood ratio family in x and if θ U1 ;α(x + R, n) isauaupperconfidencelimitforθ at confidence level (1 α), then U 1 (X; θ U1 ;α(x + R, n)) = F 1 (P ; θ U1 ;α(x + R, n),m) (21) is a UA upper (1 α, P ) tolerance limit for P. See Zacks (1970) for the proof. Letting L 1 (X; θ L1 ;α(x + R, n)) denote the analogous UA one-sided lower (1 α, P ) tolerance limit, we can then simply define an exact-based two-sided equal-tailed tolerance interval [L(X; θ L;α (X+ R, n)),u(x; θ U;α (X + R, n))] as [L 1 (X; θ L1 ;α/2(x + R, n)),u 1 (X; θ U1 ;α/2(x + R, n))]. (22) The above theory is developed in the context of randomized bounds. However, nonrandomized bounds are still exact and just as efficacious for our discussion. Thus, we proceed in obtaining nonrandomized bounds and avoid additional computational complexities, such as with the algorithm in Wright (1997). We first note that both the hypergeometric and negative hypergeometric distributions possess the monotone likelihood ratio property (see Appendix A). For the hypergeometric distribution, it is easy to construct a UA upper confidence bound by defining C UH (y) ={ :Pr {Y y} >α} ˆ UH (y; α) = arg max{pr {Y y} >α}, where ˆ UH (y; α) is an exact nonrandomized 100(1 α)% upper confidence bound for. oreover, a UA lower confidence bound is found by defining C LH (y) ={ :Pr {Y y} >α} ˆ LH (y; α) =argmin{pr {Y y} >α},

12 Hypergeometric and negative hypergeometric tolerance intervals 125 where ˆ LH (y; α) is an exact nonrandomized 100(1 α)% lower confidence bound for. Hence, [ ˆ LH (y; α/2), ˆ UH (y; α/2)] would be an exact-based nonrandomized 100(1 α)% confidence interval for. For the negative hypergeometric distribution, we proceed in a similar manner. To construct a UA upper confidence bound, define C UNH (z) ={ :Pr {Z z} >α} ˆ UNH (z; α) = arg max{pr {Z z} >α}, where ˆ UNH (z; α) is an exact nonrandomized 100(1 α)% upper confidence bound for. oreover, a UA lower confidence bound is found by defining C LNH (z) ={ :Pr {Z z} >α} ˆ LNH (z; α) =argmin{pr {Z z} >α}, where ˆ LNH (z; α) is an exact nonrandomized 100(1 α)% lower confidence bound for. Hence, [ ˆ LNH (z; α/2), ˆ UNH (z; α/2)] would be an exactbased nonrandomized 100(1 α)% confidence interval for. As stated at the end of Section 3.1, we can proceed to use any of the above confidence limits (intervals) to compute the appropriate (1 α, P ) tolerance limits (intervals) inimum Coverage Two-Sided Tolerance Intervals. As Krishnamoorthy et al point out, it is often unnecessary to control the tail percentages in practical applications (e.g., discrete quality assessment). Instead, controlling for some inner percentage of the sampled population would suffice. This amounts to Step 2a in the Hahn-Chandra method. For the binomial and Poisson distributions, Wang and Tsung (2009) provided a numerical approach to find the value α so that an exact (1 α, P ) tolerance interval will have minimum coverage probability close to the nominal level. Empirical work in Krishnamoorthy et al. (2011) regarding the binomial and Poisson distributions found that using (1 2α, P ) equal-tailed tolerance intervals yielded (minimum) coverage probabilities close to the nominal level of (1 α). In the hypergeometric and negative hypergeometric settings, we do not develop a formal numerical approach like that in Wang and Tsung (2009) for determining α. We are simply applying what Krishnamoorthy et al. (2011) suggested for α in the binomial and Poisson settings. This adjustment is incorporated into our performance study in the next section. Also, to avoid confusing the exact terminology of Wang and Tsung (2009) with the exact terminology of the intervals in the previous subsection, we will henceforth refer to the former as a coverage-adjusted approach.

