One approach might be to remove the correlations between and within the processes first and then if the processes are Gausian, employ the ordinary joint confidence interval methods available in the literature.

By "dependent" do you mean "correlated" like Seyed refers to? Correlation is quantitative: If two variables, Gaussian or not, are 100% (anti-)correlated, the confidence of the rectangle is 95%, not 90.25% or 90%

The Bonferroni correction does not refer to correlations between variables at all. It is a method to compensate for multiple testing bias.

@Rob Hooft, @Seyed Niaki:

Here is an example of two stochastic processes (or random variables) i'm considering "dependent": H(S,D) and P(S,D)

(a)

The inventory holding cost H(S,D) where S is the inventory level, and D is the random demand. In other way, H(S,D) = unit_holding_cost x max(0, S - D)

(b)

The shortage cost P(S,D) = unit_shortage_cost x max(0, D - S).

I think that H and P are not independent according to the common definition of "random variables independence". That is, Proba(H

I think that he following link should help to answer to your question: http://www.stat.columbia.edu/~fwood/w4315_fall2010/Lectures/lecture_9/lecture_9.pdf

@Nabil Belgasmi

I think you answered your question already. If you have two variables and don't know their joint distribution, Bonferroni's inequality provides you with a lower bound for that rectangle. You may look at some other probability bounds to improve Bonferroni's.

Bonferroni's is a bound for probabilities of joint events. That probability depends on correlations among variables defining those events. An application of this is in multiple comparisons, where you don't know what dependency exist between any number of tests. Here, "test rejecting when H_null is true" is the event in question.

When you have independent variables you can simply find any two probabilities such that their product is equal to a desired probability. If you choose those two probabilities to be equal, then your define a square, otherwise a rectangle.

Posted by @Fabrice Clerot:

i think that you should correct and clarify your question ; a few comments :

-1

"It is clear that if X and Y were independent and normal, than the confidence level related to R(X,Y) would be 1 - A = P(M in R(X,Y))"

this is wrong : the definition of the confidence level is linked to the probability of the CI (considered as a random variable) to cover the _true value_ (and not M, which, according to your notations is an empirical average)

-2

i do not understand why you would need normality assumptions for your claims

-3

from your notations, it is unclear if you consider a "confidence interval" as a random entity (depending on a random sample) or an _instance_ of a CI, that is a constant interval obtained by applying the formulas for the upper and lower bounds to a given instance of a sample :

[m(X, S) - zs ; m(X, S) + zs] is a confidence interval (random entity depending on the random sample S)

assuming z = 2, s = 1 and a sample S = {2, 1, 6, 3} (assimilating the sample to the sample of the values of X) which gives m(X, S) = 3

[1 5] is an instance of the above CI ; it is obviously a constant ; as such it covers or not the true (unknown) value which is also a constant : you cannot say that it has "some probability" of covering the true value (the fact that you do not know this true value does not make it, at least in this formalism, a random variable : ignorance does not imply uncertainty !). To take a simpler example, when drawing independently from a uniform distribution on [-0.1 ; 0.9], you obviously have a 0.9 probability to draw a positive number, however, if you draw -0.051(which is indeed possible ...), you are not going to say that -0.051 has a 0.9 probability to be positive !

the "probability" has to be understood from a scenario of repeated sampling of CI considered as a random variable.

-4

"stochastic process" is not a complicated way of saying "random variable"

-5

from my understanding, the Bonferroni correction follows directly from the union bound (or Boole's bound) and does not require any assumption of independence or normality

http://en.wikipedia.org/wiki/Boole%27s_inequality

-6

for the bootstrap, i would recommend you start reading "An introduction to the bootstrap"

http://www.uni-bonn.de/~kutikal/SS12/compStat/Boots/IntroductionBootstrap.pdf

Here is my answer to @Fabrice Clerot comments:

-1

(the notation is corrected now)

-2

I need normality assumptions because, the normality of "the empirical mean of X" which is a random variable depends on the asymmetry of the primary random variable "X". (Reference = page 8: biol09.biol.umontreal.ca/BIO2041e/Sujet_05-Int_confiance.pdf). That is, if the skewness of X is to high, the normality of m(X, S) will be affected. That is why i assumed that X is normal. Otherwise, a correct CI should be calculated using Bootstrapping or by performing the appropriate transformation to X (log, exp, Legendre, etc.) so that the transformed random variable X' would have better skewness and kurtosis.

