Geometrical aspects of parameter estimation of a stochastic gravitational wave background: Beyond the Fisher analysis

The maximum likelihood method is often used for parameter estimation in gravitational wave astronomy. Recently, an interesting approach was proposed by Vallisneri to evaluate the distributions of parameter estimation errors expected for the method. This approach is to statistically analyze the local peaks of the likelihood surface, and works efficiently even for signals with low signal-to-noise ratios. Focusing special attention to geometric structure of the likelihood surface, we follow the proposed approach and derive formulas for a simplified model of data analysis where the target signal has only one intrinsic parameter, along with its overall amplitude. Then we apply our formulas to correlation analysis of stochastic gravitational wave background with a power-law spectrum. We report qualitative trends of the formulas using numerical results specifically obtained for correlation analysis with two Advanced-LIGO detectors.

Figure 1

The functions $F_{\mathrm{pk}}(a)$ and $F_{\mathrm{vl}}(a)$ for densities of local peaks and valleys.

Figure 2

The overlap reduction function $\gamma$ for the two LIGO detectors ($\mathrm{\text{Hanford}}+\mathrm{\text{Livingston}}$).

Figure 3

The shape function $w_0(x)$ for the correlation analysis with the two Advanced LIGO detectors.

Figure 4

Correlation of two unit vectors $D_0=\{\widehat{k}(p),\widehat{k}(0)\}$, the ratio $D_1/\sqrt{C_{11}}$ and the inner product $Y=\{\widehat{k}(p),{\widehat{k}}_{\mathrm{oth}}\}$ related to the peak and valley abundances.

Figure 5

Correlation of noises $⟨N_0(p_1)N_0(p_2)⟩$ at two points $p=p_1$ and $p=p_2$. We have the relation $⟨N_0(p_1)N_0(p_2)⟩=\{\widehat{k}(p_1),\widehat{k}(p_2)\}$.

Figure 6

Density distribution of the local peaks ${\sigma }_{\mathrm{pk}}(p)$ and valleys ${\sigma }_{\mathrm{vl}}(p)$ for the true spectral index $p_{\mathrm{t}}=0$. We plot five curves for the intrinsic signal strengths ${\rho }_{\mathrm{t}}=0$, 1, 2, 4 and 8.

Figure 7

Profiles of the integrals $U_{\mathrm{pk}}$ (squares) and $U_{\mathrm{vl}}$ (circles). We have $U_{\mathrm{pk}}<1$ at ${\rho }_{\mathrm{t}}\lesssim 3$., and $U_{\mathrm{pk}}>1$ at ${\rho }_{\mathrm{t}}\gtrsim 3$.

Figure 8

Distribution $s_{\mathrm{pk}}({\mathcal{M}}_{I\mathrm{pk}},p)$ of the peaks height ${\mathcal{M}}_{I\mathrm{pk}}$ identified at the spectral indexes $p=-3$, 0, and 6. The intrinsic signal strength is ${\rho }_{\mathrm{t}}=4$. We have the identity ${\int }_{-\infty }^{\infty }d{\mathcal{M}}_{I\mathrm{pk}}{s}_{\mathrm{pk}}({\mathcal{M}}_i,p)={\sigma }_{\mathrm{pk}}(p)$ for the area of each curve.

Figure 9

The two dimensional density distribution ${\sigma }_{\mathrm{pk}}(\rho ,p)={\sigma }_{\mathrm{pk}}({\mathcal{M}}_{I\mathrm{pk}}=\rho ,p)$ of local peaks (left), the Fisher matrix prediction ${\sigma }_{\mathrm{pk}}(p)$ (middle) and the density of saddle points ${\sigma }_{\mathrm{vl}}(\rho ,p)$ (right). The true spectral index is $p=p_{\mathrm{t}}=0$ and the intrinsic signal strength is at ${\rho }_{\mathrm{t}}=2$ (top row), 4 (middle row), and 8 (bottom row). We show the isodensity contours for ${\sigma }_{\mathrm{pk}}=0.5$, 0.1, 0.01, and 0.0001.

Figure 10

The densities of the local peaks ${\sigma }_{\mathrm{pk}}(>{\mathcal{M}}_{I\mathrm{pk}},p)$ above given thresholds ${\mathcal{M}}_{I\mathrm{pk}}=-\infty$, 2, 4 and 6. We put ${\rho }_{\mathrm{t}}=8$.

Figure 11

Comparison of the local peak density ${\sigma }_{\mathrm{pk}}(p)$ (solid curves) with the Fisher matrix prediction (dashed curves).

Figure 12

Difference between the local peak density ${\sigma }_{\mathrm{pk}}(p)$ and the Fisher matrix prediction ${\sigma }_{\mathrm{fisher}}(p)$ for the intrinsic signal strengths at ${\rho }_{\mathrm{t}}=2$, 4, and 8.

Figure 13

Difference ${\sigma }_{\mathrm{pk}}(p)-{\sigma }_{\mathrm{fisher}}(p)$ shown with the rescaled variable $x\equiv {\rho }_{\mathrm{t}}\Delta p$. The dashed curve is for ${\rho }_{\mathrm{t}}=16$ and the thin solid one for ${\rho }_{\mathrm{t}}=32$. The thick solid one is the asymptotic limit $\eta (x)$ given in Eq. (64).

Figure 14

Absolute values for the variance and mean of the local peak density [see Eq. (110)]. The solid curves are the leading order terms $\propto {\rho }_{\mathrm{t}}^{-2}$ [Eqs. (66, 67)]. The mean value changes its sign from $+$ to $-$ around ${\rho }_{\mathrm{t}}=4$.

Figure 15

The positive bias $⟨\rho ⟩$ for the amplitude parameter $\rho$ estimated by the maximum likelihood analysis. The circles are obtained with Eq. (111) and the solid curve is their asymptotic form $1/2{\rho }_{\mathrm{t}}$ [see Eq. (68)].