Abstract

We analyze the statistical properties of the maximum likelihood estimator, least squares estimator, and Pearson’s χ2-based and Neyman’s χ2-based estimators for the estimation of decay constants and amplitudes for fluorescence lifetime imaging. Our analysis is based on the linearization of the gradient of the objective functions around true parameters. The analysis shows that only the maximum likelihood estimator based on the Poisson likelihood function yields unbiased and efficient estimation. All other estimators yield either biased or inefficient estimations. We validate our analysis by using simulations.

© 2013 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
  4. B. B. Collier and M. J. McShane, “Dynamic windowing algorithm for the fast and accurate determination of luminescence lifetimes,” Anal. Chem.84, 4725–4731 (2012).
    [CrossRef] [PubMed]
  5. H. Cramer, Mathematical Methods of Statistics (Princeton University, 1999).
  6. S. Laptenok, K. M. Mullen, J. W. Borst, I. H. M. van Stokkum, V. V. Apanasovich, and A. J. W. G. Visser, “Fluorescence lifetime imaging microscopy (FLIM) data analysis with TIMP,” J. Stat. Softw.18, 1–20 (2007).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  13. K. A. Walther, B. Papke, M. B. Sinn, K. Michel, and A. Kinkhabwala, “Precise measurement of protein interacting fractions with fluorescence lifetime imaging microscopy,” Mol. BioSyst.7, 322–336 (2011).
    [CrossRef] [PubMed]
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  19. P. Hall and B. Selinger, “Better estimates of exponential decay parameters,” J. Phys. Chem.85, 2941–2946 (1981).
    [CrossRef]
  20. K. J. Mighell, “Parameter estimation in astronomy with poisson-distributed data. I. the χγ2 statistic,” The Astrophys. J.518, 380 (1999).
    [CrossRef]
  21. J. Moré and D. Sorensen, “Computing a trust region step,” SIAM J. Sci. Comput.4, 553–572 (1983).
    [CrossRef]
  22. M. A. Branch, T. F. Coleman, and Y. Li, “A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems,” SIAM J. Sci. Comput.21, 1–23 (1999).
    [CrossRef]
  23. R. Byrd, R. Schnabel, and G. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Program.40, 247–263 (1988).
    [CrossRef]

2012 (1)

B. B. Collier and M. J. McShane, “Dynamic windowing algorithm for the fast and accurate determination of luminescence lifetimes,” Anal. Chem.84, 4725–4731 (2012).
[CrossRef] [PubMed]

2011 (1)

K. A. Walther, B. Papke, M. B. Sinn, K. Michel, and A. Kinkhabwala, “Precise measurement of protein interacting fractions with fluorescence lifetime imaging microscopy,” Mol. BioSyst.7, 322–336 (2011).
[CrossRef] [PubMed]

2009 (1)

2007 (3)

S. Kumar, C. Dunsby, P. A. A. D. Beule, D. M. Owen, U. Anand, P. M. P. Lanigan, R. K. P. Benninger, D. M. Davis, M. A. A. Neil, P. Anand, C. Benham, A. Naylor, and P. M. W. French, “Multifocal multiphoton excitation and time correlated single photon counting detection for 3-D fluorescence lifetime imaging,” Opt. Express15, 12548–12561 (2007).
[CrossRef] [PubMed]

S. Laptenok, K. M. Mullen, J. W. Borst, I. H. M. van Stokkum, V. V. Apanasovich, and A. J. W. G. Visser, “Fluorescence lifetime imaging microscopy (FLIM) data analysis with TIMP,” J. Stat. Softw.18, 1–20 (2007).

