Analysis and Interpretation of the Cramér-Rao Lower-Bound in Astrometry: One-Dimensional Case

The parameter estimation problem at hand can be presented in one dimension, in general terms, as follows: Let us consider a collection of I_i (with i = 1...n) independent and identically distributed realizations of a random variable. In this setting, it will be assumed that the {I_i : i = 1,...,n} measurements follow an underlying probability (density—if continuous, or mass—if countable) function, denoted by f_θ, which depends upon a certain target (unknown) parameter θ. Then, the parameter estimation problem reduces to finding a prescription (or statistics) such that the function is a good approximation of the underlying parameter θ that generated the data.

A standard criterion adopted in statistics to estimate θ is to consider the rule of minimum variance (denoted by Var), given by:

where is the expectation value of the argument and "arg min" represents the argument that minimizes the expression. Note that in the last equality we have assumed that is an unbiased estimator of the parameter (i.e., that ), so that under this rule we are implicitly minimizing the mean square error of the estimate with respect to the hidden true parameter θ.

Unfortunately, the general solution of equation (1) is intractable, as in principle it requires the knowledge of θ, which is the essence of the inference problem. However, there are performance bounds that characterize how far can we be from the theoretical solution in equation (1), and even scenarios where the optimal solution can be achieved in a closed-form. One of the most significant results in this field is the Cramér-Rao minimum variance bound (Rao 1945, Cramér 1946) explained below.

Let be an unbiased estimator of θ. If we define the function L(I₁,...,I_n; θ) as the likelihood of the observation given the model parameter θ, and we have n independent random variables I_i driven by the probability function f_θ, then the Cramér-Rao bound states that:

provided that we satisfy the constraint:

and where:

is called the Fisher information of the data about θ.

A powerful corollary of this result is that the minimum variance of any unbiased estimator that satisfies equation (3) is always going to be greater than the prespecified quantity given by equation (2).

Generally, this bound is not attained for the minimum variance estimator of θ; however, there exists a necessary and sufficient condition that guarantees the existence of an estimate achieving the Cramér-Rao bound, . More precisely, if we can write (see, e.g., Stuart et al. [2004], p. 12):

then it is certain that , as long as A(θ) is a function exclusively of the parameter θ (and, in particular, does not depend on the observables, I_i).² However, we must keep in mind that, unless the condition in equation (5) is satisfied, the minimum variance solution from equation (1) will have, in general, a variance strictly greater than . This important result will be further used in § 5.

The position estimation in astrometry is a slight variation of the classical parameter estimation problem introduced in § 2.1. In this article we focus on the 1D version, as it is more easily tractable from the numerical and analytical point of view, while capturing all the key elements of the problem. The extension to the 2D case will be dealt with in a forthcoming paper. However, as we shall see (see § 4.1.5 and 5), it is expected that some generalizations are possible from our 1D results which are likely to be approximately valid on the 2D scenario in some cases.

In the 1D case we have an array detector with n pixels, in which we measure the fluxes {I_i} per pixel. We will assume that the total expected (as opposed to measured) flux at each pixel on the detector, given by a function λ_i(x_c), will explicitly depend on the location of the source on the array, denoted by x_c, which is the parameter we want to determine (equivalent to the unknown parameter θ of § 2.1). Of course, this flux is not measured directly, because the actual observations, I_i, are subject to noise. However, on photon counting devices, such as CCDs, the measured flux follow a Poisson noise distribution, i.e., the I_i are random variables driven by a Poisson distribution (this determines the probability mass function f_θ introduced in § 2), with expectation value given by λ_i(x_c). At this point, we note an important difference in approach to that adopted in the work by Lee and van Altena (1983), in which they have assumed a Gaussian noise per pixel, valid for an analog detector, such as photographic plates. As we shall see, when the noise is Gaussian, a maximum likelihood parameter determination reduces to least squares, which is what they indeed adopted. Note, however, that King (1983) also adopted a least squares minimization, although for CCDs the noise is not Gaussian, and therefore a maximum likelihood solution is not equivalent to a least squares minimization (see § 5).

In order to estimate the Cramér-Rao bound in this situation, we first need to verify that our likelihood function satisfies equation (3). The likelihood function of the 1D array observations will be given by:

where , since the I_i follows a Poisson mass function distribution with mean λ_i(x_c).³ Then, we see that:

and we indeed verify that because . Hence, we can apply equations (2) and (4) to obtain the following result:

For completeness, the derivation of the Fisher information about x_c, is presented in Appendix A.

