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Consider that $P$ is the water pressure coming out from a valve A. Let $P_{dif}$ be defined as the difference between the maximum and the minimum pressure of valve A: $$P_{\text{dif}}:= P_{\text{max}} - P_{\text{min}}$$ Now, what I want to do is to estimate $P_{\text{dif}}$. 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: $$S = \{X_1 = 5, X_2 = 7, X_3 = 1\}$$ That is, $S$ contains 3 random samples drawn from the population. Therefore, I then say that $\hat{P}_{\text{dif}} = 7 – 1 = 6$.

First question: Considering a Gaussian distribution of the population parameter, how can I find the pdf, cdf of the parameter?

Second question: Would a $95\%$ confidence interval (CI) of my estimation be defined as following?: $$(pdf, mean) - 2\cdot (pdf, std) \leq (pdf, mean) \leq (pdf, mean) + 2\cdot (pdf, std) $$ where (pdf, mean) is the mean of the pdf and (pdf, std) its standard deviation.

If yes, how can I derive the (pdf, mean) and the (pdf, std)?

Edit:

As it seems quite difficult for me to find the CI of the range estimation, I am wondering whether I could use an alternative approach. whuber suggested that successive estimate should not depend strongly on the preceding ones. Say that I split my data samples into equal parts (e.g. 3) and I find one estimate for each of these sub-sets. Next, I find the 95% CI of the $\hat{P}_{dif}$ by that: $$ mean - 2\cdot std \leq mean \leq mean + 2\cdot std $$ where mean and std are the mean and std of the 3 different estimates, respectively. Would that be correct?

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  • $\begingroup$ It's difficult to give specific advice without having more information about your "estimations." Are they based on independent sets of data or are they perhaps different ways to estimate the same parameter from the same data? And what specifically is the difference between "finding the ecdf ... for a normal distribution" and then computing its mean and SD, when usually one fits values to a Normal distribution by computing the mean and SD in the first place? $\endgroup$ – whuber Feb 25 '13 at 18:54
  • $\begingroup$ I am basically deriving as many estimations as possible from the same data set by considering each time a progressively bigger part of my data set. For example, consider that I have a data set of 10 numbers and each estimation requires at least 2 numbers. Then, estimation 1 is derived by the first 2 numbers, estimation 2 by the first 4 numbers, etc. I am finding the cdf and ecdf because I want to find the probability that a particular estimation will exceed. Therefore, the CI of the ecdf will have to give me an interval where the true value of the statistic lies. $\endgroup$ – limp Mar 1 '13 at 17:35
  • $\begingroup$ In that case, your procedure is invalid, because each successive estimate depends strongly on the preceding ones. Why don't you use that last estimate and its confidence limits based on all the data? $\endgroup$ – whuber Mar 1 '13 at 17:39
  • $\begingroup$ Ok, that sounds reasonable. Let’s say that my population parameter is defined as the $max – min$ of an infinite number of samples. Consequently, an estimation will require at least 2 numbers. How can I derive the pdf and cdf in this case if I only consider the last estimate? Because the population parameter is derived by an infinite range of samples, I thought it would make sense if each estimation considers a bit more of the (finite) range of samples. $\endgroup$ – limp Mar 1 '13 at 18:01
  • $\begingroup$ I don't quite follow. First of all, a population parameter is a property of the population, not of any collection of samples of it. Second, estimating the PDF and CDF is different than estimating a particular parameter. Third, estimating the PDF (in a nonparametric setting where you haven't assumed the population's distribution has a particular mathematical form) requires access to all the data, not just one estimate of one parameter. Do you think you could edit your question to clear up these issues? $\endgroup$ – whuber Mar 1 '13 at 18:05
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This is difficult to answer, since you are using a strange terminology. But I take it (from the discussion in comments) That you are interested in the population range. But then you cannot use a normal distribution model, since the population range in that case will be $(-\infty, \infty)$, and there is nothing to estimate. You need to model with a distribution which is concentrated on the range.

The simplest such distribution is a uniform distribution on the range $(\theta_1, \theta_2)$, but is easy to construct others, like beta distributions or a restricted normal distribution. You must choose a model, but for this answer I will assume a uniform distribution. Then with your notation, it is easy to see that with probability one we have $$ \theta_1 \le P_{\text{min}} \le P_{\text{max}} \le \theta_2 $$ so that $P_{\text{dif}}= P_{\text{max}} - P_{\text{min}}\le \theta_2-\theta_1$. This makes clear that the population range cannot be smaller than the sample range, so a confidence interval you have given in the post subtraction and adding two standard deviations is not correct. To find a confidence interval you can adopt the methods from Confidence interval for Uniform($\theta$, $\theta + a$).

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