@(@\newcommand{\B}[1]{ {\bf #1} }
\newcommand{\R}[1]{ {\rm #1} }
\newcommand{\W}[1]{ \; #1 \; }@)@
This is dismod_at-20221105 documentation: Here is a link to its
current documentation
.
References
See
censoring
and the heading
Likelihoods for mixed continuous-discrete distributions
on the
wiki page
for likelihood functions.
Discussion
We use @(@
\mu
@)@ for the mean and
@(@
\delta > 0
@)@ for the standard deviation
of a Gaussian or Laplace random variable @(@
y
@)@.
We use @(@
c \leq \mu
@)@ for the value we are
censoring the random variable at.
The censored random variable is defined by
@[@
\underline{y} = \left\{ \begin{array}{ll}
c & \R{if} \; y \leq c
\\
y & \R{otherwise}
\end{array} \right.
@]@
The crucial property is that the
censored density functions (defined below)
are smooth function with respect to the mean value @(@
\mu
@)@
(but not even continuous with respect to @(@
c
@)@ or @(@
y
@)@).
Simulation Test
The file test/user/censor_density.py contains a test
of maximum likelihood estimation using the continuous-discrete densities
proposed below.
Density, G(y,mu,delta)
The Gaussian density function is given by
@[@
G( y , \mu , \delta )
=
\sqrt{ \frac{1}{ 2 \pi \delta^2 } }
\exp \left[ - \frac{1}{2} \left( \frac{y - \mu}{\delta} \right)^2 \right]
@]@
Error Function
The Error function is defined (for @(@
0 \leq x
@)@) by
@[@
\R{erf}(x)
=
\sqrt{ \frac{1}{\pi} } \int_{-x}^{+x} \exp \left( - t^2 \right) \; \R{d} t
@]@
Using he change of variables @(@
t = \sqrt{2}^{-1} (y - \mu) / \delta )
@)@
we have @(@
y = \mu + t \delta \sqrt{2}
@)@ and
@[@
\R{erf}(x)
=
\sqrt{ \frac{1}{2 \pi \delta^2} }
\int_{\mu - x \delta \sqrt{2}}^{\mu + x \delta \sqrt{2}}
\exp \left[ - \frac{1}{2} \left( \frac{y - \mu}{\delta} \right)^2 \right]
\; \R{d} y
@]@
Setting @(@
x = \sqrt{2}^{-1} ( \mu - c ) / \delta
@)@ we obtain
@[@
\R{erf}\left( \sqrt{2}^{-1} ( \mu - c ) / \delta \right)
=
\sqrt{ \frac{1}{2 \pi \delta^2} }
\int_{c}^{2 \mu - c}
\exp \left[ - \frac{1}{2} \left( \frac{y - \mu}{\delta} \right)^2 \right]
\; \R{d} y
@]@
Note that this integral is negative when @(@
c > \mu
@)@.
The Gaussian density is symmetric about @(@
y = \mu
@)@
and its integral from minus infinity to plus infinity is one.
Hence
@[@
\frac{
1 - \R{erf}\left( \sqrt{2}^{-1} ( \mu - c ) / \delta \right)
}{2}
=
\sqrt{ \frac{1}{2 \pi \delta^2} }
\int_{-\infty}^{c}
\exp \left[ - \frac{1}{2} \left( \frac{y - \mu}{\delta} \right)^2 \right]
\; \R{d} y
@]@
Censored Density, G(y,mu,delta,c)
The censored Gaussian density is defined by
@[@
G ( \underline{y} , \mu , \delta , c )
=
\left\{ \begin{array}{ll}
\left( 1 - \R{erf}\left( \sqrt{2}^{-1} (\mu - c) / \delta \right) \right) / 2
& \R{if} \; \underline{y} = c
\\
G( \underline{y} , \mu , \delta ) & \R{otherwise}
\end{array} \right.
@]@
This density function is with respect to the
Lebesgue measure plus an atom with mass one at @(@
\underline{y} = c
@)@.
Density, L(y,mu,delta)
The Laplace density function is given by
@[@
L( y , \mu , \delta )
=
\sqrt{ \frac{1}{2 \delta^2 } }
\exp \left[ - \sqrt{2} \left| \frac{y - \mu}{\delta} \right| \right]
@]@
Indefinite Integral
The indefinite integral with respect to @(@
y
@)@,
for @(@
x \leq \mu
@)@, is
@[@
\int_{-\infty}^{x} L( y , \mu , \delta ) \; \R{d} y
=
\sqrt{ \frac{1}{2 \delta^2 } }
\int_{-\infty}^{x}
\exp \left( - \sqrt{2} \frac{\mu - y}{\delta} \right) \; \R{d} y
@]@
Using @(@
c \leq \mu
@)@, we obtain
@[@
\int_{-\infty}^{c} L( y , \mu , \delta ) \; \R{d} y
=
\frac{1}{2}
\exp \left( - \sqrt{2} \frac{\mu - c}{\delta} \right)
@]@
Censored Density, L(y,mu,delta,c)
The censored Laplace density is defined by
@[@
L ( \underline{y} , \mu , \delta , c )
=
\left\{ \begin{array}{ll}
(1 / 2 )
\exp \left( - ( \mu - c ) \sqrt{2} / \delta \right)
& \R{if} \; \underline{y} = c
\\
L( \underline{y} , \mu , \delta )
& \R{otherwise}
\end{array} \right.
@]@
This density function is with respect to the
Lebesgue measure plus an atom with mass one at @(@
\underline{y} = c
@)@.
Difference Between Means
We use @(@
\overline{\underline{y}}
@)@ to
denote the mean after censoring the distribution:
@[@
\frac{ \overline{\underline{y}} - \mu }{ \delta }
=
\frac{c - \mu}{2 \delta }
\exp \left( - \sqrt{2} \frac{\mu - c}{\delta} \right)
+
\sqrt{ \frac{1}{2 \delta^2 } }
\int_c^{+\infty} \frac{y - \mu}{\delta}
\exp \left[ - \sqrt{2} \left| \frac{y - \mu}{\delta} \right| \right]
\; \R{d} y
@]@
Using integration by parts,
one can obtain a formula for @(@
\overline{\underline{y}} - \mu
@)@
in a manner similar to calculation of the
difference between means
for the Gaussian case.
Input File: omh/math/censor_density.omh