In the following section, we provide a detailed analysis of this expression and its practical implications on astrometry.

An equivalent to equation (10) in 2D has been presented, in a slightly different manner, by Lee and van Altena (1983) in the context of the expected astrometric accuracy on photographic plates (their eq. [9]). Indeed, it is easy to see that their , where we have assumed that both the source and the background follow Poisson noise (in the case of CCD detectors this is valid when the fluxes are measured in photo-e^-), and where and are the variances in the source and the background, respectively, at pixel i, j. We emphasize, however, that their equation (9) has been derived in the case of a (weighted) least squares solution (their equation [5]). This equation is valid for an analog detector, such as photographic plates, where the photographic densities in each pixel are assumed to follow Gaussian noise (see eq. [4] in Lee and van Altena [1983]). However, as will be demonstrated in § 5, in the case of digital detectors, where the noise follows a Poisson distribution, no parameter estimator can formally reach the Cramér-Rao bound.

We note that, for Poisson noise, the standard deviation of the signal is the square root of the signal and, thus, the variance is the signal itself. Thus, the interpretation of the term in equation (10) as the variance of the counts (in e^-) is important, as indicated in what follows. It is often more convenient for evaluation purposes to express and in terms of "counts" on the detector (from now on referred to as "ADUs"—analog to digital units), rather than in e^-, by introducing the so-called (inverse-) gain of the detector G in units of e^-/ADU (see, e.g., Gilliland [1992]). In this case, the source and background fluxes, and , are given by G·F and by G·B respectively, where F and B are in units of ADU. On the other hand, if is the rms deviation at pixel i (in e^-) and σ_Fi is the equivalent quantity in units of ADUs, then it is true that , and likewise for the rms deviation on the background. By putting these unit conversions into either equation (10) or (11) we see that we can express those equations with the source and the background measured in units of either e^- or ADUs, in the sense that:

To evaluate the Cramér-Rao bound if we have empirical measurements of fluxes and variances on a detector (ADUs), it would obviously be more convenient to use equation (16). However, when performing numerical experiments for a given detector setup (as done in this article, § 5.3) where we only specify flux levels for the source and the background, and where we calculate the variances associated with them, we would instead need to use equation (15) (see also eq. [21]). This is because we know that if the flux is in expressed e^-, then the variances are equal to the flux (but this is not the case if the flux is measured in ADUs). We further note that equations (15) or (16) suggest that the Cramér-Rao bound represents a mean "uncertainty over the derivate of the signal", since:

where the <  > stands for a classical type of harmonic mean over the pixels, and where . This provides an interesting connection with what will be presented in § 5.3: We note that if we have a function of one variable, y = f(x), then⁴ . If we have n independent measurements, indicated by (x_i,y_i), each with uncertainty (σ_xi,σ_yi), we know from the propagation of errors in a least squares sense that the minimum variance for the weighted mean would be given by (see, e.g., Meyer [1992], chap. 10), which is equivalent to the Cramér-Rao bound if we identify f_i = λ_i, and since the errors follow a Poisson distribution, then . We emphasize that this analogy is valid only in the 1D case (otherwise we would need to include the partial derivatives of f, and the variances in all the parameters). Indeed, as we shall see in § 5, the least squares does not reach, in general, the Cramér-Rao lower bound.

An interesting aspect of equations (10) and (11) is that the positional uncertainty is going to be dominated by the region near the center of the PSF, where its derivative is steeper. Including regions far from the central core of the PSF, where , will not contribute to a decrease in the uncertainty and, instead, will only deteriorate the overall S/N of the source by incrementally adding more noise than signal (see the paragraph following equation [28]).

It is evident that very little progress can be made in the estimation of the Cramér-Rao bound from equation (11) unless we specify a shape for the PSF. Various analytical forms have been proposed for the PSF of a point source as imaged by ground-based (King 1971) and space-based (King 1983) detectors (but see also Bendinelli et al. [1988]). Without losing too much generality, in this study we will adopt a Gaussian function which seems to be a good representation of the PSF, at least from the standpoint of astrometric accuracy on ground-based data (Méndez et al. [2010]), and which allows some simple analytical manipulation (see § 4.1). Under this assumption, we would have:

where we adopt to measure x, x_c and σ in units of arcseconds.

