Consider a sample X-1,...,X-n from an exponential distribution with un
known parameter theta. From the sample, we wish to estimate the entire
distribution, {epsilon(theta)(A) = integral(A) theta e(-theta x) dx,
A is an element of B(Borel sets)}. If an estimator epsilon(<(theta)ove
r cap>) is used for epsilon(<(theta)over cap>) we are concerned with p
roximity of epsilon(<(theta)over cap>) to epsilon(theta), under total
variation distance, D(epsilon(<(theta)over cap>),epsilon(theta)) = sup
(A is an element of B)\epsilon(theta)(A) - epsilon(<(theta)over cap>)(
A)\. For <(theta)over cap (n)> = ((X) over bar)(-1), the maximum likel
ihood estimate of theta, define d(n,1-alpha) = e(-1)((Z/n(1/2)) + [(Z
(5) - Z(3) + 6Z)/72n(3/2)]), where Z = Z(1-(alpha/2)), the 1 - (alpha/
2) percentile from the standard normal. Then, P(D(epsilon(<(theta)over
cap>n),epsilon(theta)) less than or equal to d(n,1-alpha)) = (1 - al
pha) + o(n(-3/2)). The preceding approximation to the 1 - alpha percen
tile of D (epsilon(<(theta)over cap>n), epsilon(theta)) is very accura
te even for small n. We also consider the problem of obtaining a confi
dence band for the survival function, possessing minimal maximum width
. Finally, a class of estimators of epsilon(theta) is compared to the
maximum likelihood estimator from the viewpoint of total variation dis
tance loss function.