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Merge pull request #214 from tomfaulkenberry/master
Added skew-samp, exg-mome, and wald-mome
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‎D/skew-samp.md‎

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---
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layout: definition
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mathjax: true
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author: "Thomas J. Faulkenberry"
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affiliation: "Tarleton State University"
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e_mail: "faulkenberry@tarleton.edu"
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date: 2023-10-30 12:00:00
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title: "Sample skewness"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Skewness"
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definition: "Sample skewness"
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sources:
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- authors: "Joanes, D. N. and Gill, C. A."
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year: 1998
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title: "Comparing measures of sample skewness and kurtosis"
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in: "The Statistician"
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pages: "vol. 47, part 1, pp. 183-189"
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url: "https://www.jstor.org/stable/2988433"
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def_id: "D190"
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shortcut: "skew-samp"
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username: "tomfaulkenberry"
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---
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**Definition:** Let $x = \left\lbrace x_1, \ldots, x_n \right\rbrace$ be a [sample](/D/samp) from a [random variable](/D/rvar) $X$. Then, the sample skewness of $x$ is given by
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$$ \label{eq:skew-samp}
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\hat{s} = \frac{\frac{1}{n}\sum_{i=1}^n (x_i-\bar{x})^3}{\left[\frac{1}{n}\sum_{i=1}^n(x_i-\bar{x})^2\right]^{3/2}} \; ,
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$$
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where $\bar{x}$ is the [sample mean](/D/mean-samp).

