  Kink model is developed to analyze the data where the regression function is
twostage linear but intersects at an unknown threshold. In quantile regression
with longitudinal data, previous work assumed that the unknown threshold
parameters or kink points are heterogeneous across different quantiles.
However, the location where kink effect happens tend to be the same across
different quantiles, especially in a region of neighboring quantile levels.
Ignoring such homogeneity information may lead to efficiency loss for
estimation. In view of this, we propose a composite estimator for the common
kink point by absorbing information from multiple quantiles. In addition, we
also develop a sup-likelihood-ratio test to check the kink effect at a given
quantile level. A test-inversion confidence interval for the common kink point
is also developed based on the quantile rank score test. The simulation study
shows that the proposed composite kink estimator is more competitive with the
least square estimator and the single quantile estimator. We illustrate the
practical value of this work through the analysis of a body mass index and
blood pressure data set.
