### 统计代写|工程统计代写engineering statistics代考|STA 3032

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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|ALTERNATE REPRESENTATIONS

The standard arcsine distribution SASD-I has range $(0,1)$, and SASD-II has range $(-1,+1)$. They are also written as $\frac{1}{\pi}(x(1-x))^{-1 / 2}\left(\right.$ or $\left.\left(\pi^{2} x(1-x)\right)^{-1 / 2},\left(\pi^{2} x(1-x)\right)^{-0.5}\right)$ and $\frac{1}{\pi}(1-$ $\left.x^{2}\right)^{-1 / 2}\left(\right.$ or $\left.\left(\pi^{2}\left(1-x^{2}\right)\right)^{-1 / 2},\left(\pi^{2}\left(1-x^{2}\right)\right)^{-0.5}\right)$, respectively. As $\Gamma(1 / 2)=\sqrt{\pi}$, it can also be written as $(x(1-x))^{-1 / 2} / \Gamma(1 / 2)^{2}$. Arcsine distribution with PDF
$$f(x ; R)=1 /[\pi \sqrt{x(R-x)}] \text {, for } 00 whose CDF is the inverse hyperbolic function (1 / R) \sinh ^{-1}(x / R). Geometrically, the density on [0,1] (SASD-I) gives the distribution of the projection of a random point on a circle of radius half centered at (0.50,0) to the continuous interval [0,1] on the \mathrm{X}-axis, and as projection of a random point on a centered circle (at origin) with appropriate radius for symmetric versions (e.g., SASD-II). When the domain is [-R,+R] this circle is origin-centered with radius R. Shifts of the circle on the horizontal axis results in other displaced distributions discussed below. By assuming that the circle rolls continuously at constant speed horizontally, it can be used to model the position of a particle moving in simple harmonic motion with amplitude R at a random time t. It is also used in von Neumann algebra theory. The two-parameter ASD-I has PDF$$
f(x ; a, b)=1 /[b \pi \sqrt{((x-a) / b)(1-(x-a) / b)}] \quad \text { for } \quad a<x<a+b,
$$and the corresponding ASD-II has PDF$$
f(x ; a, b)=1 /\left[b \pi \sqrt{1-((x-a) / b)^{2}}\right] \text { for } a-b<x<a+b .
$$This is a location-and-scale distribution that is symmetric around a and is U-shaped. It reduces to the SASD by the transformation Y=(X-a) / b. Differentiate w.r.t. x, and equate to zero to get the minimum at x=a, with minimum value f(a)=1 /(b \pi). In terms of the minimum value, the PDF (5.6) can be written as$$
f(x ; a, b)=f(a) / \sqrt{1-((x-a) / b)^{2}} \text { for } a-b<x<a+b .
$$Next, consider the PDF$$
f(x ; a, b)=1 /[\pi \sqrt{(x-a)(b-x)}]=\left[\pi^{2}(x-a)(b-x)\right]^{-1 / 2} \quad \text { for } \quad a<x<b
$$## 统计代写|工程统计代写engineering statistics代考|RELATION TO OTHER DISTRIBUTIONS This is a special case of the beta distribution (Chapter 4) when a=1 / 2, b=1 / 2. Hence, all properties of beta distribution are applicable to SASD-I as well. In particular, X and 1-X are identically distributed. As the range of SASD-I is (0,1), the transformation Y=-\log (X) results in log-arcsine distribution discussed in page 60. If X_{k}^{\prime} s are IID Beta-I( \left.\mathrm{I} \frac{2 k-1}{2 n}, \frac{1}{2 n}\right) random variables, the distribution of the geometric mean (GM) of them Y=\left(\prod_{k=1}^{n} X_{k}\right)^{1 / n} is SASD-I distributed for n \geq 2 ([90], [22]). This has the interpretation that the log-arcsine law is decomposable into a sum (or average) of independent log-beta random variables (so that arcsine distribution is not additively decomposable or infinite divisible). If U is \operatorname{CUNI}(0,1) then Y=-\cos (\pi U / 2) is arcsine distributed. Conversely, if X has an arcsine distribution, U=(2 / \pi) \arcsin (\sqrt{x}) has the