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Astronomical Data Analysis Software and Systems IV
ASP Conference Series, Vol. 77, 1995
R. A. Shaw, H. E. Payne, and J. J. E. Hayes, eds.
Bias­Free Parameter Estimation with Few Counts, by
Iterative Chi­Squared Minimization
K. Kearns, F. Primini, and D. Alexander
Smithsonian Astrophysical Observatory, 60 Garden St., Cambridge, MA
02138
Abstract. We present a modified ü 2 fitting technique, useful for fitting
models to binned data with few counts per bin. We demonstrate through
numerical simulations that model parameters estimated with our tech­
nique are essentially bias­free, even when the average number of counts
per bin is ¸1. This is in contrast to the results from traditional ü 2 tech­
niques, which exhibit significant biases in such cases (see, for example,
Nousek & Shue 1989; Cash 1979). Moreover, our technique can explicitly
handle bins with 0 counts, obviating the need to ignore such bins or rebin
the data. We conclude with a discussion of the problem of estimating
goodness­of­fit in the limit of few counts using our modified ü 2 statistic.
1. Introduction
When fitting models to data with few counts, two of the most common methods
used are the standard ü 2 method and the C statistic. Use of the ü 2 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
of­fit. 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 goodness­of­fit parameter which is indistinguishable from the
standard ü 2 parameter for many datasets.
2. Iterative Weighting
Iterative Weighting (IW) is an example of the class of weighted least­squares
estimators described by Wheaton et al. (1994), in which ü 2 is expressed as a
weighted sum of squared deviations,
ü 2 =
X
i
W i [O i \Gamma M i (p 1 ; p 2 :::)] 2 ;
where O i are the observed counts in bin i, M i (p 1 ; p 2 :::) are the counts predicted
by the model M with parameters (p 1 ; p 2 :::), and the weights W i are the inverses
of the true variances oe 2
i . As Wheaton et al. (1994) point out, the approximation
1

2
W i ' O \Gamma1
i leads to significant biases in the best­fit parameters, due to the strong
anti­correlation between W i and [O i \Gamma M i ] 2 . Similar biases are encountered if
the approximation W i ' M \Gamma1
i is used (Nousek & Shue 1989). The IW technique
avoids such biases by estimating W i through successive iterations, where for
each iteration, j, W j
i ' [M i (p \Lambda; j \Gamma1
1 ; p \Lambda; j \Gamma1
2 :::)] \Gamma1 , and the best­fit parameters
p \Lambda; j
1 ; p \Lambda; j
2 ::: are determined by minimization of
ü 2; j =
X
i
[O i \Gamma M i (p j
1 ; p j
2 :::)] 2
M i (p \Lambda; j \Gamma1
1 ; p \Lambda; j \Gamma1
2 :::)
:
For the first iteration, all weights are set to 1. In our sample, we find that the
minimum ü 2 values and best­fit parameters converge after about 6 iterations.
3. Data Simulation
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 power­law spectrum such that:
n i = N o
Z E 1 +i\DeltaE
E 1 +(i\Gamma1)\DeltaE
E \Gammafl dE; N = N o
Z 0:845
0:095
E \Gammafl dE
for i = 1 ! 15; \DeltaE = (0:845 \Gamma 0:095)=15; and fl = 2:0 For each ideal spectrum,
we simulate 1000 sample spectra fn i g, where fn i g are random deviates drawn
from Poisson distributions with means = n i . We then determine best­fit model
parameters N o
calc and fl calc for each simulated spectrum, using IW and Powell's
method for function minimization (Press 1988). For each N, we then compute
the average N o calc =N o and fl calc =fl; compile the distributions of minimum ü 2
for comparison with the theoretical distribution; and compute the percentage
of simulations for which the ü 2
min and \Deltaü 2 contours include N o and fl, for
comparison with the expected percentages.
4. Results
In Table 1 we compare the biases (as measured by the ratios of average best­fit
parameter values to true values) in 1000 IW runs with those found for traditional
ü 2 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 1. In Figure 2 we compare both differential
and cumulative theoretical ü 2 distributions with our observed distributions. We
apply a KS­test 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 ü 2
min + \Deltaü 2 contours include
N o and fl, for \Deltaü 2 values appropriate to various joint two­parameter confidence
levels, is shown in table 2. For most N, the measured and expected confidence
levels are in good agreement.

3
Traditional ü 2 Iterative Weighting C­Statistic
N N calc =N o fl calc =fl %cnvg N calc =N o fl calc =fl %cnvg N calc =N o fl calc =fl %cnvg
25 0.709 1.152 96 1.145 1.003 98 1.269 0.958 86
50 0.647 1.134 100 1.055 1.008 99.6 1.079 0.998 100
75 0.636 1.130 100 1.025 1.009 100 1.078 0.995 100
100 0.673 1.109 100 1.008 1.008 100 1.053 0.996 100
150 0.707 1.094 100 1.025 1.001 100 1.015 1.005 100
250 0.767 1.072 100 1.019 1.000 100 1.019 1.000 100
500 0.863 1.040 100 1.007 1.000 100 0.997 1.004 100
1000 0.937 1.017 100 1.005 1.000 100 1.001 0.999 100
Table 1. Comparison of three fitting techniques.
Minimum ü 2 plus ... (for 2 parameters of interest)
N 2.30 4.61 6.17 9.21 11.80 18.40
(68.3%) (90%) (95.4%) (99%) (99.73%) (99.99%)
25 69.8 87.0 92.1 96.8 98.4 99.9
50 68.9 88.1 93.7 97.2 98.2 98.3
75 67.7 87.8 93.5 98.4 99.3 99.8
100 68.1 89.1 94.1 98.2 99.0 99.8
150 67.0 87.1 93.6 97.7 99.3 100
250 68.0 90.3 95.6 99.2 99.8 100
500 69.1 90.1 95.5 98.9 99.6 99.9
1000 69.6 88.9 95.0 99.0 99.7 100
Table 2. Estimating Confidence Limits: Percentage of Best Fits
Within Various ü 2 Boundaries, from a total of 1000 spectral fits.
5. Conclusions
We find that unbiased parameter estimates by ü 2 minimization are possible
for binned data with few or no counts in some bins, provided the ü 2 calcula­
tion is modified slightly. Except for very small N, this modified ü 2 statistic
is distributed according to the theoretical ü 2 distribution. Goodness­of­fit can
therefore be assessed using traditional techniques. Further, this ü 2 statistic can
be used to estimate confidence levels from standard ü 2
min + \Deltaü 2 boundaries.
Acknowledgments. This work is partially supported by NASA contract
NAS5­30934.
References
Cash, W. 1979 ApJ, 228, 939
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

4
0.95
1.00
1.05
1.10
1.15
Traditional Chisq
Iterative Weighting
C­Statistic
Number of counts in spectrum
0 200 400 600 800 1000
0.6
0.8
1.0
1.2
Figure 1. Bias in best­fit parameters for three fitting techniques.
0 10 20 30 40
0
0.02
0.04
0.06
0.08
0.1
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40
0
0.02
0.04
0.06
0.08
0.1
0
0.2
0.4
0.6
0.8
1
Figure 2. Comparison of theoretical ü 2 distribution with observed
distribution for IW by KS­test, with overlaid histograms.