New estimates for the variational distance between two distributions of a sample

For a pair of samples consisting of i.i.d. random variables we obtain new upper and lower bounds for the total variation distance between their distributions. In cases of small distances these estimates are proportional to the total variation distance between distributions of elements of samples and the square root of the sample volume with coefficients depending on the distributions of elements of samples. The results are useful for estimates of the sample volume necessary for testing close hypotheses.