Minitab calculates exact (1 – α, P) nonparametric tolerance intervals, where 1 – α is the confidence level and P is the coverage (the target minimum percentage of population in the interval). The nonparametric method for tolerance intervals is a distribution free method. That is, the nonparametric tolerance interval does not depend on the parent population of your sample. Minitab uses an exact method for both one-sided and two-sided intervals.
Let X 1, X 2 , ... , X n be the ordered statistics based on a random sample from some continuously distributed population F(x;θ). Then, based on the findings of Wilks1, 2 and Robbins3, it can be shown that:
where B denotes the cumulative distribution function of the beta distribution with parameters a = r and b = n – s + 1. Thus ( Xr , Xs ) is a distribution-free tolerance interval because the coverage of the interval has a beta distribution with known parameter values, which are independent of the distribution of the parent population, F(x;θ).
Let k be the largest integer that satisfies the following:
where Y is a binomial random variable with parameters n and 1 – P. It can be shown (see Krishnamoorthy and Mathew4) that a one-sided (1 – α, P) lower tolerance bound is given by Xk . Similarly, a one-sided (1 – α, P) upper tolerance bound is given by X n - k +1. In both cases, the actual or effective coverage is given by P(Y > k).
Let k be the smallest integer that satisfies the following:
where V is a binomial random variable with parameters n and P. Thus,
where F V -1(x) is the inverse cumulative distribution function of V. It can be shown (see Krishnamoorthy and Mathew4) that a two-sided (1 – α, P) tolerance interval may be given as ( Xr , Xs ). Minitab chooses s = n - r + 1 so that r = ( n – k + 1) / 2. Both r and s are rounded down to the nearest integer. The actual or effective coverage is given by P(V < k – 1).
|1 – α||the confidence level of the tolerance interval|
| P ||the coverage of the tolerance interval (the target minimum percentage of population in the interval)|
| n ||the number of observations in the sample|
- Wilks, S. S. (1941). Sample size for tolerance limits on a normal distribution. The Annals of Mathematical Statistics, 12, 91–96.
- Wilks, S. S. (1941). Statistical prediction with special reference to the problem of tolerance limits. The Annals of Mathematical Statistics, 13, 400–409.
- Robbins, H. (1944). On distribution-free tolerance limits in random sampling. The Annals of Mathematical Statistics, 15, 214–216.
- Krishnamoorthy, K. and Mathew, T. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. Wiley, Hoboken, NJ.