统计代写|工程统计代写engineering statistics代考|MOMENTS AND CUMULANTS

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

统计代写|工程统计代写engineering statistics代考|MOMENTS AND CUMULANTS

The concept of moments is used in several applied fields like mechanics, particle physics, kinetic gas theory, etc. Moments and cumulants are important characteristics of a statistical distribution and plays an important role in understanding respective distributions. The population moments (also called raw moments or moments around zero) are mathematical expectations of powers of the random variable. They are denoted by Greek letters as $\mu_{r}^{\prime}=\mathrm{E}\left(X^{r}\right)$, and the corresponding sample moment by $m_{r}^{\prime}$. The zeroth moment being the total probability is obviously one. The first moment is the population mean (i.e., expected value of $X, \mu=E(X)$ ). There exist an alternative expectation formula for non-negative continuous distributions as $E(X)=\int_{l l}^{u l}(1-F(x)) d x$, where $l l$ is the lower and $u l$ is the upper limit. This takes the simple and more familiar form $E(X)=\int_{0}^{\infty}(1-F(x)) d x=\int_{0}^{\infty} S(x) d x$ when the range is $(0, \infty)$. Positive moments are obtained when $r$ is a positive integer, negative moments when $r$ is a negative integer, and fractional moments when $r$ is a real number. Alternate expectation formulas exist for higher-order moments as well (see [60]):
$$E\left(X^{r}\right)=\int_{0}^{\infty} r x^{r-1}(1-F(x)) d x=\int_{0}^{\infty} r x^{r-1} S(x) d x ; \quad \text { for } \quad r \geq 1 .$$
Central moments are moments around the mean, denoted by $\mu_{r}=\mathrm{E}\left((X-\mu)^{r}\right)$. As $E(X)=\mu$, the first central moment is always zero, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis.
There are several measures of dispersion available. Examples are mean absolute deviation, variance (and standard deviation (SD)), range, and coefficient of variation (CV) (the ratio of SD and the mean $\left(s / \bar{x}{n}\right)$ for a sample, and $\sigma / \mu$ for a population). A property of the variance is that the variance of a linear combination is independent of the constant term, if any. Mathematically, $\operatorname{Var}(c+b X)=|b| \operatorname{Var}(X)$ (which is devoid of the constant c). Similarly, the variance of a linear combination is the sum of the variances if the variables are independent $(\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y))$. The $\mathrm{SD}$ is the positive square root of variance, and is called volatility in finance and econometrics. It is used for data normalization as $z{i}=\left(x_{i}-\bar{x}\right) / s$ where $s$ is the $\mathrm{SD}$ of a sample $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$. Data normalized to the same scale or frequency can be combined. This technique is used in several applied fields like spectroscopy, thermodynamics, machine learning, etc. The CV quantifies relative variation within a sample or population. Very low CV values indicate relatively little variation within the groups, and very large values do not provide much useful information. It is used in bioinformatics to filter genes, which are usually combined in a serial manner. It also finds applications in manufacturing engineering, education, and psychology to compare variations among heterogeneous groups as it captures the level of variation relative to the mean.

The inverse moments (also called reciprocal moments) are mathematical expectations of negative powers of the random variable. A necessary condition for the existence of inverse moments is that $f(0)=0$, which is true for $\chi^{2}, F$, beta, Weibull, Pareto, Rayleigh, and Maxwell distributions. More specifically, $E(1 / X)$ exists for a non-negative random variable $X$ iff $\int_{0}^{\delta}(f(x) / x) d x$ converges for some small $\delta>0$. Although factorial moments can be defined in terms of Stirling numbers, they are not popular for continuous distributions. The absolute moments for random variables that take both negative and positive values are defined as $v_{k}=E\left(|X|^{k}\right)=\int_{-\infty}^{\infty}|x|^{k} f(x) d x=\int_{-\infty}^{\infty}|x|^{k} d F(x)$

统计代写|工程统计代写engineering statistics代考|SIZE-BIASED DISTRIBUTIONS

Any statistical distribution with finite mean can be extended by multiplying the PDF or PMF by $C(1+k x)$, and choosing $C$ such that the total probability becomes one ( $k$ is a user-chosen nonzero constant). This reduces to the original distribution for $k=0$ (in which case $C=1$ ). The unknown $C$ is found by summing over the range for discrete, and by integrating for continuous and mixed distributions. This is called size-biased distribution (SBD), which is a special case of weighted distributions. Consider the continuous uniform distribution with PDF $f(x ; a, b)=1 /(b-a)$ for $a<x<b$, denoted by CUNI $(a, b)$. The size-biased distribution is $g(y ; a, b, k)=[C /(b-a)](1+k y)$. This means that any discrete, continuous, or mixed distribution for which $\mu=E(X)$ exists can provide a size-biased distribution. As shown in Chapter $2, \operatorname{CUNI}(a, b)$ has mean $\mu=(a+b) / 2$. Integrate the above from $a$ to $b$, and use the above result to get $[C /(b-a)] \int_{a}^{b}(1+k y) d y=1$, from which $C=2 /[2+k(a+b)]=1 /(1+k \mu)$. Similarly, the exponential SBD is $g(y ; k, \lambda)=C \lambda(1+k y) \exp (-\lambda y)$ where $C=\lambda^{2} /(k+\lambda)$. As another example, the well-known Rayleigh and Maxwell distributions (discussed in Part II) are not actually new distributions, but simply size-biased Gaussian distributions $\mathrm{N}\left(0, a^{2}\right)$ with biasing term $x$, and $\mathrm{N}(0, k T / m)$ with biasing term $x^{2}$, respectively. Other SBDs are discussed in respective chapters.

