There are a few things to think about before you perform your bootstrapping analysis.

In bootstrapping, you assume that the samples you have represent the true distribution of the population from which you sampled.  It's difficult to say whether or not that's the case with so few observations of each of these variables, but that's for you to decide.

Since your ultimate goal is to find the 95% confidence interval (CI) of the Duration, you should consider bootstrapping the two variables in tandem.  I'm not too familiar with what these variables are, but you mention that these are yearly rates.  You have 5 years of Prevalence and 3 years of Incidence. Do the 3 years of Incidence data coincide with a subset of the 5 years of Prevalence data?  For example, are your Prevalence rates for the years 2000 - 2004 and your Incidence rates for the years 2002-2004?  If so, then you would only have three relevant data points for your calculation (years 2002-2004).  You would then sample the years of your coincident data, calculate the Duration for each sampled year, and take the mean of the Durations in the bootstrap sample.  Otherwise, you may produce incorrect CI.

You should think hard about the noise you add in your bootstrapping analysis.  If you assign a random amount of noise (i.e. white noise with a 10% variance), you should be able to justify it.  In most cases when taking observations, you would have an error associated with the observation technique.  Perhaps this is the detection limits of the instrument used or something along those lines.  You could use that error and apply the error as noise to your observed values in each bootstrap sample before calculating the Duration.

I hope this information is useful.  I agree with Ariel and Jimmy.  My answer is simply my response to how I interpreted your question.  Others may be able to help more if you would elaborate on a few items:

1)  Are these observations coincident in time?

2)  Is there error associated with your observation technique?

3)  What kind of hypothesis are you testing with your confidence interval?

Good luck with your analysis!

~Greg

It is not clear from your description why you need to add "noise" to your data? Is this a simulation study you are performing?

I agree with Ariel, i don't see a clear justification for adding noise to you data. If you can justify yhis, then it should become more clearer by varying the nouse levels to find out which noise range might be appropriate for your question of interest. 

There are a few things to think about before you perform your bootstrapping analysis.

In bootstrapping, you assume that the samples you have represent the true distribution of the population from which you sampled.  It's difficult to say whether or not that's the case with so few observations of each of these variables, but that's for you to decide.

Since your ultimate goal is to find the 95% confidence interval (CI) of the Duration, you should consider bootstrapping the two variables in tandem.  I'm not too familiar with what these variables are, but you mention that these are yearly rates.  You have 5 years of Prevalence and 3 years of Incidence. Do the 3 years of Incidence data coincide with a subset of the 5 years of Prevalence data?  For example, are your Prevalence rates for the years 2000 - 2004 and your Incidence rates for the years 2002-2004?  If so, then you would only have three relevant data points for your calculation (years 2002-2004).  You would then sample the years of your coincident data, calculate the Duration for each sampled year, and take the mean of the Durations in the bootstrap sample.  Otherwise, you may produce incorrect CI.

You should think hard about the noise you add in your bootstrapping analysis.  If you assign a random amount of noise (i.e. white noise with a 10% variance), you should be able to justify it.  In most cases when taking observations, you would have an error associated with the observation technique.  Perhaps this is the detection limits of the instrument used or something along those lines.  You could use that error and apply the error as noise to your observed values in each bootstrap sample before calculating the Duration.

I hope this information is useful.  I agree with Ariel and Jimmy.  My answer is simply my response to how I interpreted your question.  Others may be able to help more if you would elaborate on a few items:

1)  Are these observations coincident in time?

2)  Is there error associated with your observation technique?

3)  What kind of hypothesis are you testing with your confidence interval?

Good luck with your analysis!

~Greg

The reason for performing a Bootstrap analysis is to evaluate the reliability of your formal confidence limits, like your mean, standard deviation etc. Adding noise would defeat this primary purpose of performing a Bootstrap. If your sample is indeed a fair sample of the global population, your Bootstrap will indicate that, with a distribution of your mean and statistical errors for all the "re-sampled" versions. You want to draw samples out of your pristine data.

When you resampling the ECDF the best election usual is around c*sigma*N^-{1/3} where sigma is the standard deviation and N the sample size. The key is constant "c" which depends on the shape of the distribution; 1 for type-normal distribution and smaller when the kurtosis is large. Of course, the problem can change this criteria but it usual works nice.

It is a very short sample. I work Bootstrap when we catch samples n >= 7.

We 'design' a simple rule to know B (# iterations) in Bootstraping

B=100*log(n**n). B= 350 when n=5 y B=1000 when n=10 (approximate). To know how much in Gaussian or Exponential or anothe probability model you have in your data.

Make B-samples, in each simple you execute the KS test, obtain Y=0 (reject) or Y=1 (accept). The addition of Y in each iteraion divided by B result the proportion of Gaussian-model in your data.