Under the normality assumption, it can be shown that $$ \frac{\hat{\beta}{i}-\beta{i}}{s_{\hat{\beta}{i}}} \sim t{n-p} $$ although we will not derive this result. It follows that a $100(1-\alpha) \%$ confidence interval for $\beta_{i}$ is $$ \hat{\beta}{i} \pm t{n-p}(\alpha / 2) s_{\hat{\beta}{i}} $$ To test the null hypothesis $H{0}: \beta_{i}=\beta_{i 0}$, where $\beta_{i 0}$ is a fixed number, we can use the test statistic $$ t=\frac{\hat{\beta}{i}-\beta{i 0}}{s_{\hat{\beta}_{i}}} $$
We estimate $\beta$ by minimizing the penalized least squares criterion $$ H(\beta)=(\mathbf{y}-\mathbf{H} \beta)^{T}(\mathbf{y}-\mathbf{H} \beta)+\lambda|\beta|^{2} . $$ The solution is $$ \hat{\mathbf{y}}=\mathbf{H} \hat{\beta} $$
证明 .
with $\hat{\beta}$ determined by $$ -\mathbf{H}^{T}(\mathbf{y}-\mathbf{H} \hat{\beta})+\lambda \hat{\beta}=0 $$ From this it appears that we need to evaluate the $M \times M$ matrix of inner products in the transformed space. However, we can premultiply by $\mathbf{H}$ to give $$ \mathbf{H} \hat{\beta}=\left(\mathbf{H H}^{T}+\lambda \mathbf{I}\right)^{-1} \mathbf{H H}^{T} \mathbf{y} $$
The variance of a r.v. $X$ is denoted by $\operatorname{Var}(X)$ and is defined by: $$ \operatorname{Var}(X)=E(X-E X)^{2} \text {. } $$ The explicit expression of the right-hand side in (8) is taken from (6) for $g(x)=(x-E X)^{2}$. The alternative notations $\sigma^{2}(X)$ and $\sigma_{X}^{2}$ are also often used for the $\operatorname{Var}(X)$.
The course consists of an in-depth study of multivariate statistical methods (in particular, the binary logistic regression) and the impact on these methods of problems encourtered on real data as well as the possible solutions. The course will focus on the following themes
If $X$ is an integer-valued random variable, show that the frequency function is related to the cdf by $p(k)=F(k)-F(k-1)$. Show that $P(u<X \leq v)=F(v)-F(u)$ for any $u$ and $v$ in the cases that (a) $X$ is a discrete random variable and (b) $X$ is a continuous random variable. Let $A$ and $B$ be events, and let $I_{A}$ and $I_{B}$ be the associated indicator random variables. Show that and $$ I_{A \cap B}=I_{A} I_{B}=\min \left(I_{A}, I_{B}\right) $$ $$ I_{A \cap B}=I_{A} I_{B}=\min \left(I_{A}, I_{B}\right) $$ $$ I_{A \cup B}=\max \left(I_{A}, I_{B}\right) $$
证明 .
7 Find the cdf of a Bernoulli random variable. 8 Show that the binomial probabilities sum to 1 . 9 For what values of $p$ is a two-out-of-three majority decoder better than transmission of the message once? 10 Appending three extra bits to a 4-bit word in a particular way (a Hamming code) allows detection and correction of up to one error in any of the bits. If each bit has probability $.05$ of being changed during communication, and the bits are changed independently of each other, what is the probability that the word is correctly received(that is, 0 or 1 bit is in error)? How does this probability compare to the probability that the word will be transmitted correctly with no check bits, in which case all four bits would have to be transmitted correctly for the word to be correct?
Consider the binomial distribution with $n$ trials and probability $p$ of success on each trial. For what value of $k$ is $P(X=k)$ maximized? This value is called the mode of the distribution. (Hint: Consider the ratio of successive terms.)
The joint behavior of two random variables, $X$ and $Y$ is determined by the cumulative distribution function $$ F(x, y)=P(X \leq x, Y \leq y) $$ whether $X$ and $Y$ are continuous or discrete. The cdf gives the probability that the point $(X, Y)$ belongs to a semi-infinite rectangle in the plane, as shown in Figure 3.1. The probability that $(X, Y)$ belongs to a given rectangle is, from Figure 3.2, $$ \begin{aligned} P\left(x_{1}<X \leq x_{2}, y_{1}<Y \leq y_{2}\right)=& F\left(x_{2}, y_{2}\right)-F\left(x_{2}, y_{1}\right)-F\left(x_{1}, y_{2}\right) \ &+F\left(x_{1}, y_{1}\right) \end{aligned} $$
