How can we evaluate the the degree of poverty of a country, of a region, how can we compare the income distribution of two countries? The first task is to give a mathematical content to these notions so that they are measurable. The theoretical basis was devised in two papers written by Tony Atkinson in 1970 and 1987. An axiomatisation leads to the construction of welfare functions from which we can defined inequality indices and later poverty indices once a poverty level has been defined. These indices are in general integral transformations of the income distribution.
For all, all the problems are not yet solved, especially when we want to deal with inequality and poverty in European countries. Statistical data concern households which have different compositions. Equivalence  scales are then needed to make comparisons. It is different to consider income or spending, and in this last case before or after lodging expanses. This might have a non trivial effect on the results. In short, we have to know what we measure.
When considering statistical theory, it is rather easy to compare indices which are uni-dimensional statistics. When we want to compare complete distributions, the task become more difficult and one has to make use of the notion of stochastic dominance. Comparing distributions then implies to compare a series of numbers belonging to a predefined grid.
The above econometric analysis is mainly descriptive. When we want to explain the formation and the evolution of inequality and poverty, one has to introduce rather complex econometric models which explain the shape of a distribution. Quintile regression models are a first possibility.
Finally serious empirical analysis of poverty requires the use of survey data such as the UK Family  expenditure survey or the French enquˆete sur le budget des familles. The access to these data set for scientific researchers is very unequal, in particular in France.