‎P/exg-mome.md‎

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---
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layout: proof
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mathjax: true
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author: "Thomas J. Faulkenberry"
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affiliation: "Tarleton State University"
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e_mail: "faulkenberry@tarleton.edu"
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date: 2023-10-30 12:00:00
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title: "Method of moments for ex-Gaussian distributed data"
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chapter: "Probability Distributions"
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section: "Univariate continuous distributions"
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topic: "ex-Gaussian distribution"
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theorem: "Method of moments"
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sources:
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proof_id: "P424"
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shortcut: "exg-mome"
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username: "tomfaulkenberry"
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---
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**Theorem:** Let $y = \left\lbrace y_1, \ldots, y_n \right\rbrace$ be a set of observed data [independent and identically distributed](/D/iid) according to an [ex-Gaussian distribution](/D/exg) with parameters $\mu$, $\sigma$, and $\lambda$:
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$$ \label{eq:exq}
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y_i \sim \mathrm{ex-Gaussian}(\mu,\sigma,\lambda), \quad i = 1, \ldots, n \; .
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$$
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Then, the [method-of-moments estimates](/D/mome) for the parameters $\mu$, $\sigma$, and $\lambda$ are given by
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$$ \label{eq:exg-MoM}
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\begin{split}
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\hat{\mu} &= \bar{y} - \sqrt[3]{\frac{\bar{s}\cdot \bar{v}^{3/2}}{2}}\\
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\hat{\sigma} &= \sqrt{\bar{v}\cdot\left(1 - \sqrt[3]{\frac{\bar{s}^2}{4}}\right)}\\
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\hat{\lambda} &= \sqrt[3]{\frac{2}{\bar{s}\cdot \bar{v}^{3/2}}} \; ,
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\end{split}
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$$
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where $\bar{y}$ is the [sample mean](/D/mean-samp) and $\bar{v}$ is the [sample variance](/D/var-samp), and $\bar{s}$ is the [sample skewness](/D/skew-samp)
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$$ \label{eq:y-mean-var-skew}
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\begin{split}
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\bar{y} &= \frac{1}{n} \sum_{i=1}^n y_i \\
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\bar{v} &= \frac{1}{n-1} \sum_{i=1}^n (y_i - \bar{y})^2 \\
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\bar{s} &= \frac{\frac{1}{n}\sum_{i=1}^n (y_i-\bar{y})^3}{\left[\frac{1}{n}\sum_{i=1}^n(y_i-\bar{y})^2\right]^{3/2}} \; .
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\end{split}
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$$
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**Proof:** The [mean](/P/exg-mean), [variance](/P/exg-var), and [skewness](/P/exg-skew) of the [ex-Gaussian distribution](/D/exg) in terms of the parameters $\mu$, $\sigma$, and $\lambda$ are given by
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$$ \label{eq:exg-E-Var-Skew}
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\begin{split}
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\mathrm{E}(X) &= \mu + \frac{1}{\lambda} \\
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\mathrm{Var}(X) &= \sigma^2 + \frac{1}{\lambda^2}\\
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\mathrm{Skew}(X) &= \frac{2}{\lambda^3\left(\sigma^2+\frac{1}{\lambda^2}\right)^{3/2}} \; .
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\end{split}
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$$
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Thus, [matching the moments](/D/mome) requires us to solve the following system of equations for $\mu$, $\sigma$, and $\lambda$:
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$$ \label{eq:exg-mean-var-skew}
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\begin{split}
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\bar{y} &= \mu + \frac{1}{\lambda} \\
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\bar{v} &= \sigma^2 + \frac{1}{\lambda^2}\\
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\bar{s} &= \frac{2}{\lambda^3\left(\sigma^2+\frac{1}{\lambda^2}\right)^{3/2}} \; .
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\end{split}
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$$
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To this end, our first step is to substitute the second equation of \eqref{eq:exg-mean-var-skew} into the third equation:
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$$ \label{eq:lambda-s1}
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\begin{split}
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\bar{s} &= \frac{2}{\lambda^3\left(\sigma^2+\frac{1}{\lambda^2}\right)^{3/2}} \\
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&= \frac{2}{\lambda^3\cdot \bar{v}^{3/2}} \; .
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\end{split}
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$$
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Re-expressing \eqref{eq:lambda-s1} in terms of $\lambda^3$ and taking the cube root gives:
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$$ \label{eq:lambda-s2}
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\lambda = \sqrt[3]{\frac{2}{\bar{s}\cdot \bar{v}^{3/2}}} \; .
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$$
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Next, we solve the first equation of \eqref{eq:exg-mean-var-skew} for $\mu$ and substitute \eqref{eq:lambda-s2}:
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$$ \label{eq:mu-s1}
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\begin{split}
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\mu &= \bar{y} - \frac{1}{\lambda}\\
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&= \bar{y} - \sqrt[3]{\frac{\bar{s}\cdot\bar{v}^{3/2}}{2}} \; .
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\end{split}
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$$
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Finally, we solve the second equation of \eqref{eq:exg-mean-var-skew} for $\sigma$:
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$$ \label{eq:sigma-s1}
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\sigma^2 = \bar{v} - \frac{1}{\lambda^2} \; .
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$$
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Taking the square root gives and substituting \eqref{eq:lambda-s2} gives:
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$$ \label{eq:sigma-s2}
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\begin{split}
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\sigma &= \sqrt{\bar{v}-\frac{1}{\lambda^2}} \\
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&= \sqrt{\bar{v} - \left(\sqrt[3]{\frac{\bar{s}\cdot \bar{v}^{3/2}}{2}}\right)^2} \\
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&= \sqrt{\bar{v} - \bar{v}\cdot\sqrt[3]{\frac{\bar{s}^2}{4}}} \\
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&= \sqrt{\bar{v}\cdot \left(1- \sqrt[3]{\frac{\bar{s}^2}{4}}\right)} \; .
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\end{split}
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$$
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Together, \eqref{eq:mu-s1}, \eqref{eq:sigma-s2}, and \eqref{eq:lambda-s2} constitute the method-of-moment estimates of $\mu$, $\sigma$, and $\lambda$.