-3

The big confusion is here. It is okay that a "CI" of a certain parameter "m" is considered as random entity that covers the true "m" with a probability of (1 - a). An that an instance of CI (i.e. [a, b]) cannot be assumed to contains "m" with a predefined probability simply because the event "m in [a, b]" is static and is not connected to any probability (m is either inside [a, b] or no). Being convinced of all this, i conclude that any instance of a given CI does not have any importance, and does not add to the estimation of the true parameter. In other way, saying that a queue's expected waiting time is about: 20min +/- 5min give no information about the true waiting time which may be greater than 25min or less than 15min. What you said about "ignorance does not imply uncertainty" makes me think that (a) CI are interesting only if they are kept in their analytical "variable" form, (b) that a single CI instance is useless and (c) multiple CIs can provide information about the probability of existence of the true parameter "m" inside all the CIs.

-4

To my knowledge, a stochastic process adds the time axis (discrete or continuous) to the randomness of a random variable. I think it should at least inherits the complexity of the Probability Theory.

-5

The most important advantage of the Bonferroni inequality is that it does not require the independence of the events "ei" it is based on. Thus, it can be applied to construct a rectangular area with variable bounds that contains a specific random point with a predefined probability.

-6

I thought about Bootstrapping Confidence Intervals when i observed that the skewness coefficients of X and Y are too high. It is among the motivations of using such method.

@Sigfrido: That's right. Thank you for your help. There are certainly more sophisticated methods than the Bonferroni's. However, i don't think they require less assumptions than it (especially the "dependence of the random variables").

@Christian Brière: Thank you for the link.

@Nabil Belgasmi

I agree. You will need to impose some structure on the joint probability associated with your vector, if you want to improve Bonferroni's bounds. I might be useful though, because Bonferroni's gives very conservative bounds ending up with wider intervals than desirable.

@sigfrido

Do you know about some methods or inequalities that provide closer lower bound than the Bonferroni's? Proba(ev1 and ev2) >= 1 - (a1 + a2) where Proba(ev1) = 1 -a1, and so on.

Would you please comment what is said here:

-3

The big confusion is here. It is okay that a "CI" of a certain parameter "m" is considered as random entity that covers the true "m" with a probability of (1 - a). An that an instance of CI (i.e. [a, b]) cannot be assumed to contains "m" with a predefined probability simply because the event "m in [a, b]" is static and is not connected to any probability (m is either inside [a, b] or no). Being convinced of all this, i conclude that any instance of a given CI does not have any importance, and does not add to the estimation of the true parameter. In other way, saying that a queue's expected waiting time is about: 20min +/- 5min give no information about the true waiting time which may be greater than 25min or less than 15min. What you said about "ignorance does not imply uncertainty" makes me think that (a) CI are interesting only if they are kept in their analytical "variable" form, (b) that a single CI instance is useless and (c) multiple CIs can provide information about the probability of existence of the true parameter "m" inside all the CIs.

I know about two papers:

Hochberg, Y. (1988) A sharper bonferroni procedure for multiple tests of significance, Biometrika, 75, 4.

Holm, S. (1979) A simple sequentially rejective multiple test procedure, Scandinavian Journal of Statistics, 6.

Hope this helps.

@Sigfrido: thank you very much.

@Yue: Thank you.

This problem is tackled by the pharma industry. Some references are:

1. Fuchs C, Kenett RS. Multivariate tolerance regions and F-tests. Journal of Quality Technology 1987; 19, 122–131.

2. John S. A tolerance region for multivariate normal distributions. Sankhya 1963; Series A 25, 363–368.

3. Hotelling H. Multivariate quality control, illustrated by the air testing of sample bombsights. Techniques of Statistical Analysis, Eisenhart, Hastay and Wallis (Editors), McGraw-Hill, New York, USA, 1947.

4. Jackson JE. Multivariate quality control Communications in Statistics: Theory and Methods 1985; 14, 2657–2688.

5. Fuchs C, Kenett RS. Multivariate Quality Control: Theory and Application, Quality and Reliability Series, Vol. 54, Marcel Dekker, New York, USA, 1998.

6. Tsong, Y, Hammerstrom, T, Sathe P, Shah, V Statistical assessment of mean differences between two dissolution data sets, Drug Information Journal 1996 Vol. 30, pp. 1105–1112.

7. Chapters 10 and 12 in http://eu.wiley.com/WileyCDA/WileyTitle/productCd-1118456068.html

@Ron Kenett: Thank a lot.

If you want a rectangular region, Tukey's simultaneous confidence intervals may help.

See R. G. Miller, Simultaneous Statistical Inference  (Springer 1981)

You may however consider alternatives, such as confidence ellipsoids, as has already been pointed out.