# Nonparametric model

    library(serosv)

## Local estimation by polynomial

Refer to `Chapter 7.1`

Proposed model

Within the local polynomial framework, the linear predictor \(\eta(a)\) is approximated locally at one particular value \(a_0\) for age by a line (local linear) or a parabola (local quadratic).

The estimator for the \(k\)-th derivative of \(\eta(a_0)\), for \(k = 0,1,…,p\) (degree of local polynomial) is as followed:

\[ \hat{\eta}^{(k)}(a_0) = k!\hat{\beta}_k(a_0) \]

The estimator for the prevalence at age \(a_0\) is then given by

\[ \hat{\pi}(a_0) = g^{-1}\{ \hat{\beta}_0(a_0) \} \]

- Where \(g\) is the link function

The estimator for the force of infection at age \(a_0\) by assuming \(p \ge 1\) is as followed

\[ \hat{\lambda}(a_0) = \hat{\beta}_1(a_0) \delta \{ \hat{\beta}_0 (a_0) \} \]

- Where \(\delta \{ \hat{\beta}_0(a_0) \} = \frac{dg^{-1} \{ \hat{\beta}_0(a_0) \} } {d\hat{\beta}_0(a_0)}\)

Fitting data

    mump <- mumps_uk_1986_1987

Use `plot_gcv()` to show GCV curves for the nearest neighbor method (left) and constant bandwidth (right).

    plot_gcv(
       mump$age, mump$pos, mump$tot,
       nn_seq = seq(0.2, 0.8, by=0.1),
       h_seq = seq(5, 25, by=1)
     )

Use `lp_model()` to fit a local estimation by polynomials.

    lp <- lp_model(mump, kern="tcub", nn=0.7, deg=2)
    plot(lp)