13 126 D.S. Young 4 Performance Comparisons For comparing the performance of the methods discussed in Section 3, we considered the following conditions. For the hypergeometric distribution, we considered N {100, 500} and n {0.20N,0.50N}. Hence, coverage probabilities and expected widths were calculated for =0, 1,...,N. For the negative hypergeometric distribution, we again considered N {100, 500}, but with k {0.50N,0.75N}. Hence, coverage probabilities and expected widths were calculated for = k, k +1,...,N. In practice, common values for (1 α) andp are taken from the set {0.90, 0.95, 0.99}. We present results for (0.95, 0.90) tolerance limits and intervals for both the hypergeometric and negative hypergeometric distribution. We also ran limited studies at the (0.85, 0.90) and (0.95, 0.95) levels, which demonstrated similar results as to what we present for the (0.95, 0.90) setting. The figures we present for assessing the coverage probabilities and expected widths have some common characteristics that are worth noting. First is the oscillatory behavior that occurs with the coverage probabilities. This is typical for distributions that have a lattice structure, like the hypergeometric and negative hypergeometric. This has also been noted in other studies on statistical intervals for discrete distributions, such as in Brown et al. (2001), Cai and Wang (2009), and Young (2014). For our study, the oscillatory patterns become more distinct as N increases, while also the coverage probabilities tend toward the nominal level for values of not near the extremes. The expected widths for the one-sided tolerance limits were all very close to each other, regardless of the method employed. As such, we provided only a brief summary in Table 1 for the Hyp(50, 100,)and NegHyp(250, 500,) settings and for four select values of. These results were typical regardless of the conditions. Figure 1 shows the coverage probabilities for the one-sided hypergeometric tolerance limits. Coverage probabilities for the large sample (LS) Table 1: Some expected widths of one-sided upper (0.95,0.90) tolerance limits for the large sample (LS), continuity correction (CC), and exact (EX) methods. Hyp(50, 100,) NegHyp(250, 500,) LS CC EX LS CC EX

14 Hypergeometric and negative hypergeometric tolerance intervals 127 N = 100, n = 20 N = 100, n = 50 LS CC EX LS CC EX (a) (b) N = 500, n = 100 N = 500, n = 250 LS CC EX LS CC EX (c) (d) Figure 1: Coverage probabilities for the one-sided hypergeometric tolerance limits. The solid lines ( ) are for the LS method, the dashed lines ( ) are for the CC method, and the dotted lines ( )arefortheex method. and exact (EX) method appear to be similar, while the continuity correction (CC) method tends to be slightly more conservative. As N increases, all three methods are closer to the nominal level (0.95); however, there are some higher values of where the EX method tends to be less conservative than the other methods. Given their similar performance with respect to expected widths, one could reasonably use either the LS or EX methods, but there is a slightly better performance with the EX method for larger values of. Figure 2 shows the coverage probabilities for the two-sided hypergeometric tolerance intervals as well as when the coverage adjustment is applied.

15 128 D.S. Young N = 100, n = 20 (Large Sample) N = 500, n = 250 (Large Sample) (a) (b) N = 100, n = 20 (Continuity Correction) (c) N = 100, n = 20 (Exact) N = 500, n = 250 (Continuity Correction) (d) N = 500, n = 250 (Exact) (e) (f) Figure 2: Coverage probabilities for the two-sided hypergeometric tolerance intervals. The conditions used to generate the coverage probabilities are above each figure. The solid lines ( ) are for the original calculation and the dashed lines ( ) are for the coverage adjustment.

16 Hypergeometric and negative hypergeometric tolerance intervals 129 For N = 100, we see that the EX method actually is conservative relative to the other two methods. However, for N = 500, it appears to be performing closer to nominal. When comparing whether or not the coverage adjustment is applied, we observe little benefit for the case of N = 100; however, for N = 500 there appears to be a trade-off when the adjustment is not applied. Namely, there appears to be intervals of where the coverage probabilities improve with the adjustment, and intervals of where it does not. However, when we look at the expected widths of each setting in Figure 3, the coverage adjustment yields unanimously narrower intervals - especially for N = 100, n = 50 N = 100, n = 20 Expected Width LS CC EX Expected Width LS CC EX (a) (b) N = 500, n = 250 N = 500, n = 100 Expected Width LS CC EX Expected Width LS CC EX (c) (d) Figure 3: Expected widths for the two-sided hypergeometric tolerance intervals. Shading for the LS method, CC method, and EX method is noted on each figure. The solid lines ( ) are for the original calculation and the dashed lines ( ) are for the coverage adjustment.

