Conditions ensuring population-recursion termination and non-degeneracy
Establish sufficient conditions on the piecewise-linear signal that guarantee Assumptions 4.1 and 4.2 for LABS, or formulate a consistency theory that does not require these assumptions.
References
Several questions remain open. First, one could seek sufficient conditions on the signal for Assumption 4.1 and Assumption 4.2, or a formulation that does not require them.
Several questions remain open. First, one could seek sufficient conditions on the signal for Assumption 4.1 and Assumption 4.2, or a formulation that does not require them. The case in which $N$ grows with $n$ also requires separate analysis because the population recursion is then no longer a fixed finite object.
Several questions remain open. First, one could seek sufficient conditions on the signal for Assumption 4.1 and Assumption 4.2, or a formulation that does not require them. The case in which $N$ grows with $n$ also requires separate analysis because the population recursion is then no longer a fixed finite object. Second, LABS could be extended to other signal models, including piecewise-polynomial signals of higher degree and signals with both level and slope changes.
Several questions remain open. First, one could seek sufficient conditions on the signal for Assumption 4.1 and Assumption 4.2, or a formulation that does not require them. The case in which $N$ grows with $n$ also requires separate analysis because the population recursion is then no longer a fixed finite object. Second, LABS could be extended to other signal models, including piecewise-polynomial signals of higher degree and signals with both level and slope changes. Third, it would be useful to determine the smallest range of thresholds for which a solution path computed on a fixed grid is guaranteed to contain a consistent model, and to develop more efficient path-construction algorithms.
Several questions remain open. First, one could seek sufficient conditions on the signal for Assumption 4.1 and Assumption 4.2, or a formulation that does not require them. The case in which $N$ grows with $n$ also requires separate analysis because the population recursion is then no longer a fixed finite object. Second, LABS could be extended to other signal models, including piecewise-polynomial signals of higher degree and signals with both level and slope changes. Third, it would be useful to determine the smallest range of thresholds for which a solution path computed on a fixed grid is guaranteed to contain a consistent model, and to develop more efficient path-construction algorithms.
Several questions remain open. First, one could seek sufficient conditions on the signal for Assumption 4.1 and Assumption 4.2, or a formulation that does not require them. The case in which $N$ grows with $n$ also requires separate analysis because the population recursion is then no longer a fixed finite object. Second, LABS could be extended to other signal models, including piecewise-polynomial signals of higher degree and signals with both level and slope changes. Third, it would be useful to determine the smallest range of thresholds for which a solution path computed on a fixed grid is guaranteed to contain a consistent model, and to develop more efficient path-construction algorithms. Finally, further work could establish conditions under which increasing $M$ makes the assumptions of Section~\ref{sup:gridass} easier to satisfy and study how a suitable finite-sample threshold depends on $M$. The present consistency result does not address either question.