# raftery.diag

##### Raftery and Lewis's diagnostic

`raftery.diag`

is a run length control diagnostic based on a
criterion of accuracy of estimation of the quantile `q`

. It is
intended for use on a short pilot run of a Markov chain. The number
of iterations required to estimate the quantile $q$ to within an
accuracy of +/- $r$ with probability $p$ is calculated. Separate
calculations are performed for each variable within each chain.

If the number of iterations in `data`

is too small,
an error message is printed indicating the minimum length of
pilot run. The minimum length is the required sample size for a
chain with no correlation between consecutive samples. Positive
autocorrelation will increase the required sample size above this
minimum value. An estimate `I`

(the `dependence factor') of the
extent to which autocorrelation inflates the required sample size
is also provided. Values of `I`

larger than 5 indicate strong
autocorrelation which may be due to a poor choice of starting value,
high posterior correlations or `stickiness' of the MCMC algorithm.

The number of `burn in' iterations to be discarded at the beginning of the chain is also calculated.

- Keywords
- htest

##### Usage

`raftery.diag(data, q=0.025, r=0.005, s=0.95, converge.eps=0.001)`

##### Arguments

- data
- an
`mcmc`

object - q
- the quantile to be estimated.
- r
- the desired margin of error of the estimate.
- s
- the probability of obtaining an estimate in the interval (q-r,q+r).
- converge.eps
- Precision required for estimate of time to convergence.

##### Value

- A list with class
`raftery.diag`

. A print method is available for objects of this class. the contents of the list are tspar The time series parameters of `data`

params A vector containing the parameters `r`

,`s`

and`q`

Niters The number of iterations in `data`

resmatrix A 3-d array containing the results: $M$ the length of "burn in", $N$ the required sample size, $Nmin$ the minimum sample size based on zero autocorrelation and $I = (M+N)/Nmin$ the "dependence factor"

##### Note

`raftery.diag`

is based on the FORTRAN program `gibbsit'
written by Steven Lewis, and available from the Statlib archive.

##### Theory

The estimated sample size for variable U is based on the process $Z_t
= d(U_t <= u)$="" where="" $d$="" is="" the="" indicator="" function="" and="" u="" qth="" quantile="" of="" u.="" process="" $z_t$="" derived="" from="" markov="" chain="" ```
data by marginalization and truncation, but is not itself
a Markov chain. However, $Z_t$ may behave as a Markov chain if
it is sufficiently thinned out.
```

`raftery.diag`

calculates the
smallest value of thinning interval $k$ which makes the thinned
chain $Z^k_t$ behave as a Markov chain. The required sample size is
calculated from this thinned sequence. Since some data is `thrown away'
the sample size estimates are conservative.

The criterion for the number of `burn in' iterations $m$ to be
discarded, is that the conditional distribution of $Z^k_m$
given $Z_0$ should be within `converge.eps`

of the equilibrium
distribution of the chain $Z^k_t$.

##### References

Raftery, A.E. and Lewis, S.M. (1992). One long run with diagnostics:
Implementation strategies for Markov chain Monte Carlo.
*Statistical Science*, **7**, 493-497.

Raftery, A.E. and Lewis, S.M. (1995). The number of iterations,
convergence diagnostics and generic Metropolis algorithms. *In*
Practical Markov Chain Monte Carlo (W.R. Gilks, D.J. Spiegelhalter
and S. Richardson, eds.). London, U.K.: Chapman and Hall.

*Documentation reproduced from package coda, version 0.17-1, License: GPL (>= 2)*