17 130 D.S. Young values of away from the extremes. Hence, the LS and EX method with a coverage adjustment appear to have the better performance. Figure 4 shows the coverage probabilities for the one-sided negative hypergeometric tolerance limits. Coverage probabilities for the EX method appear to be closer to nominal relative to the other two methods. As N and k increase, all three methods show signs of stabilizing closer to the nominal level. As noted earlier, all three methods perform similarly with respect to expected widths, just like in the hypergeometric setting. Given these results, the EX method appears to have the better performance. N = 100, k = 50 N = 100, k = 75 LS CC EX LS CC EX (a) (b) N = 500, k = 250 N = 500, k = 375 LS CC EX LS CC EX (c) (d) Figure 4: Coverage probabilities for the one-sided negative hypergeometric tolerance limits. The solid lines ( ) are for the LS method, the dashed lines ( ) are for the CC method, and the dotted lines ( )arefortheex method.

18 Hypergeometric and negative hypergeometric tolerance intervals 131 Figure 5 shows the coverage probabilities for the two-sided negative hypergeometric tolerance intervals as well as when the coverage adjustment is applied. For N = 100, we see that the CC and EX methods are conservative, but the LS method is much more variable about the nominal level. However, for N = 500, the EX method performs closer to nominal relative to the LS and CC methods. When comparing whether or not the coverage adjustment is applied, we see that there is usually some improvement for most values of when it is applied. oreover, when we look at the expected widths of each setting in Figure 6, the coverage adjustment again yields unanimously narrower intervals - especially for values of away from the extremes. Hence, the EX method with a coverage adjustment appears to have the better performance. 5 R Functions and Examples The R package tolerance (Young, 2010) includes tools for estimating tolerance limits of various data structures, such as data from: continuous distributions (e.g., normal, Weibull, and Cauchy); discrete distributions (e.g., binomial, Poisson, and negative binomial); regression settings (e.g., linear regression, nonlinear regression, and nonparametric regression); and multivariate settings (e.g., multivariate normal and multivariate linear regression). The tolerance package now includes functions that compute the hypergeometric and negative hypergeometric tolerance intervals discussed in Section 3. They are the hypertol.int and neghypertol.int functions, respectively. Both functions have the same arguments, but some of them depend on the distribution. The arguments are summarized in Table 2. Further details on these functions can be found by typing?hypertol.int and?neghypertol.int in R. Example 1. (One-Sided Hypergeometric Tolerance Limits) Consider a manufacturing setting where a company needs to purchase plastic fasteners in bulk. The company receives N = 5000 fasteners in a given lot. When sampling n = 1000 fasteners without replacement, the company found y =15 defective units. Due to the cost of the sampling procedure, the company will reduce future inspections to m = 100 fasteners sampled without replacement. They further need to specify a one-sided upper (0.95, 0.99) tolerance limit

19 132 D.S. Young N = 100, k = 50 (Large Sample) N = 500, k = 375 (Large Sample) (a) (b) N = 100, n = 50 (Continuity Correction) N = 500, k = 375 (Continuity Correction) N = 500, k = 375 (Exact) (c) (d) N = 100, k = 50 (Exact) (e) (f) Figure 5: Coverage probabilities for the two-sided negative hypergeometric tolerance intervals. The conditions used to generate the coverage probabilities are above each figure. The solid lines ( ) are for the original method and the dashed lines ( ) are for the coverage-adjusted method.

20 Hypergeometric and negative hypergeometric tolerance intervals 133 N = 100, k = 75 N = 100, k = 50 Expected Width LS CC EX Expected Width LS CC EX (a) (b) N = 500, k = 250 N = 500, k = 375 Expected Width LS CC EX Expected Width LS CC EX (c) (d) Figure 6: Expected widths for the two-sided negative hypergeometric tolerance intervals. Shading for the LS method, CC method, and EX method is noted on each figure. The solid lines ( ) are for the original calculation and the dashed lines ( ) are for the coverage adjustment. based on this information for determining when to accept/reject the lot of fasteners. This is found using the hypertol.int function as follows: > hypertol.int(x = 15, n = 1000, N = 5000, m = 100, alpha = 0.05, P = 0.99, side = 1, method = "EX") alpha P rate p.hat 1-sided.lower 1-sided.upper Thus, the company can be 95% confident that at least 99% of all lots will have no more than 6 defects in a future sample of 100 fasteners. Note that the value of y from our earlier formulas corresponds to the x argument in the

