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Density deconvolution with Laplace errors and unknown variance

Abstract

We consider density deconvolution with zero-mean Laplace noise in the context of an error component regression model. We adapt the minimax deconvolution methods of Meister (2006) to allow estimation of the unknown noise variance. We propose a semi-uniformly consistent estimator for an ordinary-smooth target density and a modified "variance truncation device” for the unknown noise variance. We provide a simulation study and practical guidance for the choice of smoothness parameters of the ordinary-smooth target density. We apply restricted versions of our estimator to a stochastic frontier model of US banks and to a measurement error model of daily saturated fat intake.

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Notes

  1. 1.

    See Butucea and Matias (2005), Matias (2002), Meister (2006), Schwartz and Bellegem (2010), and Horrace and Parmeter (2011). See Delaigle and Van Keilegom (2021) for a good survey.

  2. 2.

    The Meister (2006) estimator is uniformly consistent relative to the target distributional family but individually relative to the noise distributional family. That is, consistency of the estimator does not hold uniformly over all noise distributions in the super-smooth family.

  3. 3.

    Horrace and Parmeter (2018) perform maximum likelihood estimation of the parametric stochastic frontier model, not deconvolution.

  4. 4.

    Neumann (1997), Johannes (2009), and Wang and Ye (2012) study deconvolution with fully unknown error distribution but require either an additional sample of the error or repeated observations, yjt.

  5. 5.

    When u is not from a parametric family, analogous sufficient conditions are that the density of u is asymmetric and does not contain a “symmetric component” (see Delaigle and Hall 2016). Since the Laplace density is symmetric, our model is actually nonparametrically identified, if we further assume that the density of u is asymmetric, a common parametric assumption in the stochastic frontier literature.

  6. 6.

    Schawrz and Van Bellegem (2010) also appeal to stability of the Normal distribution to identify the noise variance and a target density that vanishes on an interval of positive Lebesgue measure when noise is Normal. We note that their identification condition on the target density is fully compatible with the usual assumptions in the stochastic frontier model

  7. 7.

    Stable distributional families are closed to self-convolution, while the logarithm of geometric stable distributional families are closed to self-convolution.

  8. 8.

    Alternatively, b2 is identified if the density of u is asymmetric without a symmetric component, per Delaigle and Hall (2016), but here we prefer the target density be potentially symmetric.

  9. 9.

    Indeed, the deconvolution estimator of Dattner and Goldenshluger (2011) relies on very general classes of distributions for the target and noise densities that includes the Laplace–Laplace convolution as a special case, and identification and consistent target density estimation are achieved for the known noise variance case.

  10. 10.

    Butucea and Matias (2005) develop a generalized semiparametric deconvolution estimator, where the target density is ordinary smooth and the noise is super-smooth, a generalization of the parametric assumption of Normal noise. Without the parametric assumption, identification requires that the tails of the noise characteristic function decay faster than the those of the target density. We need no such assumption here, because we select a parametric family of distributions for noise.

  11. 11.

    In the section “Some useful extensions”, we propose setting \({w}_{n}={k}_{n}/{\rm{ln}}{k}_{n}\) in the case C1 and δ are not fully known.

  12. 12.

    Recall that for a Laplace distribution as defined in Assumption 2, the variance is 2b2. Hence, a natural upper bound for b2 is one-quarter the variance.

  13. 13.

    Similar to Meister (2006), it does not leverage the relative decay rates of the noise and target characteristic functions.

  14. 14.

    In Meister (2006), the bounding of the Normal variance is what leads to semi-uniformly consistency (as opposed to uniform consistency). Here, for Laplace errors, we still impose this "strong” condition for ease of proof. However, it may not be a necessary condition given Assumption 3.

  15. 15.

    The compound effect of estimating the regression function will slow the target density rate compared to pure (non-regression) deconvolution, but the final rate is not a simple algebraic sum of the rates.

  16. 16.

    … and the estimator of Meister (2006) as well.

  17. 17.

    Actually, if one wants to assume the random noise is super-smooth with similarity index s, the smoothness parameter δ of target density can be estimated as well as the s by an adaptive procedure proposed by Butucea et al. (2008).

  18. 18.

    Details see Condition 2.1 in Li and Racine (2007).

  19. 19.

    The data are publicly available on the Journal of Applied Econometrics data archive website http://qed.econ.queensu.ca/jae/2009-v24.1/feng-serletis/.

  20. 20.

    Once \({\hat{f}}_{u}\) is obtained, one can estimate the efficiency score using numerical integration on a grid of \(\hat{\varepsilon }\). To avoid an overloading of present paper, we stick to the estimation of marginal density of u.

  21. 21.

    It is interesting to note that with a skew of 1.55, this provides evidence against use of the Half-Normal distribution (Papadopoulos and Parmeter 2021).

  22. 22.

    For the Laplace distribution, δ = 2; for convolved Laplace, δ = 4. The choice δ = 3 is between Laplace and convolved Laplace.

  23. 23.

    For Laplace deconvolution, we can apply directly Example 1 in Zhang and Karunramuni (2000).

  24. 24.

    We use the package "ksdensity” in Matlab for the ErrorFree case.

  25. 25.

    We also tried larger range of δ and narrow down to this specific range by searching the minimum of Δ.

  26. 26.

    It seems to violate the independence assumption between the target variable and the measurement error.

  27. 27.

    Since the chosen smooth parameter δ = 1.5 which is different from a Laplace distribution, the proposed method is still valid even with Var(u) < Var(v) as is explained in the paragraph following Assumption 3.

  28. 28.

    Under Assumption 1, i.e., u and v are independent, the variance of Y should be the sum of the variances of u and v. Empirically, this may not be the case for real data.

  29. 29.

    One point worth mentioning is that these minimax deconvolution techniques can produce error variance estimates equal to zero as we vary the choice of C1 and δ. Recall that \({\hat{b}}_{n}^{2}\) is bound between 0 and \(0.25V(\hat{\varepsilon })\). When it happens, the deconvolution estimators will be very similar to the ErrorFree estimator.

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Correspondence to Jun Cai, William C. Horrace or Christopher F. Parmeter.

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Cai, J., Horrace, W.C. & Parmeter, C.F. Density deconvolution with Laplace errors and unknown variance. J Prod Anal (2021). https://doi.org/10.1007/s11123-021-00612-1

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Keywords

  • Stochastic frontier
  • Semi-parametric
  • Ordinary smooth