The Consistency of Estimators in a Heteroscedastic Partially Linear Model with <i>ρ</i>⁻-Mixing Errors

• Main Authors: Yu Zhang, Xinsheng Liu
• Format: Article
• Language: English
• Published: MDPI AG 2020-07-01
• Series: Symmetry
• Subjects: <i>ρ</i>⁻-mixing random variables; heteroscedastic; partially linear model; LS estimators; WLS estimators; strong consistency
• Online Access: https://www.mdpi.com/2073-8994/12/7/1188

This paper studies a heteroscedastic partially linear model based on <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>ρ</mi> <mo>−</mo></msup></mrow></semantics></math></inline-formula>-mixing random errors, stochastically dominated and with zero mean. Under some suitable conditions, the strong consistency and <inline-formula> <math display="inline"> <semantics> <mi>p</mi></semantics></math></inline-formula>-th <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>></mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> mean consistency of least squares (LS) estimators and weighted least squares (WLS) estimators for the unknown parameter are investigated, and the strong consistency and <inline-formula> <math display="inline"> <semantics> <mi>p</mi></semantics></math></inline-formula>-th <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>></mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> mean consistency of the estimators for the non-parametric component are also studied. These results include the corresponding ones of independent, negatively associated (NA), and <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>ρ</mi> <mo>*</mo></msup></mrow></semantics></math></inline-formula>-mixing random errors as special cases. At last, two simulations are presented to support the theoretical results.