21 134 D.S. Young Table 2: Arguments for the hypertol.int and neghypertol.int functions. Argument hypertol.int neghypertol.int x number of units with attribute total sample drawn to achieve in sample n units with attribute n size of the sample drawn target number of units with attribute to draw m size of a future sample future target number of units with attribute to draw N alpha P side method population size level of the test the content numeric argument taking 1 or 2 for one-sided limits or a two-sided interval an argument to specify the large sample method ("LS"), the continuity correction method ("CC"), or the exact method ("EX") R function. The above result is for the EX method. By changing the method argument, we can obtain the results for the LS and CC methods. For this example, the one-sided upper (0.95, 0.99) tolerance limit is also 6 for the LS and CC methods. Example 2. (Hypergeometric Tolerance Intervals) Vener et al. (1993) analyzed data from grant applications submitted to the National Cancer Institute in response to a February 1993 request for applications. The applications went through a triage at the National Institutes of Health. 21 members comprised a full committee, from which 5 members formed a subcommittee to review an individual application. A total of 73 applications went through this process. Each application reviewed was assigned a competitiveness score by each member, which resulted in it being classified as competitive or noncompetitive. If at least two of the five subcommittee members voted for a grant application as competitive, then it was sent to the full committee of 21 members for further review. Otherwise, the grant application was rejected. The top half of Table 3 gives the data from this triage process. Vener et al. (1993) built a hypergeometric model to estimate probabilities of possible dispositions of grant applications as a result of this triage process. Clearly, there are some limitations and assumptions made. For example, Vener et al. (1993) assumed that the reviewers were of roughly equal ability

22 Hypergeometric and negative hypergeometric tolerance intervals 135 Table 3: The peer review triage data of Vener et al. (1993) and the coal tit data of Ridiout (1999). # of Competitive Votes Frequency # of Feeders Visited Frequency and were fairly homogenous (i.e., in the long run they would each accept or reject the same percentage of applications). They also assumed that the full committee represented a gold standard for the review process. oreover, there were some subsequent decision rules after triage that fed into their overall model and analysis. For the purposes of our example, we will treat the aggregated triage data as a realization from a hypergeometric distribution. A total of y = 231 competitive votes were cast by the subcommittees for the 73 proposals, which is calculated using the data in the top half of Table 3. Thus, there were a possible n = 365 (5*73) potential competitive votes from the subcommittees. The committee (a finite population) of 21 would have the possibility of casting a total of N = 1533 (21*73) competitive votes. Using a hypergeometric distribution and clearly acknowledging the assumptions as stated in Vener et al. (1993), a(0.90, 0.90) tolerance interval is calculated as follows: > hypertol.int(x = 231, n = 365, N = 1533, m = 21, alpha = 0.10, P = 0.90, side = 2, method = "EX") alpha P rate p.hat 2-sided.lower 2-sided.upper Thus, with 90 % confidence, we would expect at least 90 % of the proposals reviewed by a full committee of m =21to have between 9 and 17 competitive votes. The above result is for the EX method, but the same tolerance interval is also obtained for the LS and CC methods. Example 3. (One-Sided Negative Hypergeometric Tolerance Limits) Ridiout (1999) analyzed data from an experiment where the memory in coal tits (a small bird found primarily throughout temperate Eurasia) was studied. The birds were released into a room that contained four feeders, of which only one contained food that was visible to the bird. The bird was removed from the room and then returned 15 minutes later. The feeders remained the

