\( \newcommand{\pen}{\lambda} \newcommand{\maxage}{\omega} \newcommand{\knotspacing}{\Delta x} \newcommand{\nintern}{n} \newcommand{\ceil}[1]{\left\lceil #1\right\rceil} \newcommand{\maxits}{J} \)

PCLM

The PCLM buildmethod fits death rates using Rizzi et al (2015)'s Penalised Composite Link Method for death rates.

ExtraInfo

The user must specify the population at risk using an extrainfo string of the form PCLM:PInfo[,ExtraInfo]. If PInfo is a number then it specifies the population per age interval in the mortality object. If PInfo is a string of the form "P=X" where X is a number then the total population is equal to X and the population per age interval will be calculated assuming the population follows a stationary distribution. If the PInfo string is not of this format then it specifies the name of a PCURVE object from which the population per mortality age interval is calculated. ExtraInfo is an optional comma-separated list of quoted strings of the form "Parameter=Value" which can be used to change the method's default settings.

Degree

By default PCLM uses linear B-Splines. The degree of splines can be changed with the extrainfo string "Degree=m".

Order

By default PCLM uses a second order penalty. The order can be changed with the extrainfo string "Order=k".

Knots

There are two ways to set the B-Spline knots. The first way is to set the maximum age using extrainfo string "MaxAge=\(\maxage\)". The internal points are then equally spaced upwards from 1 as follows \begin{eqnarray} x_0&=&0\\ x_1&=&1\\ x_i &= &x_{i-1}+\knotspacing,\quad i=2,\ldots,\nintern\\ x_{\nintern+1}&=&\maxage \end{eqnarray} where \begin{equation} \nintern=\ceil{\frac{\maxage-1}{\knotspacing}} \end{equation} and the spacing \(\knotspacing\) is specified by the extrainfo string "KnotSpacing=\(\knotspacing\)". The default values are \(\maxage=110\) and \(\knotspacing=2\). The second way is to set the knots explicitly using the extrainfo string "Knots=\(x_0\)|\(x_1\)|\(x_2\)|\(\ldots\)|\(x_\nintern\)|\(\maxage\)".

Penalty

By default PCLM automatically calculates the penalty \(\pen\) by minimizing the Akaike information criterion ("Penalty=AIC"). An alternative is the Bayesean information criterion ("Penalty=BIC"). The penalty can also be specified explicitly using "Penalty=\(\pen\)".

MaxIts

The number of iterations used to solve for the B-Spline weights can be set using the extrainfo string "MaxIts=\(\maxits\)". The default is \(\maxits=20\).