Estimation for a Class of Semiparametric Pareto Mixture Densities

Abstract

We study the estimation of a class of semiparametric mixture models, where the models have a symmetric nonparametric component and a parametric component of Pareto distribution with unknown parameters. We establish an estimation procedure by minimizing a criterion function after dealing with the jump point. We study the large sample properties of the proposed estimator, and prove consistency and asymptotic normality of the parameter estimation. For the nonparametric component, bias and variance are derived, and a rule-of-thumb bandwidth selection method is given. Simulation studies demonstrate good performance of the proposed methodology.

Appendices

Appendix: A. Regularity conditions

• (i) The kernel density function K is even, bounded, uniformly continuous, square integrable, of bounded variation and has a second-order moment.

• (ii) The first derivative $K^{\prime }$ of the kernel density function K satisfies that $K^{\prime }\in L^{1}(\mathbb {R})$ and $K^{\prime }(x)\rightarrow 0$ as $|x|\rightarrow \infty$.

• (iii) Let ω denote the square root of the modulus of continuity of K.

  $$ {{\int}_{0}^{1}}(\log(1/u))^{1/2}d\omega(u)<\infty. $$

• (iv) $h_{n}\rightarrow 0$, $nh_{n}\rightarrow \infty$ and $n{h_{n}^{4}}\rightarrow 0$;

• (v) $nh_{n}/|\log {h_{n}}|\rightarrow \infty$, $|\log {h_{n}}|/\log \log {n}\rightarrow \infty$, and there exists a constant c > 0 such that h_n ≤ ch_2n for all n ≥ 1;

• (vi) $|\log {h_{n}}|/(n{h_{n}^{3}})\rightarrow 0$.

• (vii) F is strictly increasing on $\mathbb {R}$.

• (viii) P is twice continuously differentiable. G is piecewise twice continuously differentiable. All the second derivatives of P and G are in $L^{1}(\mathbb {R})$.

Proof of Theorem 1

The proof for the first part is similar to that of lemma 1 of Hunter et al. (2007). Because d(θ₀) = 0 and $d_{n}(\boldsymbol {\hat {\theta }})\le d_{n}(\hat {\alpha },\boldsymbol {\eta }_{0})$,

$$ \begin{array}{rl} d(\boldsymbol{\hat{\theta}}) & \le [d(\boldsymbol{\hat{\theta}})-d_{n}(\boldsymbol{\hat{\theta}})] + d_{n}(\hat{\alpha},\boldsymbol{\eta}_{0})\\ & \le |d(\boldsymbol{\hat{\theta}})-d_{n}(\boldsymbol{\hat{\theta}})| + |d(\boldsymbol{\theta}_{0}) - d_{n}(\hat{\alpha},\boldsymbol{\eta}_{0})|, \end{array} $$

this inequality, together with part (iii) of lemma 2 and lemma 3, implies that $d(\boldsymbol {\hat {\theta }}) = o_{a.s.}(n^{-1/2+\gamma })$. By parts (i) and (ii) of lemma 2, $\boldsymbol {\hat {\theta }}$ converges to θ₀ almost surely. We are now to show the second part. By part (iv) of lemma 2, there exist constants r₁ > 0 and c₁ > 0 such that

$$ d(\alpha_{0}, \boldsymbol{\eta}_{0}+\boldsymbol{v})\ge c_{1}\|\boldsymbol{v}\|^{2}, \quad \text{for all } \|\boldsymbol{v}\|<r_{1}, \boldsymbol{v}\in\mathbb{R}^{3}, $$