We used expectation of a linear function $(1+k x)$ in the above formulation. This technique can be extended to higher-order polynomials acting as weights $\left(\right.$ e.g., $\left.\left(1+b x+c x^{2}\right)\right)$, as also first- or higher-order inverse moments, if the respective moments exist. Thus, if $E(1 /(a+$ $b x)$ ) exists for a distribution with PDF $f(x)$, we could form a new SBD as $g(x ; a, b, C)=$ $C f(x) /(a+b x)$ by choosing $C$ so as to make the total probability unity. This concept was introduced by Fisher (1934) [61] to model ascertainment bias in the estimation of frequencies, and extended by Rao $(1965,1984)[113,115]$. More generally, if $w(x)$ is a non-negative weight function with finite expectation $(E(w(x))<\infty)$, then $w(x) f(x) / E(w(x))$ is the PDF of weighted distribution (in the continuous case; or PMF in discrete case). It is sometimes called lengthbiased distribution when $w(x)=x$, with PDF $g(x)=x f(x) / \mu$ (because the weight $x$ acts as a length from some fixed point of reference). As a special case, we could weigh using $E\left(x^{k}\right)$ and $E\left(x^{-k}\right)$ when the respective moments of order $k$ exists with PDF $x^{\pm k} f(x) / \mu_{\pm k}^{\prime}$. This results in either distributions belonging to the same family (as in $\chi^{2}$, gamma, Pareto, Weibull, F, beta, and power laws), different known families (size-biasing exponential distribution by $E\left(x^{k}\right)$ results in gamma law, and uniform distribution in power law), or entirely new distributions (Student’s T, Laplace, Inverse Gaussian, etc.). Absolute-moment-based weighted distributions can be defined when ordinary moments do not exist as in the case of Cauchy distribution (see Section 9.4, page 124). Fractional powers can also be used to get new distributions for positive random variables. Other functions like logarithmic (for $x \geq 0$ ) and exponential can be used to get nonlinearly weighted distributions. The concept of SBD is applicable to classical (discrete and continuous) distributions as well as to truncated, transmuted, exponentiated, skewed, mixed, and other extended distributions.

统计代写|工程统计代写engineering statistics代考|LOCATION-AND-SCALE DISTRIBUTIONS

The LaS distributions are those in which the location information (central tendency) is captured in one parameter, and scale (spread and skewness) is captured in another.

Definition 1.11 A parameter $\theta$ is called a location parameter if the PDF is of the form $f(x \mp$ $\theta)$, and a scale parameter if the PDF is of the form $(1 / \theta) f(x / \theta)$.

Most of the LaS distributions are of the continuous type. Examples are the general normal, Cauchy, and double-exponential (Laplace) distributions. If $\mu$ is the mean and $\sigma$ is the standard deviation of a univariate random variable $X$, and $Z=(X-\mu) / \sigma$ results in a standard distribution (devoid of parameters; see Section 1.1.2 in page 2), we say that $X$ belongs to the LaS family. This definition can easily be extended to the multivariate case where $\mu$ is a vector and $\Sigma$ is a matrix so that $Z=(X-\mu)^{\prime} \Sigma^{-1}(X-\mu)$ is in standard form. Sample data values are standardized using the transformation $z_{k}=\left(x_{k}-\bar{x}\right) / s_{x}$, where $s_{x}$ is the sample standard deviation, which can be applied to samples from any population including $\mathrm{LaS}$ distributions. The resulting values are called $z$-values or $z$-scores.

Write the above in univariate case as $X=\mu+\sigma Z$. As $\sigma$ is positive, a linear transformation with positive slope of any standard distribution results in a location-scale family for the underlying distribution. When $\sigma=1$, we get the one-parameter location family, and when $\mu=0$, we get the scale family. The exponential, gamma, Maxwell, Pareto, Rayleigh, Weibull, and halfnormal are scale-family distributions. The CDF of $X$ and $Z$ are related as $F((x-\mu) / \sigma)=G(x)$, and the quantile functions of $X$ and $Z$ are related as $F^{-1}(p)=\mu+\sigma G^{-1}(p)$. For $X$ continuous, the densities are related as $g(x)=(1 / \sigma) f((x-\mu) / \sigma)$. Maximum likelihood estimates (MLE) of the parameters of LaS distributions have some desirable properties. They are also easy to fit using available data. Extensions of this include $\log$-location-scale (LLS) distributions, nonlinearly transformed $\mathrm{LaS}$ distributions (like trigonometric, transcendental, and other functions of it, etc.) (Jones (2015) [80], Jones and Angela (2015) [81]).

统计代写|工程统计代写engineering statistics代考|LOCATION-AND-SCALE DISTRIBUTIONS

LaS 分布是在一个参数中捕获位置信息（集中趋势）而在另一个参数中捕获尺度（分布和偏度）的分布。

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