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

2004 (2)

S. Pelet, M. J. R. Previte, L. H. Laiho, and P. T. C. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation.” Biophys. J.87, 2807–17 (2004).
[CrossRef] [PubMed]

J. Kim and J. Fessler, “Intensity-based image registration using robust correlation coefficients,” IEEE Trans. Med. Imag.23, 1430 –1444 (2004).
[CrossRef]

2002 (1)

P. J. Steinbach, R. Ionescu, and C. R. Matthews, “Analysis of kinetics using a hybrid maximum-entropy/nonlinear-least-squares method: application to protein folding,” Biophys. J.82, 2244–2255 (2002).
[CrossRef] [PubMed]

2001 (2)

T. Hauschild and M. Jentschel, “Comparison of maximum likelihood estimation and chi-square statistics applied to counting experiments,” Nucl. Instrum. Meth. A457, 384–401 (2001).
[CrossRef]

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. A. M. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem.73, 2078–2086 (2001).
[CrossRef] [PubMed]

1999 (3)

T. G., “A matrix extension of the cauchy-schwarz inequality,” Econ. Lett.63, 1–3 (1999).
[CrossRef]

K. J. Mighell, “Parameter estimation in astronomy with poisson-distributed data. I. the χγ2 statistic,” The Astrophys. J.518, 380 (1999).
[CrossRef]

M. A. Branch, T. F. Coleman, and Y. Li, “A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems,” SIAM J. Sci. Comput.21, 1–23 (1999).
[CrossRef]

1996 (1)

J. Fessler, “Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography,” IEEE Trans. Image Process.5, 493 –506 (1996).
[CrossRef] [PubMed]

1993 (1)

J. Tellinghuisen and C. W. Wilkerson, “Bias and precision in the estimation of exponential decay parameters from sparse data,” Anal. Chem.65, 1240–1246 (1993).
[CrossRef]

1988 (1)

R. Byrd, R. Schnabel, and G. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Program.40, 247–263 (1988).
[CrossRef]

1983 (1)

J. Moré and D. Sorensen, “Computing a trust region step,” SIAM J. Sci. Comput.4, 553–572 (1983).
[CrossRef]

1981 (1)

P. Hall and B. Selinger, “Better estimates of exponential decay parameters,” J. Phys. Chem.85, 2941–2946 (1981).
[CrossRef]

Ameloot, M.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Anand, P.

Anand, U.

Apanasovich, V. V.

S. Laptenok, K. M. Mullen, J. W. Borst, I. H. M. van Stokkum, V. V. Apanasovich, and A. J. W. G. Visser, “Fluorescence lifetime imaging microscopy (FLIM) data analysis with TIMP,” J. Stat. Softw.18, 1–20 (2007).

Basaric, N.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Benham, C.

Benninger, R. K. P.

Beule, P. A. A. D.

Boens, N.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Borst, J. W.

S. Laptenok, K. M. Mullen, J. W. Borst, I. H. M. van Stokkum, V. V. Apanasovich, and A. J. W. G. Visser, “Fluorescence lifetime imaging microscopy (FLIM) data analysis with TIMP,” J. Stat. Softw.18, 1–20 (2007).

Branch, M. A.

M. A. Branch, T. F. Coleman, and Y. Li, “A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems,” SIAM J. Sci. Comput.21, 1–23 (1999).
[CrossRef]

Byrd, R.

R. Byrd, R. Schnabel, and G. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Program.40, 247–263 (1988).
[CrossRef]

Coleman, T. F.

M. A. Branch, T. F. Coleman, and Y. Li, “A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems,” SIAM J. Sci. Comput.21, 1–23 (1999).
[CrossRef]

Collier, B. B.

B. B. Collier and M. J. McShane, “Dynamic windowing algorithm for the fast and accurate determination of luminescence lifetimes,” Anal. Chem.84, 4725–4731 (2012).
[CrossRef] [PubMed]

Cotlet, M.

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. A. M. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem.73, 2078–2086 (2001).
[CrossRef] [PubMed]

Cramer, H.

H. Cramer, Mathematical Methods of Statistics (Princeton University, 1999).

Davis, D. M.

De Schryver, F. C.

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. A. M. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem.73, 2078–2086 (2001).
[CrossRef] [PubMed]

Dunsby, C.

Engelborghs, Y.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Fessler, J.

