Regression towards the mode

Semi‐linear mode regression

“…Assumption 3.6(b) requires at least four derivatives for K, which is slightly stronger than needed for parametric mode estimation (see Kemp and Santos Silva, 2012). For the sequences of strictly positive real numbers {h v } n and {h} n , we have, Assumption 3.5 entails classic nonparametric smoothness conditions because the analogue estimates the unknown functions in question with the kernel method.…”

“…if Assumption 3.5 is true with p(e|X, V ) admitting at least three derivatives around the origin, provided φ admits at least two derivatives (see Kemp and Santos Silva, 2012). However, this rate accelerates as the functions of Assumption 3.5 become more differentiable, and β − β ≈ O p (n −1/2 ) will be fulfilled if the functions in questions admit a large number of derivatives.…”

“…To resolve this issue, a first-stage estimator is retrieved, first by simulated annealing starting from the null coefficient and using a budget of 500 iterations, and then by refining the annealing solution by hill climbing (Goldfeld et al, 1966) using 50 iterations. The estimated effect of primogenture 10 The RTME uses K(t) = (35/32) * (1 − t 2 ) 3 1[|t| ≤ 1] and the bandwidths rule proposed by Kemp and Santos Silva (2012). 10 Table 1 summarizes the estimates for the institutional variables and RET(v) = (exp(φ(v + 3) − φ(v)) − 1) × 100, which is the percentage change in GDP per capita at Educ = v due to a 3 percentage point increase in the college-educated population.…”

“…1 Finally, mode regression is judicious for summarizing conditional relationships with heavily skewed distributions. For instance, if φ is known up to a finite-dimensional parameter, β can be estimated consistently at a rate equal to n 2/7 using the regression toward the mode estimator (hereafter RTME) described by Kemp and Santos Silva (2012). Fully nonparametric methods for estimating conditional modes have been developed by Parzen (1962), Chernos (1964), Collomb et al (1987), Bickel and Fan (1996), Quintela Del Rio and Vieu (1997) and Chen et al (2016), amongst others.…”

Fast Mode Regression in Big Data Analysis

“…Lee ([5], [6]) explored direct inference for mode regression and focused on the case where the dependent variable is truncated. [3] proposed a semi-parametric mode regression estimator for the case in which the variable of interest is unbounded, continuous and observable over its entire support and [10] proposed an ExpectationMaximization algorithm in order to estimate the regression coefficients of modal linear regression. All these existing mode regression methods involve either semi-parametric or non-parametric estimation of regression parameters or have slow convergence rate or are subject to bandwidth selection with little, if any, use in practice.…”