In this case, it is easy to show that the derivative of g_i in equation (12) can be written in a closed-form, as follows:

where:

with and .

Combining the results of equations (12), (18), (19) and (20) in equation (11) and converting to ADUs, we finally arrive at the following exact expression for the Cramér-Rao lower-bound in 1D for a Gaussian PSF:

where we have assumed, for simplicity, that the background is uniform (and equal to B) under the PSF of the object. As noted by Winick (1986), this expression makes it explicit that the Cramér-Rao bound depends on F and B separately, and not just on the ratio F/B. If we adopt σ and Δx in units of arcseconds in the sky, then the square root of equation (21) gives us the Cramér-Rao bound in units of arcseconds directly.

In Figure 1 we show the results of evaluating equation (21) under the experimental setting proposed by Winick (1986), i.e., assuming a fixed set of F and B values (and therefore a constant ratio F/B) for different values of the detector pixel size Δx. In this figure we introduce the "Full-Width at Half-Maximum" (FWHM), usually termed as "image quality" at astronomical observing sites, which is related to the Gaussian σ through . Figure 1 is equivalent to Figure 1 in Winick (1986).

Zoom In Zoom Out Reset image size

We note that, for a given continuous pixel x-coordinate on the array, the corresponding (integer) pixel ID, i, on the array, is given by:

where the function INT represents the integer part of the argument. In this article we adopt that pixel ID i = 1 has pixel coordinates 0.5 ≤ x < 1.5, pixel ID i = 2 has pixel coordinates 1.5 ≤ x < 2.5, and so on, following the convention of the IRAF package.⁵ In this scheme, each pixel has width 1.0 in pixel x-coordinates, centered at x = FLOAT(i), with upper/lower pixel boundaries given by x_± = FLOAT(i) ± 0.5 (where the FLOAT function converts an integer into a real number). The relationship between pixel x-coordinates and "physical" coordinates (projected onto the sky, as measured from the origin of the array), is given by px = [(x - 0.5)·Δx] arcseconds.

An interesting feature is the effect on the predicted Cramér-Rao bound of pixel decentering of the source. Figure 1 shows the effect of pixel decentering for a FWHM = 0.5'' as a function of Δx. The fact that the Cramér-Rao bound depends on the location of the source itself was already pointed-out by Winick (1986), but this is evident from equations (10), (11) and (21), all of which explicitly depend on x_c. Of course, the effect is symmetrical with respect to the pixel center, so the impact of a decentering of, e.g., +0.125 pixels with respect to the pixel center is exactly the same as that of a decentering of −0.125 pixels, and (as long as the array properly samples the source), i.e., the effect is periodic (such that the Cramér-Rao bound is the same if the source is placed at x ± FLOAT(n), where n is an arbitrary integer).

The important role of image decentering in the case of undersampled images (such as those of HST; see below) has been demonstrated by Anderson & King (2000). They refer to the "pixel-phase error" as the systematic error in the derived center if a centering algorithm is used that does not properly account for the distribution of signal among pixels when the source is not centered in a pixel. On the other hand, the Cramér-Rao bound described here represents instead the precision, or random error, statistically attainable with an ideal, unbiased, centering algorithm.

The dotted line in Figure 1 is centered on a pixel, near the center of the array, the double-dotted-dashed line is for an offset of 0.125 pixels, and the dot-dashed line is for a pixel decentering of 0.25 pixels. As can be seen, the loss of astrometric accuracy as the pixel size increases is less severe when the target is not at the center of a given pixel, but, rather, when it is offset from it. This is actually an intuitive result: For a given pixel size, when the source is not centered on a pixel, its flux is spread among more neighboring pixels, and therefore the source can be located more precisely. This result also implies that, for undersampled systems, it is a good practice to "dither" the source a bit (even a fraction of a pixel) so that, in the end, the average Cramér-Rao bound is better than if the source were located at the center of a pixel, a well-known technique applied, e.g., to HST images (Anderson & King 2000; Fruchter & Hook 2002). To quantify the effect of dithering, in Table 1 we show the Cramér-Rao bound for a source with a FWHM = 0.5'' observed through detectors of pixel size Δx = 0.5, 0.7 and 0.9'', and when the source is centered and slightly decentered by the amounts indicated in the table. As can be seen from this table, for a pixel size that matches the FWHM, the change on the Cramér-Rao bound is small as a function of pixel offset. In the case of a 0.7'' pixel size, a dither pattern including offsets of 0.125 and 0.25 pixels, plus a central pointing, yields an average Cramér-Rao bound of ∼6.0 mas, which represents an almost 18% improvement over the (single) centered Cramér-Rao bound. For a 0.9'' detector the effect is even more dramatic, yielding an almost 40% improvement.

