Local Linear Quantile Regression

Early Puberty in Internationally Adopted Girls: Hormonal and Clinical Markers of Puberty in 276 Girls Examined Biannually over Two Years

“…Reference curves for BMI and the reproductive hormones as a function of age were obtained for the Danish reference population using locally weighted regression quantiles [17]. This allowed taking into account the highly skewed distributions of the reproductive hormones and also that some hormones had a considerable number of values below the detection limits of the assays.…”

Efficient Regressions via Optimally Combining Quantile Information

“…Given samples ${false\{false(X_t,Y_{t}false)false\}}_{t=1}^n$, we can estimate Q Y (τ| x ) by the local linear quantile regression [Yu and Jones (1998)]: $$true({\mathrm{Q̂}}_{Y}false(\tau false|xfalse),\mathrm{b̂}true)=\underset{false(a,bfalse)}{\text{argmin}}\sum _{t=1}^n{\rho }_{\tau }false\{Y_t-a-bfalse(X_t-xfalse)false\}Ktrue(\frac{X_t-x}{h}true),$$ for a kernel function K (·) and bandwidth h . From (30), we propose the WQAE of m ( x ): $${\mathrm{m̂}}_{\mathrm{normalW}\mathrm{normalQ}\mathrm{normalA}\mathrm{normalE}}false(xfalse|\omega false)=\sum _{j=1}^k{\omega }_j{\mathrm{Q̂}}_{Y}false({\tau }_{j}false|xfalse).$$ …”

“…In the absence of Gaussianity, asymptotic results of m̂ LS ( x ) generally still hold but this estimator is less efficient in terms of mean-squared error than estimators that exploit the distributional information. For heavy-tailed data, local quantile regression is a robust estimation method; see, e.g., Yu and Jones (1998). The local median regression estimator, denoted by m̂ LAD ( x ), corresponds to τ = 0.5 in (31).…”

“…Then the conditional median of | Y − m ( X )| given X is σ( X ), and we can apply local median quantile regression to estimate σ(·). We propose the procedure: Use (32) with the uniform weight to obtain the preliminary estimator m̂ (·).Compute Y t − m̂ ( X t ) and estimate σ(·) by local linear median quantile regression: $$false(\mathrm{\sigma ̂}false(xfalse),\mathrm{b̂}false)=\underset{\sigma ,b}{\text{argmin}}\sum _{t=1}^n{\rho }_{0.5}false\{false|Y_t-\mathrm{m̂}false(X_{t}false)false|-\sigma -bfalse(X_t-xfalse)false\}Ktrue(\frac{X_t-x}{\ell }true).$$ For the bandwidth ℓ, following Yu and Jones (1998), we use ℓ = ℓ LS (π/2) 1/5 , where ℓ LS is the plug-in bandwidth [Ruppert, Sheather and Wand (1995)] for local linear LS regression based on the data ( X i , | Y i − m̂ ( X i )| 2 ), i = 1, …, n .Compute the errors ε̂ t = [ Y t − m̂ ( X t )]/σ̂( X t ) and obtain the estimator f̂ ε ( Q̂ ε (τ)) as in the parametric regression case 1 above.Use (14) to obtain ω̂ 1 , …, ω̂ k and symmetrize them: ${\mathrm{\omega ̂}}_j^{*}=false({\mathrm{\omega ̂}}_j+{\mathrm{\omega ̂}}_{k+1-j}false)/2$, j = 1, …, k . …”

“…Compute Y t − m̂ ( X t ) and estimate σ(·) by local linear median quantile regression: $$false(\mathrm{\sigma ̂}false(xfalse),\mathrm{b̂}false)=\underset{\sigma ,b}{\text{argmin}}\sum _{t=1}^n{\rho }_{0.5}false\{false|Y_t-\mathrm{m̂}false(X_{t}false)false|-\sigma -bfalse(X_t-xfalse)false\}Ktrue(\frac{X_t-x}{\ell }true).$$ For the bandwidth ℓ, following Yu and Jones (1998), we use ℓ = ℓ LS (π/2) 1/5 , where ℓ LS is the plug-in bandwidth [Ruppert, Sheather and Wand (1995)] for local linear LS regression based on the data ( X i , | Y i − m̂ ( X i )| 2 ), i = 1, …, n .…”