Re: Some Paradox on Confidence Interval on Population Parameter!
- From: "MathCraft Consulting" <mathcraft@xxxxxxxxxxxxx>
- Date: Thu, 24 May 2007 02:11:52 GMT
What is it you are actually trying to measure?
If you are simply measuring the Yi on any given day *AND* if Xi is being
counted without error, then Yi is indeed your exact average failure rate. It
will change, of course, each day. But, on any given day, you will have the
exact average for that day. No confidence interval is involved.
On the other hand, if what you are really trying to measure is the
theoretical long-term average failure rate, then each day is only a sample.
Yi is NOT a population parameter in that case, because the "population"
parameter would be the average Yi measured over the entire population of
days, whatever that happened to be. If you assume that the underlying
failure rate is not changing over time (at least, not over the time that you
are collecting data), and that failures on one day do not depend upon
failures on any other day, then you might assume that the number of failures
in any given day follows a Poisson distribution, and what you are really
trying to estimate is the Poisson parameter lambda for the daily number of
failures. You certainly have an appropriate confidence interval in that
case, which will depend upon how many days you have measured.
If you have to worry about changing -- perhaps increasing -- failure rate
over the time you are collecting data, you would need a different
statistical model than Poisson. Also... if there is the possibility that Xi
may involve error, then Yi is again uncertain, and any confidence interval
would be affected by that.
MathCraft Consulting
Dayton, Ohio
"undiscern" <undiscern@xxxxxxxxx> wrote in message
news:1179795706.860642.172310@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
parameter the probability of failure on a particular day or on anyIs it the traffic lights on a particular day? Or the traffic lights on
any day? Is the
day?
Thank you. But is there any difference? The number of traffic lights
in a city is a constant N.
I have failure data for everyday and am tracking the Rate of change in
Yi. Yi = Xi/N
GRAPH 1
-------
SUM *
Yi | * *
| * *
| *
| *
|
| *
|*
|______________________________
1 2 3 4 5 6 7 8 9 10 11 12 ... (Calendar Day 2005)
GRAPH 2
-------
Ri,
Rate of Change of
Yi |
|
|
|
|
|
| * * *
|* * **
| ***
|______________________________
1 2 3 4 5 6 7 8 9 10 11 12 ... (Calendar Day 2005)
Frequentist
------------
Since Yi is the population parameter for day i, it is a constant
according to frequentist. Hence there should not be any confidence
interval on CI.
Since Yi has no interval, Ri has no interval too?
If Ri has no interval, how do we know that there is indeed a shift in
Ri?
Bayesian
---------
Yi though is a population parameter, it is not a constant and has a
distribution. For example if Y1 ~ N (2,1) Y2 ~ N(3, 1.2) ...
Implying Graph 1 plot of Yi is the mean of its distribution. Therefore
the 95% confidence interval for Y1 is 2 +- 1.96 Sqrt(1/M).
Is M = N???
.
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