统计代写|STAC58 Statistical inference

MY-ASSIGNMENTEXPERT™可以为您提供 utstat.utoronto STAC58 Statistical inference统计推断的代写代考和辅导服务！

STAC58课程简介

The course surveys the various approaches that have been considered for the development of a theory of statistical reasoning. These include likelihood methods, Bayesian methods and frequentism. These are compared with respect to their strengths and weaknesses. To use statistics successfully requires a clear understanding of the meaning of various concepts such as likelihood, confidence, p-value, belief, etc. This is the purpose of the course.

Prerequisites 

Evaluation
A midterm worth 40% and a final worth 60%.

References
The following texts will be used in the course.
Probability and Statistics; The Science of Uncertainty by M. Evans and J. Rosenthal – this is the text used for STAB52F/B57S and we will be focusing on chapters 6, 7, 8 and 9 but only some this will be review. The book is available for download on my website.`
Measuring Statistical Evidence Using Relative Belief by M. Evans – this book gives a general discussion of the different approaches to inference. The book is available electronically from the University of Toronto Library.

STAC58 Statistical inference HELP（EXAM HELP， ONLINE TUTOR）

(15 points) Let $\left(X_1, \cdots, X_n\right)$ be a sample from $N\left(0, \sigma_X^2\right)$, and let $\left(Y_1, \cdots, Y_m\right)$ be a sample from $N\left(0, \sigma_Y^2\right)$, independent of the $\left(X_1, \cdots, X_n\right)$. Define $\lambda=\sigma_Y^2 / \sigma_X^2$.

(b). To construct the rejection region of the LRT, we need to find the distribution of the test statistic under the null and alternative hypotheses. Under the null hypothesis $\$ H_{-} 0: \backslash$ lambda $=1 \$$, the test statistic
$$
2 \log \frac{L(\lambda=1)}{L(\lambda)}=\frac{n m}{n+m} \log \left(1+\frac{(n+m-2) S^2}{\sigma_X^2}\right),
$$
where $\$ S^{\wedge} 2 \$$ is the pooled sample variance given by
$$
S^2=\frac{(n-1) S_X^2+(m-1) S_Y^2}{n+m-2}
$$
with $\$ S_{-} X^{\wedge} 2 \$$ and $\$ S_{-} Y^{\wedge} 2 \$$ being the sample variances of the $\$ X_{-} \mathrm{i} \$$ and $\$ Y j \$$, respectively. Under the alternative hypothesis $\$ H_{-} 1$ : $\backslash$ lambda $>1 \$$, the test statistic has an $\$ F \$$ distribution with degrees of freedom $\$ d f _1=n+m-$ $2 \$$ and $\$ \mathrm{df} 2=\mathrm{n}-1 \$$, given by
$$
F=\frac{(n+m-2) S^2}{(n-1) \sigma_X^2}
$$

Thus, the rejection region of the LRT of part (a) is given by
$$
\frac{(n+m-2) S^2}{(n-1) \sigma_X^2}>F_{d f_1, d f_2, 1-\alpha},
$$
where $\$ F _\left{d f _1, d f _2,1 \text {-\alpha} } \$ \text { is the } \$(1 \text {-\alpha) } \$ \text {-quantile of the } \$ F \$\right.$ distribution with degrees of freedom $\$ \mathrm{df}_{-} 1 \$$ and $\$ \mathrm{df} 2 \$$.

(c). To find a $\$ 1$-\alpha $\$$ confidence interval for $\$ \backslash$ lambda $\$$, we can use the pivotal quantity
$$
\frac{S_Y^2}{S_X^2} \sim \frac{\sigma_Y^2}{\sigma_X^2} \cdot \frac{\chi_{m-1}^2}{\chi_{n-1}^2}
$$
which follows a non-central \$F\$ distribution with degrees of freedom $\$ m-1 \$$ and $\$ n-1 \$$ and non-centrality parameter $\$ \backslash$ delta $=\backslash f r a c{\backslash a m b d a}$ ${1+\backslash$ lambda $} \backslash c d o t \backslash$ frac ${m-1}{n-1} \$$. Thus, a $\$ 1$-\alpha $\$$ confidence interval for $\$ \backslash$ lambda $\$$ is given by
$$
\left[\frac{S_Y^2}{S_X^2} \cdot \frac{1}{F_{m-1, n-1,1-\frac{\alpha}{2}}\left(1+\frac{\lambda}{1+\lambda} \cdot \frac{m-1}{n-1}\right)}, \frac{S_Y^2}{S_X^2} \cdot F_{n-1, m-1,1-\frac{\alpha}{2}}\left(1+\frac{\lambda}{1+\lambda} \cdot \frac{n-1}{m-1}\right)\right]
$$

MY-ASSIGNMENTEXPERT™可以为您提供UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN MATH2940 linear algebra线性代数课程的代写代考和辅导服务！ 请认准MY-ASSIGNMENTEXPERT™. MY-ASSIGNMENTEXPERT™为您的留学生涯保驾护航。