(4.1)

where ∥⋅∥ stands for Euclidean norm. Since d is Lipschitz continuous, and $\hat {\alpha }-\alpha _{0}$ converges to 0 with rate (n/h_n)^− 1/2, it follows that $|d(\alpha _{0},\boldsymbol {\hat {\eta }}) - d(\hat {\alpha },\boldsymbol {\hat {\eta }})|=o_{a.s.}(n^{-1/2})$. Note that

$$ |d(\alpha_{0},\boldsymbol{\hat{\eta}})| \le |d(\alpha_{0},\boldsymbol{\hat{\eta}}) - d(\hat{\alpha},\boldsymbol{\hat{\eta}})| + d(\hat{\alpha},\boldsymbol{\hat{\eta}}), $$

where $(\hat {\alpha },\boldsymbol {\hat {\eta }}) = \boldsymbol {\hat {\theta }}$. Therefore, $|d(\alpha _{0},\boldsymbol {\hat {\eta }})| = o_{a.s.}(n^{-1/2+\gamma })$ for all γ > 0. Then inequality (4.1) implies the desired rate for $|\boldsymbol {\hat {\eta }}-\boldsymbol {\eta }_{0}|$ and $|\boldsymbol {\hat {\theta }}-\boldsymbol {\theta }_{0}|$.

Proof of Theorem 2

Recall that we use dot and double dot to denote the derivatives of a function with respect to η. Since $\boldsymbol {\hat {\eta }}$ is the minimizer of $d_{n}(\hat {\alpha },\boldsymbol {\eta })$, $\dot {d_{n}}(\hat {\alpha },\boldsymbol {\hat {\eta }})=0$. We do a Taylor expansion for $\dot {d_{n}}(\hat {\alpha },\boldsymbol {\hat {\eta }})$ around $(\hat {\alpha },\boldsymbol {\eta }_{0})$:

$$ \ddot{d}_{n}(\hat{\alpha},\boldsymbol{\eta}^{*}) \sqrt{n}(\boldsymbol{\hat{\eta}}-\boldsymbol{\eta}_{0}) = -\sqrt{n}\dot{d}_{n}(\hat{\alpha},\boldsymbol{\eta}_{0}), $$

where η^∗ lies in the line segment connecting $\boldsymbol {\hat {\eta }}$ and η₀.

The regularity condition (viii) implies that $\dot {d}_{n}(\alpha ,\boldsymbol {\eta })$ is Lipschitz continuous with respect α. Hence there exists a constant c₂ which is independent of n such that

$$ |\dot{d}_{n}(\alpha_{1},\boldsymbol{\eta}_{0}) - \dot{d}_{n}(\alpha_{2},\boldsymbol{\eta}_{0})|\le c_{2}|\alpha_{1}-\alpha_{2}|. $$

Since $\hat {\alpha }-\alpha _{0}$ converges to 0 with a rate (n/h_n)^− 1/2,

$$ |\dot{d}_{n}(\hat{\alpha},\boldsymbol{\eta}_{0}) - \dot{d}_{n}(\alpha_{0},\boldsymbol{\eta}_{0})|=o_{p}(n^{-1/2}). $$

By the proof the Theorem 3.2 of Bordes and Vandekerkhove (2010),

$$ \dot{d}_{n}(\boldsymbol{\theta}_{0}) = \frac{2}{n}\sum\limits_{i=1}^{n} H(X_{i};\boldsymbol{\theta}_{0},\hat{F}_{n}) \dot{H}(X_{i};\boldsymbol{\theta}_{0},F) +o_{a.s.}(n^{-1/2}). $$

It follows that

$$ \dot{d}_{n} (\hat{\alpha},\boldsymbol{\eta}_{0}) = \frac{2}{n}\sum\limits_{i=1}^{n} H(X_{i};\boldsymbol{\theta}_{0},\hat{F}_{n}) \dot{H}(X_{i};\boldsymbol{\theta}_{0},F) +o_{p}(n^{-1/2}). $$

Similar to the proof the Theorem 3.2 of Bordes and Vandekerkhove (2010), we can show that

$$ \ddot{d}_{n}(\hat{\alpha},\boldsymbol{\eta}^{*}) \rightarrow -\boldsymbol{J}(\boldsymbol{\theta}_{0}) \text{ a.s. as } n\rightarrow\infty. $$

The remaining part of the proof is similar to that of Theorem 3.2 of Bordes and Vandekerkhove (2010) and is omitted here.