For simple random sampling, I know that the probability of each point being part of the sample should be equal. Also, any sample of size say $k$ should be equally likely. In the sampling procedure I am using right now, the probability of each point being part of the sample comes out to be the same. However, if a point is chosen, some points are then more probable to be part of the sample. My question is, how important is the second criterion for the sample to be called simple random sample, and does such a sample have a technical name?
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$\begingroup$ The correlation might be an issue. If you give away more details about the dependency and which parameters you want to estimate, a more specific answer might be possible. Consider e.g. the linear regression problem where the errors of the data points are correlated. Estimates for the slope parameters are still unbiased but confidence intervals are not valid anymore. $\endgroup$– Georg SchnabelCommented Mar 4, 2014 at 19:03
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$\begingroup$ @Georg The goal is not to estimate any parameters but to construct a random sample. About the dependency, I can say that if a point is selected, then it is much more likelier that a few other points might also get selected. Would I need to find a way to correct this or is correlation between points in a sample not a big issue if the points apriori have equal probability of being part of the sample? $\endgroup$– user1150989Commented Mar 4, 2014 at 19:09
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$\begingroup$ systematic sampling and stratified sampling are examples for methods distinct from simple random sampling. $\endgroup$– Georg SchnabelCommented Mar 4, 2014 at 19:21
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$\begingroup$ Are there any that account for correlation between the samples? $\endgroup$– user1150989Commented Mar 4, 2014 at 19:25
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$\begingroup$ There are a number of designs in which the probability of selection for a unit depends on the units previously selected. Classification of your procedure won't be possible until you supply a detailed description. $\endgroup$– Steve SamuelsCommented Mar 5, 2014 at 2:47
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What you'd get in the end will effectively be a cluster sample. For SRS, the definition is that any sample of size $n$ has the probability of selection ${N \choose n}^{-1}$ where $N$ is the population size. That the selection probability of every unit is $n/N$ (epsem: equal probability of selection method) is a consequence of this definition, and unrelated to the sample being SRS. In my sampling course, I asked students to give at least three sampling designs that would be epsem, but won't be SRS.
In general, if you can compute the probabilities of selection of a single unit and a pair of units, you can estimate the totals using Horvitz-Thompson estimator, $$t[y] = \sum_{j\mbox{ in sample}} w_j y_j, \quad w_j = \pi_j^{-1}, \quad \pi_j = {\rm Prob}[\mbox{select unit $j$ into the sample}],$$ and you can estimate its variance via Yates-Grundy-Sen estimator $$ v\{t[y]\} = \frac{1}{2} \sum_j \sum_k \frac{\pi_j\pi_k - \pi_{jk}}{\pi_{jk}} \Bigl( \frac{y_j}{\pi_j} - \frac{y_k}{\pi_k} \Bigr)^2 $$ where $\pi_{jk}={\rm Prob}[\mbox{ both units $j$ and $k$ are in the sample }]$. Other statistics follow as functions of totals, e.g., the mean is the ratio of the estimated total of your variable of intereset $y$ to the population size (denoted here as $t[1]$ to highlight the uniformity of notation; also, there are cases where the population size is unknown beforehand, and needs to be estimated, and this formula can handle it better): $$ \bar y = \frac{ t[y]}{t[1]} $$ For SRS, $\pi_{jk} = \frac{n(n-1)}{N(N-1)}$, and the above variance formula simplifies considerably. If you have correlated samples, then some of the $\pi_{jk}$ will be greater than those in SRS, and some will be smaller. You need to be careful with not letting any of the $\pi_{jk}$ drop down to zero, as that would clearly blow things up (and that's what happens with systematic sampling).
If all of that is gibberish to you, then you need (a) to read a good sampling book, and (b) consider hiring a high quality survey consultant (unless you are OK with publishing an irreproducible result that won't hold if somebody tries to reproduce your findings in an independent study).
