I am looking for an interval to contain a fixed central proportion, for
example, P = 95%, of a bivariate distribution, for example, linear regression, at a
given significance level, for example, alpha = .95, where the intervals are
simultaneous in both the independent variable and the fixed central proportion.
I believe this is a tolerance interval on the distribution of future
observations, the intervals being probabilistically simultaneous in possible values of
the independent variable, where future values of the dependent variable are
based on the same linear regression model, and that have the property that the
interval contains 100%P of the normal distribution that is centered at B + Bx
for any x and any or fixed P with confidence coefficient 1 - alpha.
In terms of prediction, a future value of the dependent variable, given the
independent variable x, should lie in this interval with probability at least P
to the sampling distribution of future observations.
Formally, the confidence 1 - alpha refers to the sample from which the
regression line is estimated and the probability at least P to the sampling
distributions of future observations. If for a single regression line one asserts the
the proportion of future observations falling within the given tolerance
limits, for any x, is at least P, and similar statements are made repeatedly for
different regression lines, then for 100(1 - alpha)% of the different
regression lines the statements will be correct.
Can anyone direct me to literature or text that describes how to estimate
this interval? Early 1960ish literature acknowledge this question but do not
provide solutions.
Prediction intervals for the next observation, or the next k observations, or
the next k of m observations, or all future observations are not appropriate
because prediction intervals are not independent and the number of future
observations is unknown and an exact probability statement is not possible.
Thank you.
Stan Alekman
Let charm be thy spear, wit thine armor.
Stanley L. Alekman PhD
Pharmaceutical Consultant
S.L. Alekman Associates, Inc.
Inverness, Illinois
847-358-6975
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