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You describe a sequence of rejection samples. By definition, after selecting $k$ objects in the process of obtaining a weighted sample of $m$ objects without replacement from a list of $n$ objects, you draw one of the remaining $n-k$ objects according to their relative weights. In your description of the alternative sampling scheme, at the same stage you ...

One approach to choosing the cutoff value $\epsilon$ for ABC rejection sampling is the following (similar to Aniko's answer). Simulate several test data sets from known parameter values which are vaguely similar to your observed data (e.g. by performing ABC with a relatively large $\epsilon$). From the ABC output for a test data set, some criterion of ...

An acceptance rate of $1\%$ is extremely low. You have to aim for an acceptance rate of around $15\%-30\%$. In the normal case the optimal rate is the famous $0.234$. However, this is a tricky diagnostic tool since this "optimal rate" can also be achieved despite a terrible sampling (for example when you sample well from an entry and terribly from another ...

This is the fundamental point of weighting a sample to population. You weight each individual in your sample based on known demographic features of the population such that the weights of each demographic group in the sample add up to population totals. See any book on sampling theory and practice - no, it's not reject inference. I recommend Thomas ...

Based on your edit, it appears that you are looking for guidance in selecting the tolerance parameter $\epsilon$ for ABC sampling. I don't know much about the topic, but $\epsilon$ should be small. A simple possibility is to select several different values and see whether the resulting posterior distributions look similar (based on new sets of samples). The ...