1
$\begingroup$

Consider that $P$ is the water pressure coming out of a valve $A$. Let $P_{dif}$ be the difference between the maximum and the minimum pressure of valve $A$:

$$P_{dif}≔P_{max}-P_{min}$$

Now, what I want to do is estimate $P_{dif}$. In order to do that, I take a number of water pressure samples from valve $A$.

Let $S$ be a set of $3$ measured samples:

$$S = {5,7,1}$$

That is, $S$ contains 3 random samples of $P$, therefore, by placing S in ascending order, I can estimate $P_{dif}$ like that:

$$\hat{P}_{dif} = S_{(n:3)}-S_{(n:1)} = 7-1 = 6$$

Questions:

  • What is the probability of exceeding this estimation ($\hat{P}_{dif}$). That is, what is the probability that the population parameter ($P_{dif}$) will exceed the estimation ($\hat{P}_{dif}$).
  • Reliability of the estimation ($\hat{P}_{dif}$): this refers to the probability that the estimation is wrong (it may be possible to find this by deriving the confidence interval of the sample range, but I don't know how).

Any help towards this will be greatly appreciated.

Improve this question
$\endgroup$
5
  • $\begingroup$ The population parameter is the greatest possible difference in water pressures that could be measured. Under many reasonable assumptions, it is virtually certain that the range in any finite set of measurements will be less than this value. If you wish to derive a confidence interval, you need to supply additional assumptions about the distribution of water pressures. $\endgroup$
    – whuber ♦
    Commented Mar 4, 2013 at 20:14
  • $\begingroup$ @whuber: What sort of assumptions do I need to supply? Would it help if I add a figure of the water pressure samples (they are actually more than 3, however I've mentioned just 3 in the description to make it more readable)? Regarding the probability of exceedance, do you have any thoughts? Thanks! $\endgroup$
    – limp
    Commented Mar 4, 2013 at 21:44
  • $\begingroup$ You need to make assumptions about the upper and lower tails of the distribution of water pressures (as well as assuming that the true range $P_{dif}$ is finite). $\endgroup$
    – whuber ♦
    Commented Mar 4, 2013 at 21:46
  • $\begingroup$ Do you suggest that I need to create a kernel density transformed representation from the measured $P$ values? If we assume that the range of $P_{dif}$ refers to a specific time interval (e.g. 1 year), and that we are interested in the water pressure only at the time of sampling, doesn't this make it finite? When you say that I need to make assumptions about the tails of the distribution, could you please be a bit more specific? $\endgroup$
    – limp
    Commented Mar 4, 2013 at 22:28
  • $\begingroup$ No, I'm not suggesting a KDE, because that means you are assuming the tails have the shape of the kernel. "Tails" is pretty specific: have a look at extreme value theory and search the extreme-value tag for relevant examples. $\endgroup$
    – whuber ♦
    Commented Mar 4, 2013 at 22:36

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.