J. Kim and J. Fessler, “Intensity-based image registration using robust correlation coefficients,” IEEE Trans. Med. Imag.23, 1430 –1444 (2004).
[CrossRef]

J. Fessler, “Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography,” IEEE Trans. Image Process.5, 493 –506 (1996).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++ - The Art of Scientific Computing (Cambridge University, 2002).

French, P. M. W.

G., T.

T. G., “A matrix extension of the cauchy-schwarz inequality,” Econ. Lett.63, 1–3 (1999).
[CrossRef]

Gensch, T.

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. A. M. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem.73, 2078–2086 (2001).
[CrossRef] [PubMed]

Gratton, E.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Grecco, H. E.

Gryczynski, I.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Hall, P.

P. Hall and B. Selinger, “Better estimates of exponential decay parameters,” J. Phys. Chem.85, 2941–2946 (1981).
[CrossRef]

Hauschild, T.

T. Hauschild and M. Jentschel, “Comparison of maximum likelihood estimation and chi-square statistics applied to counting experiments,” Nucl. Instrum. Meth. A457, 384–401 (2001).
[CrossRef]

Hofkens, J.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. A. M. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem.73, 2078–2086 (2001).
[CrossRef] [PubMed]

Ionescu, R.

P. J. Steinbach, R. Ionescu, and C. R. Matthews, “Analysis of kinetics using a hybrid maximum-entropy/nonlinear-least-squares method: application to protein folding,” Biophys. J.82, 2244–2255 (2002).
[CrossRef] [PubMed]

Jentschel, M.

T. Hauschild and M. Jentschel, “Comparison of maximum likelihood estimation and chi-square statistics applied to counting experiments,” Nucl. Instrum. Meth. A457, 384–401 (2001).
[CrossRef]

Kim, J.

J. Kim and J. Fessler, “Intensity-based image registration using robust correlation coefficients,” IEEE Trans. Med. Imag.23, 1430 –1444 (2004).
[CrossRef]

Kinkhabwala, A.

K. A. Walther, B. Papke, M. B. Sinn, K. Michel, and A. Kinkhabwala, “Precise measurement of protein interacting fractions with fluorescence lifetime imaging microscopy,” Mol. BioSyst.7, 322–336 (2011).
[CrossRef] [PubMed]

Krajcarski, D. T.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Kumar, S.

Laiho, L. H.

S. Pelet, M. J. R. Previte, L. H. Laiho, and P. T. C. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation.” Biophys. J.87, 2807–17 (2004).
[CrossRef] [PubMed]

Lakowicz, J. R.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Kluwer Academic/Plenum, 1999).
[CrossRef]

Lanigan, P. M. P.

Laptenok, S.

S. Laptenok, K. M. Mullen, J. W. Borst, I. H. M. van Stokkum, V. V. Apanasovich, and A. J. W. G. Visser, “Fluorescence lifetime imaging microscopy (FLIM) data analysis with TIMP,” J. Stat. Softw.18, 1–20 (2007).

Lefèvre, J.-P.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Li, Y.

M. A. Branch, T. F. Coleman, and Y. Li, “A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems,” SIAM J. Sci. Comput.21, 1–23 (1999).
[CrossRef]

Malak, H.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Matthews, C. R.

P. J. Steinbach, R. Ionescu, and C. R. Matthews, “Analysis of kinetics using a hybrid maximum-entropy/nonlinear-least-squares method: application to protein folding,” Biophys. J.82, 2244–2255 (2002).
[CrossRef] [PubMed]

Maus, M.

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. A. M. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem.73, 2078–2086 (2001).
[CrossRef] [PubMed]

McShane, M. J.

B. B. Collier and M. J. McShane, “Dynamic windowing algorithm for the fast and accurate determination of luminescence lifetimes,” Anal. Chem.84, 4725–4731 (2012).
[CrossRef] [PubMed]

Michel, K.

