From 40a54529ad239986232facd616329923729388ac Mon Sep 17 00:00:00 2001 From: Chi-Kuang Yeh Date: Tue, 21 May 2024 00:03:18 -0600 Subject: [PATCH] Update README file to include the description about SLSE, --- README.Rmd | 6 +++++- README.md | 26 ++++++++++++++++---------- 2 files changed, 21 insertions(+), 11 deletions(-) diff --git a/README.Rmd b/README.Rmd index 2b6e6d5..69d7d99 100644 --- a/README.Rmd +++ b/README.Rmd @@ -64,8 +64,12 @@ y_i-\eta(\boldsymbol{x_i};\boldsymbol{\theta})\\ y_i^2-\eta^2(\boldsymbol{x_i};\boldsymbol{\theta})-\sigma^2 \end{pmatrix}. ``` -Note that $`W(\boldsymbol{x_i})`$ is a $`2\times 2`$ non-negative semi-definite matrix which may or may not depend on $\boldsymbol{x_i}$ \Wang and Leblanc (2008). It is clear that SLSE is a natural extension of the OLSE which is defined based on the first-order difference function (i.e. $`y_i-\mathbb{E}[y_i]=y_i-\eta(\boldsymbol{x_i};\boldsymbol{\theta})`$). On the other hand, SLSE is defined using not only the first-order difference function, but also second-order difference function (i.e. $`y_i^2-\mathbb{E}[y_i^2]=y_i^2-(\eta^2(\boldsymbol{x_i};\boldsymbol{\theta})+\sigma^2))`$. One might think about the downsides of the SLSE after talking about the advantages of SLSE over OLSE. SLSE does have its disadvantages indeed. It is not a linear estimator and there is no closed-form solution. It requires more computational resources compared to the OLSE due to the nonlinearity. However, numerical results can be easily computed for SLSE nowadays. As a result, SLSE is a powerful alternative estimator to be considered in research studies and real-life applications. +### Comparison between ordinary least-squares and second order least-squares estimators + +Note that $`W(\boldsymbol{x_i})`$ is a $`2\times 2`$ non-negative semi-definite matrix which may or may not depend on $\boldsymbol{x_i}$ (Wang and Leblanc, 2008). It is clear that SLSE is a natural extension of the OLSE which is defined based on the first-order difference function (i.e. $`y_i-\mathbb{E}[y_i]=y_i-\eta(\boldsymbol{x_i};\boldsymbol{\theta})`$). On the other hand, SLSE is defined using not only the first-order difference function, but also second-order difference function (i.e. $`y_i^2-\mathbb{E}[y_i^2]=y_i^2-(\eta^2(\boldsymbol{x_i};\boldsymbol{\theta})+\sigma^2))`$. One might think about the downsides of SLSE after discussing its advantages over OLSE. SLSE does have its disadvantages. It is not a linear estimator, and there is no closed-form solution. It requires more computational resources compared to OLSE due to its nonlinearity. However, numerical results can be easily computed for SLSE nowadays. As a result, SLSE is a powerful alternative estimator to be considered in research studies and real-life applications. + +In particular, if we set the skewness parameter $t$ to be zero, the resulting optimal designs under SLSE and OLSE **will be the same**! ## Examples diff --git a/README.md b/README.md index 32a768b..1752c28 100644 --- a/README.md +++ b/README.md @@ -3,7 +3,7 @@ estimator ================ *Chi-Kuang Yeh, Julie Zhou* -*May 20, 2024* +*May 21, 2024* @@ -69,21 +69,27 @@ y_i^2-\eta^2(\boldsymbol{x_i};\boldsymbol{\theta})-\sigma^2 \end{pmatrix}. ``` +### Comparison between ordinary least-squares and second order least-squares estimators + Note that $`W(\boldsymbol{x_i})`$ is a $`2\times 2`$ non-negative semi-definite matrix which may or may not depend on $\boldsymbol{x_i}$ -and Leblanc (2008). It is clear that SLSE is a natural extension of the -OLSE which is defined based on the first-order difference function +(Wang and Leblanc, 2008). It is clear that SLSE is a natural extension +of the OLSE which is defined based on the first-order difference +function (i.e. $`y_i-\mathbb{E}[y_i]=y_i-\eta(\boldsymbol{x_i};\boldsymbol{\theta})`$). On the other hand, SLSE is defined using not only the first-order difference function, but also second-order difference function (i.e. $`y_i^2-\mathbb{E}[y_i^2]=y_i^2-(\eta^2(\boldsymbol{x_i};\boldsymbol{\theta})+\sigma^2))`$. -One might think about the downsides of the SLSE after talking about the -advantages of SLSE over OLSE. SLSE does have its disadvantages indeed. -It is not a linear estimator and there is no closed-form solution. It -requires more computational resources compared to the OLSE due to the -nonlinearity. However, numerical results can be easily computed for SLSE -nowadays. As a result, SLSE is a powerful alternative estimator to be -considered in research studies and real-life applications. +One might think about the downsides of SLSE after discussing its +advantages over OLSE. SLSE does have its disadvantages. It is not a +linear estimator, and there is no closed-form solution. It requires more +computational resources compared to OLSE due to its nonlinearity. +However, numerical results can be easily computed for SLSE nowadays. As +a result, SLSE is a powerful alternative estimator to be considered in +research studies and real-life applications. + +In particular, if we set the skewness parameter $t$ to be zero, the +resulting optimal designs under SLSE and OLSE **will be the same**! ## Examples