23 136 D.S. Young same, but the food in the filled feeder was hidden. The number of distinct feeders visited by each bird was recorded. A total of 19 birds were in the experiment, such that each bird was used 5-15 times, resulting in 207 different trials. There were 20 instances where the bird gave up searching for the food and, thus, were censored observations. Jolliffe and Jolliffe (1997) applied the E algorithm to incorporate the incomplete data and estimate different models for the probability distribution of the number of looks. Ridiout (1999) used a (generalized) negative hypergeometric distribution to estimate the probabilities. For our purposes, we will focus on the complete-data portion (i.e., 187 trials) of the analysis in Ridiout (1999). These data are given in the bottom half of Table 3. There are again some limitations and assumptions made. For example, the negative hypergeometric model in Ridiout (1999) ignored the fact that all of the data arose from multiple measurements on 19 birds. There could also be, say, a learning or fatigue effect that occurs across these multiple trials. For the purposes of our example, we will treat the aggregated coal tit data as a realization from a negative hypergeometric distribution. A total of z = 306 visits were made to the feeders for the 187 (complete) trials, which is calculated using the data in the bottom half of Table 3. Since we are interested in when the coal tit arrives at the feeder with food, the total number of successes is k = 187. Thus, there were a possible N =748visits to feeders when aggregating across all of the birds. Using a negative hypergeometric distribution and clearly acknowledging the assumptions as stated in Ridiout (1999), we are interested in finding a one-sided upper (0.85, 0.90) tolerance limit for the total number of feeders visited by a bird; i.e., m =1. This is found using the neghypertol.int function as follows: > neghypertol.int(x = 306, n = 187, N = 748, m = 1, alpha = 0.15, P = 0.90, side = 1, method="ex") alpha P rate p.hat 1-sided.lower 1-sided.upper Thus, with 85% confidence, we would expect at least 90% of the birds to visit no more than 3 feeders in total to find the one with the food. Note that the values of k and z from our earlier formulas correspond to the n and x arguments, respectively, in the R function. The above result is for the EX method, but the one-sided upper (0.85, 0.90) tolerance limit is also 3 for the LS and CC methods. Example 4. (Negative Hypergeometric Tolerance Intervals) We next consider the example in Zhang and Johnson (2011) for planning a sample survey

24 Hypergeometric and negative hypergeometric tolerance intervals 137 that utilizes random digit dialing. Consider a sampling frame that is a list of both residential and non-residential telephone numbers. Suppose that a researcher has banks of N = 100 telephone numbers (the sampling frame), from which they randomly sample one-at-a-time a sequence of telephone numbers (the primary sampling units). From a previous bank of 100, the researcher found that z =21calls were necessary until k =15residential numbers were reached. For workload planning purposes, the researcher would like to know with 95% confidence, the total number of calls necessary to reach their target for 85% of all such lists. A (0.95, 0.85) tolerance interval is calculated as follows: > neghypertol.int(x = 21, n = 15, N = 100, m = 15, alpha = 0.05, P = 0.85, side = 2, method = "EX") alpha P rate p.hat 2-sided.lower 2-sided.upper Thus, with 95% confidence, the researcher can expect that at least 85% of all lists will require between 15 and 36 total calls to make contact with 15 residential units. The above result is for the EX method. For this example, the (0.95, 0.85) tolerance intervals for the LS and CC methods are [15, 32] and [15, 34], respectively. 6 Discussion In this paper, we have provided a rigorous development of constructing one-sided tolerance limits and two-sided tolerance intervals for hypergeometric and negative hypergeometric variables. The construction of such tolerance limits when sampling without replacements has not been handled in the literature. For one-sided tolerance limits and two-sided equal-tailed tolerance intervals, we applied the approach of Hahn and Chandra (1981). We also leveraged the numerical results of Krishnamoorthy et al. (2011) (based on the methodology of Wang and Tsung (2009)) to provide a coverage adjustment to the equal-tailed tolerance intervals as a way to estimate two-sided tolerance intervals that control an inner percentage of the sampled population. The tolerance limits in all of these procedures depend on confidence intervals for, the unknown number of elements possessing an attribute of interest in the population. We compared their performance based on three different approaches: a large sample approach, an approach with a continuity correction, and an exact method based on nonrandomization. From our comparisons, we found that the exact method typically performs better with respect to coverage probabilities and expected widths. We have also included