‎P/wald-mome.md‎

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---
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layout: proof
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mathjax: true
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author: "Thomas J. Faulkenberry"
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affiliation: "Tarleton State University"
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e_mail: "faulkenberry@tarleton.edu"
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date: 2023-10-30 12:00:00
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title: "Method of moments for Wald distributed data"
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chapter: "Probability Distributions"
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section: "Univariate continuous distributions"
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topic: "Wald distribution"
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theorem: "Method of moments"
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sources:
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proof_id: "P423"
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shortcut: "wald-mome"
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username: "tomfaulkenberry"
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---
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**Theorem:** Let $y = \left\lbrace y_1, \ldots, y_n \right\rbrace$ be a set of observed data [independent and identically distributed](/D/iid) according to a [Wald distribution](/D/wald) with drift rate $\gamma$ and threshold $\alpha$:
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$$ \label{eq:wald}
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y_i \sim \mathrm{Wald}(\gamma,\alpha), \quad i = 1, \ldots, n \; .
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$$
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Then, the [method-of-moments estimates](/D/mome) for the parameters $\gamma$ and $\alpha$ are given by
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$$ \label{eq:wald-MoM}
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\begin{split}
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\hat{\gamma} &= \sqrt{\frac{\bar{y}}{\bar{v}}} \\
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\hat{\alpha} &= \sqrt{\frac{\bar{y}^3}{\bar{v}}}
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\end{split}
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$$
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where $\bar{y}$ is the [sample mean](/D/mean-samp) and $\bar{v}$ is the [unbiased sample variance](/D/var-samp):
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$$ \label{eq:y-mean-var}
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\begin{split}
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\bar{y} &= \frac{1}{n} \sum_{i=1}^n y_i \\
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\bar{v} &= \frac{1}{n-1} \sum_{i=1}^n (y_i - \bar{y})^2 \; .
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\end{split}
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$$
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**Proof:** The [mean](/P/wald-mean) and [variance](/P/wald-var) of the [Wald distribution](/D/wald) in terms of the parameters $\gamma$ and $\alpha$ are given by
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$$ \label{eq:wald-E-Var}
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\begin{split}
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\mathrm{E}(X) &= \frac{\alpha}{\gamma} \\
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\mathrm{Var}(X) &= \frac{\alpha}{\gamma^3} \; .
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\end{split}
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$$
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Thus, [matching the moments](/D/mome) requires us to solve the following system of equations for $\gamma$ and $\alpha$:
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$$ \label{eq:wald-mean-var}
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\begin{split}
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\bar{y} &= \frac{\alpha}{\gamma} \\
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\bar{v} &= \frac{\alpha}{\gamma^3} \; .
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\end{split}
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$$
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To this end, our first step is to express the second equation of \eqref{eq:wald-mean-var} as follows:
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$$ \label{eq:gamma-s1}
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\begin{split}
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\bar{v} &= \frac{\alpha}{\gamma^3} \\
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& = \frac{\alpha}{\gamma} \cdot \gamma^{-2}\\
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& = \bar{y} \cdot \gamma^{-2} \; .
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\end{split}
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$$
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Rearranging \eqref{eq:gamma-s1} gives
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$$ \label{eq:gamma-s2}
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\gamma^2 = \frac{\bar{y}}{\bar{v}} \; ,
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$$
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or equivalently,
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$$ \label{eq:gamma-s3}
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\gamma = \sqrt{\frac{\bar{y}}{\bar{v}}} \; .
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$$
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Our final step is to solve the first equation of \eqref{eq:wald-mean-var} for $\alpha$ and substitute \eqref{eq:gamma-s3} for $\gamma$:
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$$ \label{eq:alpha-s1}
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\begin{split}
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\alpha & = \bar{y} \cdot \gamma \\
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& = \bar{y} \cdot \sqrt{\frac{\bar{y}}{\bar{v}}}\\
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&= \sqrt{\bar{y}^2} \cdot \sqrt{\frac{\bar{y}}{\bar{v}}}\\
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&= \sqrt{\frac{\bar{y}^3}{\bar{v}}} \; .
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\end{split}
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$$
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Together, \eqref{eq:gamma-s3} and \eqref{eq:alpha-s1} constitute the method-of-moment estimates of $\gamma$ and $\alpha$.

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