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K. Kearns, F. Primini, and D. Alexander
Smithsonian Astrophysical Observatory,
60 Garden St.,
Cambridge, MA 02138
When fitting models to data with few counts, two of the most common
methods used are the standard method and the C statistic. Use
of the
method requires that one avoid bins with 0 counts by
either ignoring them or rebinning, and produces significantly biased
results for data with few counts. The C-statistic gives unbiased results
but is difficult to interpret in terms of goodness-of-fit.
Neither approach is
ideal, though each is useful in some cases. The Iterative Weighting
Technique which we investigate here both addresses the
deficiencies inherent in using the standard method for data with few
counts, and provides a goodness-of-fit parameter which is
indistinguishable from the standard
parameter for many datasets.
Iterative Weighting (IW) is an example of the class of weighted
least-squares estimators described by Wheaton et al. (1994), in
which is expressed as a weighted sum of squared deviations,
where are the observed counts in bin i,
are
the counts predicted by the model M with parameters
,
and the weights
are the inverses of the true variances
. As Wheaton et al. (1994) point out, the
approximation
leads to significant biases in the
best-fit parameters, due to the strong anti-correlation between
and
. Similar
biases are encountered if the approximation
is
used (Nousek & Shue 1989).
The IW technique avoids such biases by estimating
through
successive iterations, where for each iteration, j,
, and the best-fit
parameters
are determined by
minimization of
For the first iteration, all weights are set to 1.
In our sample, we find that the minimum values
and
best-fit parameters converge after about 6 iterations.
To demonstrate the IW technique, we repeat the simple numerical experiment of Nousek & Shue (1989). For a range of total counts, N, from 25 to 1000, we generate an ideal power-law spectrum such that:
for and
For each ideal spectrum, we simulate 1000 sample spectra {n
}, where
{n
} are random deviates drawn from Poisson distributions with means
=
. We then determine best-fit model parameters
and
for each simulated spectrum, using IW
and Powell's method for function minimization (Press 1988). For
each N, we then compute
the average
and
; compile the
distributions of minimum
for comparison with the theoretical
distribution; and compute the percentage of simulations for which the
and
contours include
and
,
for comparison with the expected percentages.
Table: Comparison of three fitting techniques.
we compare the biases (as measured by the ratios of average best-fit
parameter values to true values) in 1000 IW runs with those found for
traditional and the C statistic by Nousek & Shue in 250 runs.
We find that the IW biases are comparable to those encountered using
the C statistic for all N. These results are displayed in
Figure
.
In Figure
we compare both differential and cumulative
theoretical
distributions with our observed
distributions. We apply a KS-test to the cumulative distributions and
find that at N=25 the match is poor, but by
N=100 the two distributions are in good agreement.
The percentage of simulations for which the
contours include
and
,
for
values appropriate to various joint two-parameter
confidence levels, is shown in
table
.
Table: Estimating Confidence Limits: Percentage of Best Fits Within Various
Boundaries, from a total of 1000 spectral fits.
For most N, the measured and expected confidence levels are in good agreement.
Figure: Bias in best-fit parameters for three fitting techniques.
Original PostScript figure (13 kB)
Figure: Comparison of theoretical distribution with observed
distribution for IW by KS-test, with overlaid histograms.
Original PostScript figures
(56 kB),
(56 kB)
We find that unbiased parameter estimates by minimization are
possible for binned data with few or no counts in some bins, provided the
calculation is modified slightly. Except for
very small N, this modified
statistic is distributed
according to the theoretical
distribution. Goodness-of-fit can
therefore be assessed using traditional techniques. Further,
this
statistic can be used to estimate
confidence levels from standard
boundaries.
This work is partially supported by NASA contract NAS5-30934.
Nousek, J. A., & Shue, D. R. 1989, ApJ, 342, 1207
Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1986, Numerical Recipes (New York, Cambridge University Press)
Wheaton, W. A. et al. 1995, ApJ, submitted