1 Introduction
2 Haar wavelet and discrete wavelet transform
2.1 Haar wavelet
2.2 Haar Discrete Wavelet Transformation (HDWT)
3 Estimation and transformation of AR(1) models
4 Simulation study
\(x_0=1\)
|
\(\varphi =0.99\)
| MSE |
\(\varphi =0.8\)
| MSE |
\(\varphi =0.7\)
| MSE |
---|---|---|---|---|---|---|
Average \(\hat{\varphi }_\mathbf{x}\) | 0.969821 | 1.5819 | 0.786203 | 4.3048 | 0.687516 | 5.6160 |
Average \(\hat{\varphi }_\mathbf{y}\) | 0.954043 | 2.5735 | 0.775157 | 5.6511 | 0.676731 | 7.7051 |
\(x_0=0.1\)
|
\(\varphi =0.99\)
| MSE |
\(\varphi =0.8\)
| MSE |
\(\varphi =0.7\)
| MSE |
---|---|---|---|---|---|---|
Average \(\hat{\varphi }_\mathbf{x}\) | 0.971516 | 1.4638 | 0.782047 | 4.2449 | 0.687871 | 5.3416 |
Average \(\hat{\varphi }_\mathbf{y}\) | 0.955874 | 2.3352 | 0.772372 | 5.4061 | 0.679346 | 7.5531 |
\(x_0=1\)
|
\(\varphi =0.99\)
| MSE |
\(\varphi =0.8\)
| MSE |
\(\varphi =0.7\)
| MSE |
---|---|---|---|---|---|---|
Average \(\hat{\varphi }_\mathbf{x}\) | 0.968205 | 0.0368 | 0.803024 | 0.0741 | 0.672014 | 0.1725 |
Average \(\hat{\varphi }_\mathbf{y}\) | 0.960740 | 0.0396 | 0.802245 | 0.0729 | 0.675860 | 0.15026 |
\(x_0=0.1\)
|
\(\varphi =0.99\)
| MSE |
\(\varphi =0.8\)
| MSE |
\(\varphi =0.7\)
| MSE |
---|---|---|---|---|---|---|
Average \(\hat{\varphi }_\mathbf{x}\) | 0.969563 | 0.0174 | 0.794335 | 0.0658 | 0.699146 | 0.0858 |
Average \(\hat{\varphi }_\mathbf{y}\) | 0.960106 | 0.0289 | 0.788669 | 0.0827 | 0.700260 | 0.0693 |
\(\ell _2\) error |
\(\varphi =0.7\)
|
\(x_0\)
| MSPE (8) | MSPE (17) | MSPE (12) |
---|---|---|---|---|---|
\(\hat{\varphi }_\mathbf{x}=0.699146\)
|
\(\hat{\varphi }_\mathbf{y}= {0.700260}\)
| 1 | 0.0007 | 319.2027 | 383.8687 |
\(\hat{\varphi }_\mathbf{x}=0.672014\)
|
\(\hat{\varphi }_\mathbf{y}= {0.675860}\)
| 0.1 | 0.7611 | 319.3887 | 376.3713 |
\(\ell _\infty\) error |
\(\varphi =0.7\)
|
\(x_0\)
| MSPE (8) | MSPE (17) | MSPE (12) |
---|---|---|---|---|---|
\(\hat{\varphi }_\mathbf{x}=0.699146\)
|
\(\hat{\varphi }_\mathbf{y}= {0.700260}\)
| 1 | 0.0067 | 4.9695 | 5.4833 |
\(\hat{\varphi }_\mathbf{x}=0.672014\)
|
\(\hat{\varphi }_\mathbf{y}= {0.675860}\)
| 0.1 | 0.2129 | 4.8487 | 5.2666 |