U(0,1) distribution. Differentiate w.r.t. u to get |\partial y / \partial u|= (\pi / 2) \sin (\pi u / 2), so that |\partial u / \partial y|=(2 / \pi) / \sqrt{1-\cos ^{2}(\pi u / 2)}=(2 / \pi) / \sqrt{1-y^{2}}. An alternate way to state this is as follows. If U \sim \operatorname{CUNI}(-\pi, \pi), the distribution of Y=\cos (u) is SASD-I (see below). Similarly, the SASD-II is related to the U(0,1) distribution as X=\cos (\pi u), because |\partial x / \partial u|=\pi \sin (\pi u)=\pi \sqrt{1-\cos ^{2}(\pi u)}=\pi \sqrt{1-x^{2}}. The transformation Y=2 X-1 and Y=X^{2} when applied to SASD-II results in SASD-I. Similarly, if X \sim SASD-I, then Y=\sqrt{X} is SASD-II. If \Phi 0 denotes the CDF of a normal distribution, \Phi^{-1}(F(x)) \sim \mathrm{N}(0,1) where F(x) denotes the CDF of ASD. Problem 5.2 If X \sim SASD-I prove that Y=1 / X has PDF f(y)=1 /[\pi y \sqrt{y-1}] for y>1. Example 5.3 Distribution of \cos (\mathrm{X}) If X \sim \operatorname{CUNI}(-\pi, \pi), find the distribution of Y= \cos (X) Solution 5.4 As X \sim \operatorname{CUNI}(-\pi, \pi), F(x)=1 / 2 \pi. From y=\cos (x), we get |d y / d x|= \sin (x)=\sqrt{1-\cos ^{2}(x)}=\sqrt{1-y^{2}}, so that f(y)=(1 / 2 \pi)\left(1 / \sqrt{1-y^{2}}\right). Since the equation y=\cos (x) has two solutions in -\pi, \pi as x_{1}=\cos ^{-1}(y) and x_{2}=2 \pi-x_{1}, the PDF becomes f(y)=1 /\left(\pi \sqrt{1-y^{2}}\right), which is SASD-II. ## 统计代写|工程统计代写engineering statistics代考|PROPERTIES OF ARCSINE DISTRIBUTION The SASD-I is a special case of beta type-I distribution. It is symmetric around the mean ( 1 / 2) and is concave upward (the probability decreases and then increases), but satisfies the log-convex property. As x \rightarrow 0 or x \rightarrow 1 the PDF f(x) \rightarrow \infty. Put Y=X-\frac{1}{2} to get$$
f(y)=\frac{1}{\pi \sqrt{(y+1 / 2)(1 / 2-y)}}, \quad-1 / 2<y<1 / 2 .
$$As (1 / 2+y)(1 / 2-y)=\left(1 / 4-y^{2}\right), the PDF becomes f(y)=(2 / \pi) / \sqrt{\left(1-4 y^{2}\right)}, for -1 / 2<y<1 / 2. The mean is 0.5 and variance is 0.125 for the standard arcsine distribution (see below). As the distribution is symmetric, coefficient of skewness is zero. The kurtosis coefficient is \beta_{2}=3 / 2. Thus, it is always platykurtic. Note that the density is maximum when x is near 0 or 1 with the center as a cusp (U-shaped), and minimum at x=0.5 with minimum value 2 / \pi. Hence, there are two modes (bimodal) that are symmetrically placed in the tails. This is the reason why it is platykurtic. Arcsine distribution is the exact antithesis of bell-shaped laws because (i) the mean coincides with the minimum (whereas mean coincides with maximum for bell-shaped laws), (ii) lower and upper limits correspond to asymptotes (density rises up to \infty ) (whereas bellshaped laws tail off to zero), (iii) bimodal (bell-shaped laws are unimodal), and (iv) statistical measures are more prone to outliers as the peaks are away from the mean (samples from bellshaped distributions have lesser chance of outliers). Due to these peculiarities, the convergence of central limit theorem to normality is slow. The hazard function is given by$$
1 / h(x ; a, b)=b \sqrt{1-[(x-a) / b]^{2}} \arccos ((x-a) / b)

## 统计代写|工程统计代写engineering statistics代考|PROPERTIES OF ARCSINE DISTRIBUTION

SASD-I 是 beta I 型分布的一个特例。它围绕均值对称（1/2)并且向上凹（概率先减小后增大），但满足对数凸性质。作为X→0或者X→1PDF格式F(X)→∞. 放是=X−12要得到

F(是)=1圆周率(是+1/2)(1/2−是),−1/2<是<1/2.

1/H(X;一个,b)=b1−[(X−一个)/b]2阿尔科斯⁡((X−一个)/b)

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