One-Order-Statistic Conditional Estimators of Shape Parameters of Limited and Pareto Distributions and Scale Parameters of Type II Asymptotic Distributions of Smallest and Largest Values

One-order-statistic estimators are derived for the shape parameter K of the limited distribution function F1(x, ?, K) = 1 - (? - x)K and the Pareto distribution function F2(y, ?, K) = 1 - (y - ?)-K, given the location parameters ? and ?, respectively. Similar estimators are derived for the scale parameters v1 and Vn, of the Type II asymptotic distributions of smallest and largest values, F3(w, v1, K) = 1 - exp-(w/v1)-K] and F4(z, vn K) = exp -(z/vn)-K], given the shape parameter K and assuming the location parameter is zero. The one-order-statistic estimators are K?|? = -1/Cmn 1n(? - xmn) for the limited distribution, K?|? = 1/Cmn 1n(ymn - ?) for the Pareto distribution, ?1|K = Cmn-1/K Wmn and ?n|K = Cmn-1/K Zn-m+1,n for the Type II distributions of smallest and largest values, where Xmn, Ymn, Wmn, Zmn are the mth order statistics of samples of size n from the respective distributions and Cmn is the coefficient for a one-order-statistic estimator of the scale parameter of an exponential distribution, which has been tabled in an earlier paper. It is shown that exact confidence bounds can be easily derived for these parameters using exact confidence bounds for the scale parameter of the exponential distribution. Use of the estimators is illustrated by numerical examples.