K. A. Walther, B. Papke, M. B. Sinn, K. Michel, and A. Kinkhabwala, “Precise measurement of protein interacting fractions with fluorescence lifetime imaging microscopy,” Mol. BioSyst.7, 322–336 (2011).
[CrossRef] [PubMed]

Mighell, K. J.

K. J. Mighell, “Parameter estimation in astronomy with poisson-distributed data. I. the χγ2 statistic,” The Astrophys. J.518, 380 (1999).
[CrossRef]

Miura, A.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Moré, J.

J. Moré and D. Sorensen, “Computing a trust region step,” SIAM J. Sci. Comput.4, 553–572 (1983).
[CrossRef]

Mullen, K. M.

S. Laptenok, K. M. Mullen, J. W. Borst, I. H. M. van Stokkum, V. V. Apanasovich, and A. J. W. G. Visser, “Fluorescence lifetime imaging microscopy (FLIM) data analysis with TIMP,” J. Stat. Softw.18, 1–20 (2007).

Naylor, A.

Neil, M. A. A.

Owen, D. M.

Papke, B.

K. A. Walther, B. Papke, M. B. Sinn, K. Michel, and A. Kinkhabwala, “Precise measurement of protein interacting fractions with fluorescence lifetime imaging microscopy,” Mol. BioSyst.7, 322–336 (2011).
[CrossRef] [PubMed]

Pelet, S.

S. Pelet, M. J. R. Previte, L. H. Laiho, and P. T. C. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation.” Biophys. J.87, 2807–17 (2004).
[CrossRef] [PubMed]

Phillips, D.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Pouget, J.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++ - The Art of Scientific Computing (Cambridge University, 2002).

Previte, M. J. R.

S. Pelet, M. J. R. Previte, L. H. Laiho, and P. T. C. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation.” Biophys. J.87, 2807–17 (2004).
[CrossRef] [PubMed]

Qin, W.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Roda-Navarro, P.

Rumbles, G.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Schaffer, J.

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. A. M. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem.73, 2078–2086 (2001).
[CrossRef] [PubMed]

Schnabel, R.

R. Byrd, R. Schnabel, and G. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Program.40, 247–263 (1988).
[CrossRef]

Seidel, C. A. M.

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. A. M. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem.73, 2078–2086 (2001).
[CrossRef] [PubMed]

Selinger, B.

P. Hall and B. Selinger, “Better estimates of exponential decay parameters,” J. Phys. Chem.85, 2941–2946 (1981).
[CrossRef]

Shultz, G.

R. Byrd, R. Schnabel, and G. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Program.40, 247–263 (1988).
[CrossRef]

Sillen, A.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Silva, N. D.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Sinn, M. B.

K. A. Walther, B. Papke, M. B. Sinn, K. Michel, and A. Kinkhabwala, “Precise measurement of protein interacting fractions with fluorescence lifetime imaging microscopy,” Mol. BioSyst.7, 322–336 (2011).
[CrossRef] [PubMed]

So, P. T. C.

S. Pelet, M. J. R. Previte, L. H. Laiho, and P. T. C. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation.” Biophys. J.87, 2807–17 (2004).
[CrossRef] [PubMed]

Sorensen, D.

J. Moré and D. Sorensen, “Computing a trust region step,” SIAM J. Sci. Comput.4, 553–572 (1983).
[CrossRef]

Steinbach, P. J.

P. J. Steinbach, R. Ionescu, and C. R. Matthews, “Analysis of kinetics using a hybrid maximum-entropy/nonlinear-least-squares method: application to protein folding,” Biophys. J.82, 2244–2255 (2002).
[CrossRef] [PubMed]

Szabo, A. G.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Tamai, N.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Tellinghuisen, J.

J. Tellinghuisen and C. W. Wilkerson, “Bias and precision in the estimation of exponential decay parameters from sparse data,” Anal. Chem.65, 1240–1246 (1993).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++ - The Art of Scientific Computing (Cambridge University, 2002).

Valeur, B.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

van Hoek, A.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

van Stokkum, I. H. M.