Figure 1 also shows that, while the dithering technique offers a better asymptotic trend among all the values at large Δx (i.e., in the low-resolution regime), eventually, when Δx becomes too large, the astrometric accuracy deteriorates regardless of the relative position of the source with respect to the center of a pixel (even with dithering), as expected.

Under certain circumstances, the summation in the denominator of the RHS of equation (10) can be approximated into an integral, which allows us to explore the behavior of the Cramér-Rao bound in a more explicit manner. Indeed, we see from equations (12) and (18) that the application of the mean-value theorem when Δx/σ ≪ 1 implies that . In this case, for a Gaussian PSF, it is easy to show that:

Replacing the RHS of equation (29) into the RHS of equation (10) we have:

Let us have a closer look at the (dimensionless) and terms in the RHS of equation (30). For our Gaussian function we will have:

which, in the small pixel size approximation becomes:

where is the (dimensionless) maximum flux, which will occur at a certain pixel j that satisfies the condition x_j - Δx/2 ≤ x_c < x_j + Δx/2.

Equation (32) prompts us to define a new function, which describes the distribution across pixels, in the small-pixel approximation, of the square of the flux that appears in equation (30), as:

where is a proper normalization factor (in units of arcsec^-1; see below).

Combining equations (33) and (34) in equation (30), and considering that Δx is very small, we have:

Taking into account the definition of in equation (33), from equation (34) we find that , where we have used the fact that, since Δx is very small, then x_j ≈ x_c. Replacing this approximation and equation (33) into equation (35) we get for the Cramér-Rao bound in the small-pixel approximation:

where we note that (the trivial definition of) integral I has units of arcsec³.

This formula is only valid under the assumption of small pixels, so, in a general application, equation (10) should be used instead, or equation (21) for a Gaussian PSF. However, the truly interesting aspect of this equation is that it can be explicitly evaluated in two extreme cases: When the detection is dominated by the source, and when it is dominated by the background. This is done in the next subsections.

4.1.1. Weak Source

When the background dominates, then . In this case we would have:

If we assume an approximately constant background under the PSF of the target then:

Therefore, in this case, replacing I into equation (36) and re-arranging terms, the Cramér-Rao bound becomes:

The (pedagogical) use of this equation is that it allows us to draw some basic conclusions regarding the expected positional accuracy in this regime. First, because we predict, in general, a rather large positional uncertainty, as expected, due to the low S/N of the source. Furthermore, the accuracy will improve proportionally to , in agreement with equation (10) of Lee and van Altena (1983) (compare also the first line of eq. [39] with eq. 7 in Auer and van Altena [1978]). Furthermore, as intuitively expected, the accuracy deteriorates for a larger background, coarser pixel size, or for lower-quality images (or sites), but relatively slowly: Only as the square root of these parameters.

4.1.2. Strong Source

In this case, the signal from the source dominates over the background, i.e., , and the approximation for I becomes:

Evaluating the definite integral we have:

Replacing this value for I into equation (36) we end up with:

We see that, in this regime, the ultimate positional accuracy is proportional to , similar to what was found by Lee and van Altena (1983) (see their eq. [13]). Again, the accuracy deteriorates slowly for coarser pixel size and for lesser-quality images, but in this case the background level, formally, plays no role in the expected accuracy.

4.1.3. Limiting Cases as a Function of Total Flux

Equations (39) and (42) are a bit misleading because, by definition, F_max depends itself on the adopted value for Δx and, therefore, needs to be evaluated in each particular case. However, in the small pixel approximation, one can find an approximate relationship between the total flux F (which is independent of Δx) and F_max. Indeed, assuming, as done before, that x_j ≈ x_c, then:

Replacing equation (44) into equations (39) and (42) we have:

