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 F
₁
(x, ω, K) = 1 - (ω - x)
^K
and the Pareto distribution function F
₂
(y, ϵ, K) = 1 - (y - ϵ)
^-K
, given the location parameters ω and ϵ, respectively. Similar estimators are derived for the scale parameters v
₁
and V
_n
, of the Type II asymptotic distributions of smallest and largest values, F
₃
(w, v
₁
, K) = 1 - exp[-(w/v
₁
)
^-K
] and F
₄
(z, v
_n
K) = exp [-(z/v
_n
)
^-K
], given the shape parameter K and assuming the location parameter is zero. The one-order-statistic estimators are K|ω = -1/C
_mn
1n(ω - x
_mn
) for the limited distribution, K|ϵ = 1/C
_mn
1n(y
_mn
- ϵ) for the Pareto distribution, v
₁
|K = C
_mn
^-1/K
W
_mn
and v
_n
|K = C
_mn
^-1/K
Z
_n-m+1,n
for the Type II distributions of smallest and largest values, where X
_mn
, Y
_mn
, Wmn, Z
_mn
are the mth order statistics of samples of size n from the respective distributions and C
_mn
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.