Mixture Quantiles Estimated by Constrained Linear Regression

Abstract: The paper considers the problem of modeling a univariate random variable. Main contributions: (i) Suggested a new family of distributions with quantile defined by a linear combination of some basis quantiles. This family of distributions has a high shape flexibility and covers commonly used distributions. (ii) Proposed an efficient estimation method by constrained linear regression (linearity with respect to the parameters). Various types of constraints and regularizations are readily available to reduce model flexibility and improve out-of-sample performance for small datasets . (iii) Proved that the estimator is asymptotically a minimum Wasserstein distance estimator and is asymptotically normal. The estimation method can also be viewed as the best fit in quantile-quantile plot. (iv) Case study demonstrated numerical efficiency of the approach (estimated distribution of historical drawdowns of SP500 Index).