Narrow Results

In this paper, we provide an asymptotic expression for mean integrated squared error (MISE) of nonlinear wavelet density estimator for a truncation model. It is assumed that the lifetime observations form a stationary α-mixing sequence. Unlike for kernel estimator, the MISE expression of the nonlinear wavelet estimator is not affected by the presence of discontinuities in the curves. Also, we establish asymptotic normality of the nonlinear wavelet estimator.

In this paper, we construct a nonlinear wavelet estimator of conditional density function for a left truncation model. We provide an asymptotic expression for the mean integrated squared error (MISE) of the estimator. It is assumed that the lifetime observations form a stationary α-mixing sequence. Unlike for kernel estimator, the MISE expression of the nonlinear wavelet estimator is not affected by the presence of discontinuities in the curves.

In this paper, we discuss the global L₂-error of the nonlinear wavelet estimator of density in the Besov space for the truncation model when the data exhibit strong mixing assumption, and prove that the estimator can achieve the optimal rate of convergence, which is similar to that in the complete and independent data case with term-by-term thresholding of the empirical wavelet coefficients (D. L. Donoho, I. M. Johnstone, G. Kerkyacharian and D. Picard, Density estimation by wavelet thresholding, Ann. Statist.24 (1996) 508–539). In addition, the conclusion shows that the convergence rate of the nonlinear estimator is faster than that of its linear estimator in some range.

Conditional probability distribution models have been widely used in economics and finance. In this chapter, we introduce two closely related popular methods to estimate conditional distribution models—Maximum Likelihood Estimation (MLE) and Quasi-MLE (QMLE). MLE is a parameter estimator that maximizes the model likelihood function of the random sample when the conditional distribution model is correctly specified, and QMLE is a parameter estimator that maximizes the model likelihood function of the random sample when the conditional distribution model is misspecified. Because the score function is an MDS and the dynamic Information Matrix (IM) equality holds when a conditional distribution model is correctly specified, the asymptotic properties of MLE is analogous to those of the OLS estimator when the regression disturbance is an MDS with conditional homoskedasticity, and we can use the Wald test, LM test and Likelihood Ratio (LR) test for hypothesis testing, where the LR test is analogous to the J · F test statistic. On the other hand, when the conditional distribution model is misspecified, the score function has mean zero, but it may no longer be an MDS and the dynamic IM equality may fail. As a result, the asymptotic properties of QMLE are analogous to those of the OLS estimator when the regression disturbance displays serial correlation and/or conditional heteroskedasticity. Robust Wald tests and LM tests can be constructed for hypothesis testing, but the LR test can no longer be used, for a reason similar to the failure of the F-test statistic when the regression disturbance displays serial correlation and/or conditional heteroskedasticity. We discuss methods to test the MDS property of the score function, and the dynamic IM equality, and correct specification of a conditional distribution model.