Page 150 - Contributed Paper Session (CPS) - Volume 3

CPS1973 Matúš M. et al.
group LASSO proposed in Jacob et al. (2009) which allows to implement any
arbitrary hierarchical structure into the model. The idea is to replace

in the sum in (5) by its decomposition
into latent parameters, such that

and with some well defined restrictions of the form , for some
, ∈ {0, . . . , − 1} and all = 1, . . . , one can enforce the required form
of the changepoint hierarchy in the model (for instance, if there is a jump in
the j-th order derivative of , changepoints will also occur in higher order
derivatives but not in the lower order derivatives).

3. Results
The nonparametric regression models with changepoints being detected
and estimated by using the -norm regularization approaches are
1
investigated for various -type penalty forms. Theoretical results are derived
1
with respect to the quality of the final estimate and also with respect to the
quality of the changepoint detection performance. To be specific:
• under some necessary regularity assumptions, some technical conditions,
and some minor changepoint restrictions, the consistency of the model
estimation is proved such that

,
for a well defined constant and the vector of unknown param
;
• under some necessary regularity assumptions and some technical
conditions on the number of estimated changepoints the consistency of
the detection is proved such that
, for N → ,

for some well defined non-increasing positive sequence > 0, where

∈ ℕ is the total number of true changepoint locations t in the model
∗
k
with their corresponding estimates ̂ ;

• under some necessary regularity assumptions and some technical
conditions it is proved that the proposed methodology recovers all
existing changepoints with probability tending to one as sample size
increases (in a sense that the number of estimated locations is at least );
• the consistency of the model estimation and changepoint detection is also
proved (in an analogous sense as above) for estimating the conditional
quantiles thus, for the -norm objective function in (3) being replaced
2
with the quantile check function () = ( − {<0} ), for some ∈

(0,1);
139 | I S I W S C 2 0 1 9