S. Laptenok, K. M. Mullen, J. W. Borst, I. H. M. van Stokkum, V. V. Apanasovich, and A. J. W. G. Visser, “Fluorescence lifetime imaging microscopy (FLIM) data analysis with TIMP,” J. Stat. Softw.18, 1–20 (2007).

Van Trees, H.

H. Van Trees, Detection, Estimation, and Modulation Theory, Part 1 (John Wiley & Sons, 2001).

vandeVen, M.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Verveer, P. J.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++ - The Art of Scientific Computing (Cambridge University, 2002).

Visser, A. J. W. G.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

S. Laptenok, K. M. Mullen, J. W. Borst, I. H. M. van Stokkum, V. V. Apanasovich, and A. J. W. G. Visser, “Fluorescence lifetime imaging microscopy (FLIM) data analysis with TIMP,” J. Stat. Softw.18, 1–20 (2007).

Walther, K. A.

K. A. Walther, B. Papke, M. B. Sinn, K. Michel, and A. Kinkhabwala, “Precise measurement of protein interacting fractions with fluorescence lifetime imaging microscopy,” Mol. BioSyst.7, 322–336 (2011).
[CrossRef] [PubMed]

Wilkerson, C. W.

J. Tellinghuisen and C. W. Wilkerson, “Bias and precision in the estimation of exponential decay parameters from sparse data,” Anal. Chem.65, 1240–1246 (1993).
[CrossRef]

Willaert, K.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

Anal. Chem. (4)

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[CrossRef] [PubMed]

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007).
[CrossRef] [PubMed]

M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. A. M. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem.73, 2078–2086 (2001).
[CrossRef] [PubMed]

J. Tellinghuisen and C. W. Wilkerson, “Bias and precision in the estimation of exponential decay parameters from sparse data,” Anal. Chem.65, 1240–1246 (1993).
[CrossRef]

Biophys. J. (2)

S. Pelet, M. J. R. Previte, L. H. Laiho, and P. T. C. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation.” Biophys. J.87, 2807–17 (2004).
[CrossRef] [PubMed]

P. J. Steinbach, R. Ionescu, and C. R. Matthews, “Analysis of kinetics using a hybrid maximum-entropy/nonlinear-least-squares method: application to protein folding,” Biophys. J.82, 2244–2255 (2002).
[CrossRef] [PubMed]

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T. G., “A matrix extension of the cauchy-schwarz inequality,” Econ. Lett.63, 1–3 (1999).
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J. Kim and J. Fessler, “Intensity-based image registration using robust correlation coefficients,” IEEE Trans. Med. Imag.23, 1430 –1444 (2004).
[CrossRef]

J. Phys. Chem. (1)

P. Hall and B. Selinger, “Better estimates of exponential decay parameters,” J. Phys. Chem.85, 2941–2946 (1981).
[CrossRef]

J. Stat. Softw. (1)

S. Laptenok, K. M. Mullen, J. W. Borst, I. H. M. van Stokkum, V. V. Apanasovich, and A. J. W. G. Visser, “Fluorescence lifetime imaging microscopy (FLIM) data analysis with TIMP,” J. Stat. Softw.18, 1–20 (2007).

Math. Program. (1)

R. Byrd, R. Schnabel, and G. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Program.40, 247–263 (1988).
[CrossRef]

Mol. BioSyst. (1)

K. A. Walther, B. Papke, M. B. Sinn, K. Michel, and A. Kinkhabwala, “Precise measurement of protein interacting fractions with fluorescence lifetime imaging microscopy,” Mol. BioSyst.7, 322–336 (2011).
[CrossRef] [PubMed]

Nucl. Instrum. Meth. A (1)

T. Hauschild and M. Jentschel, “Comparison of maximum likelihood estimation and chi-square statistics applied to counting experiments,” Nucl. Instrum. Meth. A457, 384–401 (2001).
[CrossRef]

Opt. Express (2)

SIAM J. Sci. Comput. (2)