25 138 D.S. Young functions for computing these tolerance limits in the R package tolerance (Young, 2010). While we demonstrated the relative performance of the intervals between the different methods, we note that future research could be done to further improve coverage probabilities, especially for smaller N. odificationstothe probability-matching approach of Cai and Wang (2009) might be possible and could potentially improve coverage probabilities for the hypergeometric and negative hypergeometric tolerance intervals discussed here. However, we note that neither distribution belongs to the natural discrete exponential family and, hence, the approach of Cai and Wang (2009) does not directly apply. oreover, their probability-matching approach involves fairly complex forms and are only fully-developed for the one-sided setting given the difficulty of extending the methodology to the two-sided setting. Thus, the trade-off between the complexity with such an approach and the gains in performance measures on the intervals would need to be closely considered. Acknowledgements. We are grateful to three anonymous referees and an Associate Editor for numerous helpful comments during the preparation of this article. We would also like to thank Thomas athew for some suggestions on an earlier version of this work. References brown, l.d., cai, t.t. and dasgupta, a. (2001). Interval estimation for a binomial proportion. Statist. Sci. 16, cai, t.t. and wang, h. (2009). Tolerance intervals for discrete distributions in exponential families. Statist. Sinica 19, eichenberger, p., hulliger, b. and potterat, j. (2011). Two measures for sample size determination. Survey Research ethods 5, guenther, w.c. (1975). The inverse hypergeometric - a useful model. Stat. Neerl. 29, hahn, g.j. and chandra, r. (1981). Tolerance intervals for Poisson and binomial random variables. Journal of Quality Technology 13, hahn, g.j. and meeker, w.q. (1991). Statistical Intervals: A Guide for Practitioners. Wiley-Interscience, New York. johnson, n.l., kemp, a.w. and kotz, s. (2005). Univariate Discrete Distributions, 3rd edn. Wiley, Hoboken. jolliffe, i.t. and jolliffe, a.r. (1997). odelling memory in coal tits: An illustration of the E algorithm. Biometrics 53, krishnamoorthy, k. and mathew, t. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. Wiley, Hoboken. krishnamoorthy, k., xia, y. and xie, f. (2011). A simple approximate procedure for constructing binomial and Poisson tolerance intervals. Comm. Statist. Theory ethods 40, lehmann, e.l. and romano, j.p. (2005). Testing Statistical Hypotheses. Springer.

26 Hypergeometric and negative hypergeometric tolerance intervals 139 mathew, t. and young, d.s. (2013). Fiducial-based tolerance intervals for some discrete distributions. Comput. Statist. Data Anal. 61, miller, g.k. and fridell, s.l. (2007). A forgotten discrete distribution? Reviving the negative hypergeometric model. Amer. Statist. 61, montgomery, d.c. (2013). Introduction to Statistical Quality Control, 7th edn. Wiley, New Jersey. newcombe, r.g. (1998). Two-sided confidence intervals for the single proportion: Comparison of seven methods. Stat. ed. 17, r development core team (2013) R: A Language and Environment for Statistical Comridiout, m.s. (1999). emory in coal tits: An alternative model. Biometrics 55, puting, Vienna. ISBN tian, m., tang, m.l., ng, h.k.t. and chen, p.s. (2009). A comparative study of confidence intervals for negative binomial proportions. J. Stat. Comput. Simul. 79, vener, k.j., feuer, e.j. and gorelic, l. (1993). A statistical model validating triage for the peer review process: Keeping the competitive applications in the review pipeline. The Federation of American Societies for Experimental Biology Journal 7, wang, h. and tsung, f. (2009). Tolerance intervals with improved coverage probabilities for binomial and Poisson variables. Technometrics 51, wilks, s.s. (1941). Determination of sample sizes for setting tolerance limits. The Annals of athematical Statistics 12, wilks, s.s. (1942). Statistical prediction with special reference to the problem of tolerance limits. The Annals of athematical Statistics 13, wright, t. (1997). A simple algorithm for tighter exact upper confidence bounds with rare attributes in finite universes. Statist. Probab. Lett. 36, young, d.s. (2010). tolerance: An R package for estimating tolerance intervals. Journal of Statistical Software 36, young, d.s. (2014). A procedure for approximate negative binomial tolerance intervals. J. Stat. Comput. Simul. 84, zacks, s. (1970). Uniformly most accurate upper tolerance limits for monotone likelihood ratio families of discrete distributions. J. Amer. Statist. Assoc. 65, zaslavsky, b.g. (2007). Calculation of tolerance limits and sample size determination for clinical trials with dichotomous outcomes. J. Biopharm. Statist. 17, zhang, l. and johnson, w.d. (2011) Approximate confidence intervals for a parameter of the negative hypergeometric distribution. In Proceedings of the Section on Survey Research ethods, pages American Statistical Association. Appendix A onotone Likelihood Ratio Property For a likelihood function L(θ; X), define Λ(θ 1,θ 2 ; X) = L(θ 1; X) L(θ 2 ; X) (A.1) to be the ratio between the likelihood evaluated at θ 1 and θ 2,whereθ 1 θ 2.