Interestingly, we now see that the positional accuracy should deteriorate linearly with the FWHM of the image for strong sources, but it will not otherwise depend on the array pixel size. However, for weak sources, the dependence on the FWHM is not only steeper but, also, the finer the pixel size, the larger the expected positional uncertainty. This latter result should also be intuitive since, when we are dominated by the background, the more pixels we have under the PSF, and the larger the contribution of the background will be, to the point of significantly perturbing the final positional accuracy. Equivalently, note that the term B/Δx is the background per unit area (see eq. [23]): As this value becomes larger, we indeed expect a larger positional uncertainty, as shown by the first line of equation (45). The behavior depicted by equation (45) is similar to that found by King (1983): For sources where the background dominates, his equation (23) predicts that the positional uncertainty goes as , whereas when the background is negligible, his equation (24) shows that the positional uncertainty increases as . Similarly, Lindegren (1978) (see also van Altena & Fomalont [2013], p. 127) finds that for a seeing-limited image with no background, the limiting astrometric precision goes as FWHM/(S/N), which is equivalent to equation (45) for the case F >  > B, when (see eq. [28]). All these predictions coincide with those from our equation (45). Finally, and as already noted in § 4, the general trends seen in Figure 2 are well explained by equation (45).

4.1.4. Range of Use of the High Resolution Cramér-Rao Bound

Besides their qualitative usefulness, what is the range of applicability of equations (39) and (42), or (45)? For illustration purposes, in Table 2 we compare the values predicted by these equations, with the "exact" prediction from equation (21), for some representative values of the parameters. From this table we can see that, even for relatively large pixels, in comparison with the FWHM, these equations predict very reasonable values in comparison with the "exact" value, as long as we respect the conditions for F/B under which equations (39) and (42) were derived. The second and third lines show that, indeed, as predicted by equation (45), the Cramér-Rao bound does not depend on B. The third and fourth lines show that in the high-S/N regime the Cramér-Rao bound goes linearly with the FWHM, while a comparison of the first and fifth line demonstrate that in the low-S/N case the Cramér-Rao bound varies as the ratio FWHM²/Δx, as shown by equation (45). Finally, in the last two lines of the table, we show a case when equation (45) fails miserably, i.e., for an intermediate S/N value.

4.1.5. A Simplified Extension to the 2D Case

Given the flux values I₁,...,I_n, the maximum-likelihood (ML) estimate of the position x_c is obtained through the following rule:

where "arg max" represents the argument that maximizes the expression. Imposing the first order condition on this optimization problem, it reduces to satisfying the condition , and, consequently, we can work with the general expression in equation (7). We note from that equation that the term in the brackets given by is independent of x_c, and consequently its derivative is zero. Therefore the ML condition becomes:

where we have used the high-resolution approximation, equation (29). In general, this condition does not offer a closed-form solution (note that the dependency of on x_c could be quite complex). Consequently, numerical gradient-descent methods need to be implemented. Nevertheless, in the signal-dominated regime, when , it is straightforward to show that equation (48) reduces to the classical moment solution, i.e.:

In general, the formal solution to equation (48) in the high-resolution regime offers a simple relationship to implement a recursive algorithm that satisfies the ML estimate, given by:

where the weights .

To conclude this subsection, we must mention that it is well-known that in the setting of independent and identically distributed measurements, the ML estimate is asymptotically unbiased and efficient (Stuart et al. 2004, chap. 18). In other words, as the number of samples increases, the variance tends to the Cramér-Rao bound and consequently the ML is asymptotically optimal (in the sense of achieving the minimum variance bound). However, the astrometry setting is different as the measurements (fluxes) follow a Poisson distribution with parameters that are position-dependent (see § 2.2), and consequently the ML is not necessarily efficient in this statistical sense.

Given the flux values I₁,...,I_n, the (weighted) least square (LS) estimate of the position x_c is given by the following decision-rule:

where and b_i(x_c) ≡ Var(I_i) = λ_i(x_c) for all i.⁸ The unweighted LS estimate assumes instead that b_i = 1, where no variance normalization is considered at all.

If we define , then the first-order condition over the unweighted LS estimate implies that:

Hence, again using the high-resolution regime (eq. [29]), this condition implies that:

and, consequently, the problem reduces to solve the equation:

Note that this expression is the counterpart of equation (50) for the ML estimate, emphasizing that these two techniques offer different estimates of the position in the parametric setting of astrometry. Again, the solution of equation (54) involves the use of numerical methods, as, in general, no closed-form solution can be derived from it.

In this section we present some results from numerical experiments for the standard deviation of the astrometric position of a 1D Gaussian source sampled by a linear detector. The goal of these experiments is to compare the performance of the ML and LS solutions described in the previous subsections to the theoretical Cramér-Rao bound under different assumptions regarding the detector and the source.