J. Moré and D. Sorensen, “Computing a trust region step,” SIAM J. Sci. Comput.4, 553–572 (1983).
[CrossRef]

M. A. Branch, T. F. Coleman, and Y. Li, “A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems,” SIAM J. Sci. Comput.21, 1–23 (1999).
[CrossRef]

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K. J. Mighell, “Parameter estimation in astronomy with poisson-distributed data. I. the χγ2 statistic,” The Astrophys. J.518, 380 (1999).
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Other (4)

H. Cramer, Mathematical Methods of Statistics (Princeton University, 1999).

J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Kluwer Academic/Plenum, 1999).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++ - The Art of Scientific Computing (Cambridge University, 2002).

H. Van Trees, Detection, Estimation, and Modulation Theory, Part 1 (John Wiley & Sons, 2001).

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Figures (6)

Fig. 1
Fig. 1

Mean of the estimated parameters for different photon counts: (a) τ1; (b) τ2; (c) A1; (d) A2;

Fig. 2
Fig. 2

Variance in the estimation of parameters for different photon counts: (a) τ1; (b) τ2; (c) A1; (d) A2;

Fig. 3
Fig. 3

Mean of the estimated parameters for different initial parameter values (high photon counts): (a) τ1 (true value=1); (b) τ2 (true value=4); (c) A1 (true value=72); (d) A2 (true value=62);

Fig. 4
Fig. 4

Variance in the estimation of parameters for different initial parameter values (high photon counts): (a) τ1; (b) τ2; (c) A1; (d) A2;

Fig. 5
Fig. 5

Mean of the estimated parameters for different initial parameter values (low photon counts and similar decay parameters): (a) τ1 (true value=1); (b) τ2 (true value=2); (c) A1 (true value=28); (d) A2 (true value=18);

Fig. 6
Fig. 6

Variance in the estimation of parameters for different initial parameter values (low photon counts and similar decay parameters): (a) τ1; (b) τ2; (c) A1; (d) A2;

Equations (39)

