Electronic Journal of Statistics

Laplace deconvolution with noisy observations

Felix Abramovich, Marianna Pensky, and Yves Rozenholc

Full-text: Open access


In the present paper we consider Laplace deconvolution problem for discrete noisy data observed on an interval whose length $T_{n}$ may increase with the sample size. Although this problem arises in a variety of applications, to the best of our knowledge, it has been given very little attention by the statistical community. Our objective is to fill the gap and provide statistical analysis of Laplace deconvolution problem with noisy discrete data. The main contribution of the paper is an explicit construction of an asymptotically rate-optimal (in the minimax sense) Laplace deconvolution estimator which is adaptive to the regularity of the unknown function. We show that the original Laplace deconvolution problem can be reduced to nonparametric estimation of a regression function and its derivatives on the interval of growing length $T_{n}$. Whereas the forms of the estimators remain standard, the choices of the parameters and the minimax convergence rates, which are expressed in terms of $T_{n}^{2}/n$ in this case, are affected by the asymptotic growth of the length of the interval.

We derive an adaptive kernel estimator of the function of interest, and establish its asymptotic minimaxity over a range of Sobolev classes. We illustrate the theory by examples of construction of explicit expressions of Laplace deconvolution estimators. A simulation study shows that, in addition to providing asymptotic optimality as the number of observations tends to infinity, the proposed estimator demonstrates good performance in finite sample examples.

Article information

Electron. J. Statist. Volume 7 (2013), 1094-1128.

First available in Project Euclid: 22 April 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G20: Asymptotic properties

Adaptivity kernel estimation minimax rates Volterra equation Laplace convolution


Abramovich, Felix; Pensky, Marianna; Rozenholc, Yves. Laplace deconvolution with noisy observations. Electron. J. Statist. 7 (2013), 1094--1128. doi:10.1214/13-EJS796. https://projecteuclid.org/euclid.ejs/1366639033