Basically, we start by adopting a set of values for the gain and readout noise of the detector, as well as its pixel size and number of pixels. For a source with a Gaussian PSF, we specify its width, the maximum flux at the central pixel, and its center location (x_c). For the background, we adopt a certain (fixed) value per pixel. The total flux of the source (in ADUs) is obtained through equation (43) by direct integration of the Gaussian PSF given the adopted values for Δx, x_c and σ. We generate many possible "observations" for the same combination of parameters, using a random-number generator driven by a Poisson distribution using the poidev, gammaln, and ran1 routines explained in Press et al. (1992). We note that we transform all the ADUs (source and background) to units of e^- using the adopted gain before randomizing the data. When generating the data, we also consider the read-out noise and the "digitization noise" (see, e.g., Gilliland [1992]). On output we have an array of pixel positions (1 to n) and their corresponding fluxes.

After generating a (large) number of simulations for a given set of parameters, we compute the value of x_c using a ML and a LS (weighted and unweighted) procedure. Because we are operating in the 1D case, where we are estimating the object's position exclusively, all the other parameters are assumed to be known, and fed to the routine that searched for the value of x_c, using either equation (47) in the case of ML or equation (51) in the case of LS. To estimate the optimum number of simulations required to obtain a stable solution, we computed a very large number of simulations for our first set, and then calculated the mean of the recovered x_c and its standard deviation, σ_xc, as a function of the number of simulations. The optimum value for the number of simulations should render a set of values (x_c, σ_xc) that do not depend appreciably (within their statistical uncertainty) on adding more simulations. For our simulations, this number turned out to be ∼250 simulations per set.

In Table 3 we present the standard deviation from the simulations, σ_xc, for a number of representative cases, as well as the calculated Cramér-Rao minimum variance bound based on equation (21), denoted by σ_CR. In all cases we adopted an array size of 100 pixels that properly covers the PSF, a pixel size of 0.2'', a detector RON = 5e^-, a (fixed) sky background of 300 ADU per pixel, and a Gaussian source with a FWHM = 1^''.

To our surprise, the results in Table 3 indicate that the performance of the ML and LS estimators are almost equal to the Cramér-Rao lower bound. This is a remarkable result, and it validates the predictive power of the Cramér-Rao bound and its use as a benchmark indicator in astrometry. From the theoretical side, we formally showed (§ 5, eq. [46]) that the Cramér-Rao bound for astrometry can not be achieved. Our numerical results suggest then that both the ML and the LS methods (which are widely used in astrometry) are very efficient in the sense of asymptotically (with the number of measurements) approaching the Cramér-Rao bound. Proving this conjecture is an interesting topic of further work which will help us to explain the results obtained, and further consolidate the use the Cramér-Rao bound for performance analysis. A possible explanation, valid in the 1D case, has been proposed in § 3.1: This explanation, however, is not valid in a general situation where one needs to simultaneously estimate astrometric and photometric parameters, the subject of which will be further explored in a forthcoming paper. We also note from Table 3 that the ML is as good as, or even better in some cases, than the LS and WLS estimators.

We finally compare the predictions of the Cramér-Rao lower bound to the performance of some 2D digital centering algorithms applied to simulated stellar images, reported by Stone (1989). As argued in § 4.1.5 we should expect that our results from the 1D case should not significantly differ from those of a 2D case when the source is symmetrical. Indeed, for his simulations, Stone (1989) adopted symmetrical Gaussian sources (his eq. [1]) on a flat background. We have read approximately from his Figures 2–5 the results for the minimum rms (over different profile fitting methods) of the positional uncertainty from his 2D numerical simulations for some representative values of the total flux F. These are tabulated in our Table 3 under the label σ_St, in units of arcseconds. For each of these values we have computed the predicted Cramér-Rao bound using our equation (21), and adopting the same values for F, B, FWHM, and Δx used by Stone (1989) for each of his simulations. These values are denoted by σ_CR in Table 3, also in units of arcseconds. As can be seen from this table, our predictions are extremely encouraging: In all cases our computed Cramér-Rao bound is smaller than (although typically close to) the results from the "measured" standard deviation of the position (derived from the 2D simulations). Furthermore, as indicated in § 5.1 (especially eq. [50]), when F >  > B we would expect that the ML estimate approaches the moment solution, which is indeed the case as seen from Table 4 (see also Figs. 2 and 3 for large "counts in image" in Stone [1989]).