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y ( t i ) = Poisson { λ ( t i ; θ ) } , i = 0 , . , N 1 ,
λ ( t i ; θ ) = g ( t i ; θ ) * h ( t i ) ,
g ( t ; θ ) = j = 0 M 1 A j e ( t τ j ) ,
θ ^ = argmin θ > 0 Φ ( θ ; Y ) .
L ( θ ; Y ) = i = 0 N 1 λ i ( θ ) y i log λ i ( θ ) + log y i ! .
Φ L S ( θ ; Y ) = 1 2 i = 0 N 1 ( y i λ i ( θ ) ) 2 .
Φ P ( θ ; Y ) = 1 2 i = 0 N 1 ( y i λ i ( θ ) ) 2 λ i ( θ ) .
Φ N ( θ ; Y ) = 1 2 i = 0 N 1 ( y i λ i ( θ ) ) 2 max { y i , 1 } .
θ Φ ( θ ^ ; Y ) = θ Φ ( θ ; Y ) | θ = θ ^ = 0 ,
θ Φ ( θ ^ ; Y ) θ Φ ( θ ; Y ) + [ θ [ θ Φ ( θ ; Y ) ] T ] ( θ ^ θ ) .
θ ^ θ [ θ [ θ Φ ( θ ; Y ) ] T ] 1 θ Φ ( θ , Y ) .
E [ θ ^ ] θ E [ [ θ [ θ Φ ( θ ; Y ) ] T ] 1 θ Φ ( θ ; Y ) ] H 1 E [ θ Φ ( θ ; Y ) ] ,
H = E [ θ [ θ Φ ( θ ; Y ) ] T ] .
Cov [ θ ^ ] H 1 E [ θ Φ ( θ ; Y ) θ Φ ( θ ; Y ) T ] H 1
Var [ θ ^ j ] J j j ,
F j k E [ log p ( Y ; θ ) θ j log p ( Y ; θ ) θ k ] = E [ 2 log p ( Y ; θ ) θ j θ k ] .
F j k = i = 0 N 1 1 λ i ( θ ) λ i ( θ ) θ j λ i ( θ ) θ k .
E [ L ( θ ; Y ) ) θ j ] = E [ i = 0 N 1 ( y i λ i ( θ ) 1 ) λ i ( θ ) θ j ] = 0 .
E [ [ θ L ( θ , Y ) ] [ θ L ( θ , Y ) ] T ] = E [ θ [ θ L ( θ , Y ) ] T ] .
E [ Φ L S ( θ ; Y ) θ j ] = E [ i ( y i λ i ( θ ) ) λ i ( θ ) θ j ] = 0 .
H j k = i = 0 N 1 λ i ( θ ) θ j λ i ( θ ) θ k ,
K j k = i = 0 N 1 λ i ( θ ) λ i ( θ ) θ j λ i ( θ ) θ k .
Cov { θ ^ L S } H 1 K H 1 F 1 ,
E [ Φ p ( θ ; Y ) θ j ] = E [ 1 2 i = 0 N 1 2 ( y i λ i ( θ ) ) λ i ( θ ) ( y i λ i ( θ ) ) 2 λ i ( θ ) 2 λ i ( θ ) θ j ] = 1 2 i = 0 N 1 1 λ i ( θ ) λ i ( θ ) θ j .
H j k P = i = 0 N 1 [ 1 λ i ( θ ) + 1 λ i ( θ ) 2 ] λ i ( θ ) θ j λ i ( θ ) θ k 1 2 1 λ i ( θ ) 2 λ i ( θ ) θ j θ k .
E [ Φ N ( θ ; Y ) θ j ] = E [ i = 0 N 1 ( y i λ i ( θ ) max { y i , 1 } ) λ i ( θ ) θ j ] i = 0 N 1 ( 1 E [ λ i ( θ ) max { y i , 1 } ] ) λ i ( θ ) θ j i = 0 N 1 [ 1 λ i ( θ ) + λ i ( θ ) e λ i ( θ ) ] λ i ( θ ) θ j ,
E [ 1 max { y i , 1 } ] e λ i ( θ ) + 1 λ i ( θ ) 1 .
H j k N = E [ i = 0 N 1 1 max { y i , 1 } λ i ( θ ) θ j λ i ( θ ) θ k ( y i λ i ( θ ) max { y i , 1 } ) 2 λ i ( θ ) θ j θ k ] , i = 0 N 1 [ e λ i ( θ ) + 1 λ i ( θ ) ] λ i ( θ ) θ j λ i ( θ ) θ k + [ 1 λ i ( θ ) + λ i ( θ ) e λ i ( θ ) ] 2 λ i ( θ ) θ j θ k .
H j k M = i = 0 N 1 ( 1 y i λ i ( θ ) ) 2 λ i ( θ ) θ j θ k + y i λ i ( θ ) 2 λ i ( θ ) θ j λ i ( θ ) θ k .
λ i ( θ ) = ( A 1 e t i τ 1 + A 2 e t i τ 2 ) * h ( t i ) .
B T B [ B T A ] [ A T A ] 1 [ A T B ] 0 ,
B i j = 1 λ i ( θ ) λ i ( θ ) θ j ,
A i j = λ i ( θ ) λ i ( θ ) θ j , i = 0 , , N 1 , j = 0 , , 2 M 1 .
F = B T B .
Cov { θ ^ L S } [ A T B ] 1 [ A T A ] [ B T A ] 1 F 1 Cov { θ ^ M L E } ,
λ i ( θ ) = A e t i τ * i ( t i ) .
λ i ( θ ) τ = t i τ 2 λ i ( θ ) .
h p i = 0 N 1 t i 2 τ 4 λ i ( θ ) .
E [ τ ^ ] τ 1 h p 1 2 i = 0 N 1 1 λ i ( θ ) λ i ( θ ) τ = 1 2 A i = 0 N 1 t i τ 2 i = 0 N 1 t i 2 τ 4 e t i τ * i ( t i ) .

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