| 91年 6月19日 10:30~12:00 應用數學系 (選考) | 誠實是我們珍視的美德， 我們喜愛「拒絕作弊，堅守正直」的你！ |
| 科目： 統 計 |
|
1.
|
(二十分) Let X1, . . . , Xn
be a random sample from a population with a continuous distribution function
F, and let  denote
the empirical distribution of the sample. |
|
| (a) | Show that
converges almost surely to F (x). |
|
| (b) | Show that the distribution of the test statistic Dn
= supx | (x)
- F (x)| is the same for all continuous F . |
|
|
2.
|
(二十分) Suppose that we observe Y
∼ Nn(μ,  ),
μ , 
> 0, where V is a p- dimensional subspace of Rn.
Suppose that there exist subspaces Ui such that Ui
 Uj for i 
j and such that μ = 
. Let Pｚ denote the projection onto the subspace Z. |
|
| (a) | Show that  for
all  . |
|
| (b) |
Show that the F-statistic for testing that where kj is the dimension of Uj
, |
|
|
3.
|
(二十分) Let X1, . . . ,Xn
be a random sample from a distribution with mean μ, variance
, skewness  , and kurtosis
 . Let 
and  be the sample mean and
sample variance of the  . |
|
| (a) | Show that  . |
|
| (b) | What is the limiting distribution of 
? |
|
|
4.
|
(十分) Consider one parameter exponential
family of X given by 
, and of its size-biased version X* given by 
. |
|
| (a) |
Show that the change in Fisher's information due to size-bias becomes |
|
| (b) | Show that X* ∼ Gamma( ,
k + 1) when X ∼ Gamma(, k).
Further show that the size-bias version of gamma is more informative for
the scale parameter than the original gamma. |
|
|
5.
|
(二十分) The coefficient of crowding |
|
| (a) |
Let μ(r) denote the factorial moment of order r of X. Show that |
|
| (b) | Show that  |
|
|
6.
|
(十分) Consider the bivariate distribution of X and Y defined
as follows: Let U and V be independent N (0, 1)
random variables. Let X = U if UV |
|
| Show that | ||
| (a) | X and Y are each N (0,1), but their joint distribution is not bivariate normal; | |
| (b) | X 2 and Y2 are indepe ndent, but X and Y are not. | |
= 0 is given by 
and 
is the ordinary least squares estimator of 
.
of order r for a non-negative integer-valued random variable X
is defined by
.
.
0 and X = －U if UV < 0.--- END---