**Resarch Reports**
# No 2007:15:

On curve estimation under order restrictions

*Kjell Pettersson *()

**Abstract:** Robust regression is of interest in many problems where
assumptions of a parametric function may be inadequate. In this thesis, we
study regression problems where the assumptions concern only whether the
curve is increasing or decreasing. Examples in economics and public health
are given. In a forthcoming paper, the estimation methods presented here
will be the basis for likelihood based surveillance systems for detecting
changes in monotonicity. Maximum likelihood estimators are thus derived.
Distributions belonging to the regular exponential family, for example the
normal and Poisson distributions, are considered. The approach is
semiparametric, since the regression function is nonparametric and the
family of distributions is parametric. In Paper I, “Unimodal Regression in
the Two-parameter Exponential Family with Constant or Known Dispersion
Parameter”, we suggest and study methods based on the restriction that the
curve has a peak (or, equivalently, a trough). This is of interest for
example in turning point detection. Properties of the method are described
and examples are given. The starting point for Paper II, “Semiparametric
Estimation of Outbreak Regression”, was the situation at the outbreak of a
disease. A regression may be constant before the outbreak. At the onset,
there is an increase. We construct a maximum likelihood estimator for a
regression which is constant at first but then starts to increase at an
unknown time. The consistency of the estimator is proved. The method is
applied to Swedish influenza data and some of its properties are
demonstrated by a simulation study.

**Keywords:** Non-parametric; Order restrictions; Two-parameter exponential family; Known dispersion parameter; Poisson distribution; Constant Base-line; Monotonic change; Exponential family; (follow links to similar papers)

**JEL-Codes:** C10; (follow links to similar papers)

32 pages, February 4, 2008

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