On the Analysis of Optical Mapping Data - University of Wisconsin ...

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Figure 2.3 The non-parametric MLE of π based on the Grenander estimator. The smooth
grey curve is based on a naive zero truncated kernel density estimate of h.
the so called Grenander estimator (see van der Vaart, 1998, Chapter 24). To see this, let G
be the (known) CDF of Y . This happens to be exponential in this case, but any monotone
decreasing density is sufficient. Consider the quantities of interest in a scale transformed by
G, i.e., Ỹ = G(Y ), ˜X = G(X) and
)
˜π (ỹ) = P
(Z = 1|Ỹ = ỹ = π(G −1 (ỹ))
Let ˜h be density of ˜X. Since Ỹ ∼ U (0, 1) with constant density, ˜h(˜x) ∝ ˜π(˜x). Since G and
hence G −1 are monotone increasing transformations,
π ↑ =⇒ ˜π = π ◦ G −1 ↑ =⇒ ˜h ↑
Hence, the MLE of ˜h, based on ˜X i ’s, is given by the Grenander estimator. The MLE of ˜π is
proportional to that of ˜h, with the constant of proportionality obtained from the fact that
lim ey→1 ˜π(ỹ) = 1. The estimator is inconsistent at ỹ = 1, i.e. y = ∞, but that is not of
interest to us. The MLE of π in the original scale is given by ̂π = ̂˜π ◦ G.
Parametric estimation: Parametric forms of π are naturally easier to work with in practice.
Obtaining maximum likelihood estimates in that case is straightforward in principle;
most of the difficulty arises in obtaining the normalizing constant
K(π) =
∫ ∞
0
π(t) g(t) dt