Everything depends on the estimator.

An estimator is a function on observations

A=f(X_1,X_2,...,X_n) and is itself a random variable. So, everything depends on the form

of E[A] -the expectation of A. If ,for example, f(...) is a linear combination of X_i's,

E[A]=Sum c_i E[X_i] ,  the violation of iid property is not important. Otherwise

you should investigate E[A] .

Ilya

Everything depends on the estimator.

An estimator is a function on observations

A=f(X_1,X_2,...,X_n) and is itself a random variable. So, everything depends on the form

of E[A] -the expectation of A. If ,for example, f(...) is a linear combination of X_i's,

E[A]=Sum c_i E[X_i] ,  the violation of iid property is not important. Otherwise

you should investigate E[A] .

Ilya

Yes. the sample mean will be biased.

Since your sample is not iid, it is a non-random sample of the population,   in which all individuals, or instances, were not equally likely to have been selected.  The  sample is collected in such a way that some members of the intended population are less likely to be included than others.  Such sampling  bias can lead to an over- or under-estimation of the corresponding parameter in the population. Almost every sample in practice is biased because it is practically impossible to ensure a perfectly random sample. If the degree of misrepresentation is small, the sample can be treated as a reasonable approximation to a random sample. Also, if the group that is underrepresented does not differ markedly from the other groups in the quantity being measured, then a random sample can still be a reasonable approximation.

If entire segments of the population are excluded from a sample, then there are no adjustments that can produce estimates that are representative of the entire population. But if some groups are underrepresented and the degree of underrepresentation can be quantified, then sample weights can correct the bias.

For example, a hypothetical population might include 10 million men and 10 million women. Suppose that a biased sample of 100 patients included 20 men and 80 women. A researcher could correct for this imbalance by attaching a weight of 2.5 for each male and 0.625 for each female. This would adjust any estimates to achieve the same expected value as a sample that included exactly 50 men and 50 women, unless men and women differed in their likelihood of taking part in the survey.

Another way is using stratified sampling.  Assume that we need to estimate average number of votes for each candidate in an election. Assume that country has 3 towns: Town A has 1 million factory workers, Town B has 2 million office workers and Town C has 3 million retirees. We can choose to get a random sample of size 60 over entire population but there is some chance that the random sample turns out to be not well balanced across these towns and hence is biased causing a significant error in estimation. Instead if we choose to take a random sample of 10, 20 and 30 from Town A, B and C respectively then we can produce a smaller error in estimation for the same total size of sample.

Stratified sampling strategies

1.Proportionate allocation uses a sampling fraction in each of the strata that is proportional to that of the total population. For instance, if the population consists of X total individuals, m of which are male and f female (and where m + f = X), then the relative size of the two samples (x1 = m/X males, x2 = f/X females) should reflect this proportion.

2.Optimum allocation (or Disproportionate allocation) - Each stratum is proportionate to the standard deviation of the distribution of the variable. Larger samples are taken in the strata with the greatest variability to generate the least possible sampling variance

Pankaj -

You mention "...a highly heterogeneous populations which can be classified into number of subpopulations and samples are drawn from them."  Perhaps what you are getting at is that you need to stratify.  Stratified random sampling can be very helpful.  If you aren't sure of the categories ahead of time, then there is poststratification.   I just checked the internet to see what was available on poststratification, and saw the followng by Pennsylvania State University, and they often have nice explanations, so I assume this one will give you a good idea as to my suggestion. 

 6.3 Poststratification and further topics on stratification

 https://onlinecourses.science.psu.edu/stat506/node/29

 I took a quick look at part of it, and saw where it said this:

"Poststratification (stratification after the selection of a sample) is often appropriate when a simple random sample is not properly balanced by the representation."

Sorry if I have misunderstood what you want, but I thought this might be it. 

Cheers - Jim

PS -  I see that Yuanzhang Li has a good discussion of stratification above. 

 --  Anyway, I think that this may possibly be what you had in mind, though I'm not certain about the original question.

-  revised from original response

Pankaj -

I think that perhaps what you are looking into might have an interesting impact on variance.  

(First, perhaps as an aside, note that for stratified random sampling, Cochran, W.G(1977), Sampling Techniques, 3rd ed., John Wiley & Sons, section 5A.2, pages 117-119, mentions bias in sample estimates when the stratum totals, Nh, are in error.  This may be of interest in your work, though maybe somewhat off topic.)   

You might consider what lack of independence might mean for variance estimation in cluster sampling (design-based), or relatedly for model-based estimation.  See page 243 in Cochran, W.G(1977), Sampling Techniques, 3rd ed., John Wiley & Sons, on variance for cluster sampling.  The variance between elements in the same unit, as shown there, can be modeled analogously to that indicated by Fairfield Smith, H. (1938) in an agricultural setting, as explored by Ken Brewer and others.  Perhaps see https://www.researchgate.net/publication/261596397_Ken_Brewer_and_the_coefficient_of_heteroscedasticity_as_used_in_sample_survey_inference.

Brewer, KRW (2002), Combined survey sampling inference: Weighing Basu's elephants, Arnold: London and Oxford University Press, in the middle of page 111, states that the exponent in that model, which is written as twice the coefficient of heteroscedasticity that he uses, is such that heteroscedasticity matches the classical ratio estimator when those elements are INDEPENDENT of one another, and then goes on to say that heteroscedasticity is higher when there is more dependence between those elements.  To be lower, and approach the OLS level - Brewer says "...would imply that components within such aggregations  differed among themselves more than they differed from components of other units."  Here, to correspond to Cochran's discussion of cluster sampling in Chapter 9A, say, pages 249 - 259, we could say "element" instead of "component," and "cluster unit" instead of "unit." 

Cluster sampling can have less efficiency than simple random sampling, by the way, but be better logistically for collecting data, for example, when travel is involved.

In stratified random sampling, we want the "units" (here, meaning the lowest level, analogous to the "element" level) within a stratum to have less variance among them than the difference between strata, to give it a favorable deff (see Leslie Kish and others), meaning making it more efficient than simple random sampling. 

Anyway, Cochran and Brewer might be interesting references for you. 

Also, the paper at the following link relates this modeling to unequal probability selection:

"Using unequal probability sampling in business surveys to limit anticipated variances of regression estimators," ICES III, Anders Holmberg:

https://www.amstat.org/meetings/ices/2007/proceedings/ICES2007-000162.PDF

Cheers - Jim

PS - This may be of interest to you:

http://www.stat.cmu.edu/~cshalizi/uADA/13/reminders/uncorrelated-vs-independent.pdf

However, Brewer, KRW (2002), Combined survey sampling inference: Weighing Basu's elephants, Arnold: London and Oxford University Press, on page 111 says that an explanation for the level of heteroscedasticity has to do with the level of independence between what I'll call subunits within a unit.  He explains in terms of formerly smaller retail stores run as a larger unit, which may be independently run, or they may be "...tightly controlled by...central management."   

... 

Note: above revised from original 

Article Ken Brewer and the coefficient of heteroscedasticity as used...