Export citation


  • [1] Ameloot, M. and Hendrickx, H. (1983). Extension of the performance of Laplace deconvolution in the analysis of fluorescence decay curves., Biophys. Journ., 44, 27–38.
  • [2] Ameloot, M., Hendrickx, H., Herreman, W., Pottel, H., Van Cauwelaert, F. and van der Meer, W. (1984). Effect of orientational order on the decay of the fluorescence anisotropy in membrane suspensions. Experimental verification on unilamellar vesicles and lipid/alpha-lactalbumin complexes., Biophys. Journ., 46, 525–539.
  • [3] Bisdas, S., Konstantinou, G.N., Lee, P.S., Thng, C.H., Wagenblast, J., Baghi, M. and Koh, T.S. (2007). Dynamic contrast-enhanced CT of head and neck tumors: perfusion measurements using a distributed-parameter tracer kinetic model. Initial results and comparison with deconvolution- based analysis., Physics in Medicine and Biology, 52, 6181–6196.
  • [4] Bunea, F., Tsybakov, A. and Wegkamp, M.H. (2007). Aggregation for Gaussian regression., Ann. Statist. 35, 1674–1697.
  • [5] Cao, M.,Liang, Y., Shen, C., Miller, K.D. and Stantz, K.M. (2010). Developing DCE-CT to quantify intra-tumor heterogeneity in breast tumors with differing angiogenic phenotype., IEEE Trans. Medic. Imag., 29, 1089–1092.
  • [6] Carroll, R. J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density., J. Amer. Statist. Assoc. 83, 1184–1186.
  • [7] Cinzori, A.C. and Lamm, P.K. (2000). Future polynomial regularization of ill-posed Volterra equations., SIAM J. Numer. Anal., 37, 949–979.
  • [8] Comte, F. (2001) Adaptive estimation of the spectrum of a stationary Gaussian sequence., Bernoulli, 7, 267–298.
  • [9] Comte, F., Rozenholc, Y. and Taupin, M.L. (2006). Penalized contrast estimator for adaptive density deconvolution., Canad. J. Statist., 3, 431–452.
  • [10] Comte, F., Rozenholc, Y. and Taupin, M.L. (2007). Finite sample penalization in adaptive density deconvolution., J. Stat. Comput. Simul., 77, 977–1000.
  • [11] Cuenod, C.A., Fournier, L., Balvay, D. and Guinebretire, J.M. (2006). Tumor angiogenesis: pathophysiology and implications for contrast-enhanced MRI and CT assessment., Abdom. Imaging, 31, 188-193.
  • [12] Cuenod, C-A., Favetto, B., Genon-Catalot, V., Rozenholc, Y. and Samson, A. (2011). Parameter estimation and change-point detection from Dynamic Contrast Enhanced MRI data using stochastic differential equations., Math. Biosci., 233-1, 68–76.
  • [13] Delaigle, A., Hall, P. and Meister, A. (2008). On deconvolution with repeated measurements., Ann. Statist., 36, 665-685.
  • [14] Dey, A.K., Martin, C.F. and Ruymgaart, F.H. (1998). Input recovery from noisy output data, using regularized inversion of Laplace transform., IEEE Trans. Inform. Theory, 44, 1125–1130.
  • [15] Diggle, P. J. and Hall, P. (1993). A Fourier approach to nonparametric deconvolution of a density estimate., J. Roy. Statist. Soc. Ser. B, 55 523–531.
  • [16] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problem., Ann. Statist., 19, 1257-1272.
  • [17] Fan, J. and Koo, J. (2002). Wavelet deconvolution., IEEE Trans. Inform. Theory, 48, 734–747.
  • [18] Gasser, T. and Müller, H-G. (1984). Estimating regression functions and their derivatives by the kernel method., Scand. J. Statist., 11, 171–185.
  • [19] Gasser, T., Müller, H-G. and Mammitzsch, V. (1985). Kernels for nonparametric kernel estimation., J. Roy. Statist. Soc. Ser. B, 47, 238–252.
  • [20] Gafni, A., Modlin, R. L. and Brand, L. (1975). Analysis of fluorescence decay curves by means of the Laplace transformation., Biophys. J., 15, 263–280.
  • [21] Gendre, X. (2013). Model selection and estimation of a component in additive regression., ESAIM: Probability and Statistics, to appear.
  • [22] Goh, V., Halligan, S., Hugill, J.A., Gartner, L. and Bartram, C.I. (2005). Quantitative colorectal cancer perfusion measurement using dynamic contrastenhanced multidetector-row computed tomography: effect of acquisition time and implications for protocols., J. Comput. Assist. Tomogr., 29, 59–63.
  • [23] Goh, V. and Padhani, A. R. (2007). Functional imaging of colorectal cancer angiogenesis., Lancet Oncol., 8, 245–255.
  • [24] Gradshtein, I.S. and Ryzhik, I.M. (1980)., Tables of Integrals, Series, and Products. Academic Press, New York.
  • [25] Gripenberg, G., Londen, S.O. and Staffans, O. (1990)., Volterra Integral and Functional Equations. Cambridge University Press, Cambridge.
  • [26] Johnstone, I.M., Kerkyacharian, G., Picard, D. and Raimondo, M. (2004). Wavelet deconvolution in a periodic setting., J. Roy. Statist. Soc. Ser. B, 66, 547–573 (with discussion, 627–657).
  • [27] Lakowicz, J.R. (2006)., Principles of Fluorescence Spectroscopy. Kluwer Academic, New York.
  • [28] Lamm, P. (1996). Approximation of ill-posed Volterra problems via predictor-corrector regularization methods., SIAM J. Appl. Math., 56, 524–541.
  • [29] Laurent, B. and Massart, P. (1998). Adaptive estimation of a quadratic functional by model selection., Ann. Statist., 28, 1302–1338.
  • [30] LePage, W.R. (1961)., Complex Variables and the Laplace Transform for Engineers. Dover, New-York.
  • [31] Lepski, O.V. (1991). Asymptotic mimimax adaptive estimation. I: Upper bounds. Optimally adaptive estimates., Theory Probab. Appl., 36, 654–659.
  • [32] Lepski, O.V., Mammen, E. and Spokoiny, V.G. (1997). Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors, Ann. Statist., 25, 929–947.
  • [33] Lien, T.N., Trong, D.D. and Dinh, A.P.N. (2008). Laguerre polynomials and the inverse Laplace transform using discrete data, J. Math. Anal. Appl., 337, 1302–1314.
  • [34] Maleknejad, K., Mollapourasl, R. and Alizadeh, M. (2007). Numerical solution of Volterra type integral equation of the first kind with wavelet basis., Appl. Math.Comput., 194, 400–405.
  • [35] McKinnon, A. E., Szabo, A. G. and Miller, D. R. (1977). The deconvolution of photoluminescence data., J. Phys. Chem., 81, 1564–1570.
  • [36] Meister, A. (2009)., Deconvolution Problems in Nonparametric Statistics (Lecture Notes in Statistics). Springer-Verlag, Berlin.
  • [37] Miles, K. A. (2003). Functional CT imaging in oncology., Eur. Radiol., 13 - suppl. 5, M134-8.
  • [38] O’Connor, D. V., Ware, W. R. and Andre, J. C. (1979). Deconvolution of fluorescence decay curves. A critical comparison of techniques., J. Phys. Chem., 83, 1333–1343.
  • [39] Padhani, A. R. and Harvey, C. J. (2005). Angiogenesis imaging in the management of prostate cancer., Nat. Clin. Pract. Urol., 2, 596–607.
  • [40] Pensky, M. and Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution., Ann. Statist., 27, 2033–2053.
  • [41] Polyanin, A.D. and Manzhirov, A.V. (1998)., Handbook of Integral Equations, CRC Press, Boca Raton, Florida.
  • [42] Stefanski, L. and Carroll, R. J. (1990). Deconvoluting kernel density estimators., Statistics, 21, 169–184.
  • [43] Tsybakov, A.B. (2009)., Introduction to Nonparametric Estimation, Springer, New York.
  • [44] Weeks, W.T. (1966). Numerical inversion of Laplace transforms using Laguerre functions., J. Assoc. Comput. Machinery, 13, 419–429.