Theory of Statistics - George Mason University

198 2 Distribution Theory and Statistical Models
where µ is a 2-vector of constants and Σ is a 2 ×2 positive definite matrix
of constants.
Show that this family is of the exponential class, and express the density
in the canonical (or natural) form of the exponential class.
2.17. Show that both G and G in Exmaple 2.3 on page 179 are groups.
2.18. a) Show that each of the distributions listed in Table 2.4 is a location
family.
b) Show that each of the distributions listed in Table 2.5 is a scale family.
2.19. Show that the likelihood function and the PDF for a location familiy are
the same function. Produce graphs similar to those in Figure 1.2 for the
exponential(α, θ0) family.
2.20. Show that these distributions are not location-scale families:
the usual one-parameter exponential family, the binomial family, and the
Poisson family.
2.21. Show that a full rank exponential family is complete.
2.22. Prove Theorem 2.1.
2.23. Show that the normal, the Cauchy, and the Poisson families of distributions
are all infinitely divisible.
2.24. Show that the Poisson family of distributions is not stable.
2.25. Show that the normal and the Cauchy families of distributions are stable,
and show that their indexes of stability are 2 and 1 respectively.
2.26. Express the PDF of the curved normal family N(µ, µ 2 ) in the canonical
form of an exponential family.
2.27. Higher dimensions.
a) Let the random variable X have a uniform distribution within the
ball x2 ≤ 1 in IR d . This is a spherical distribution. Now, for given
0 < δ < 1, show that
Pr(X > 1 − δ) → 1
as d increases without bound.
b) Let the random variable X have a d-variate standard normal distribution
distribution, Nd(0, Id). Determine
lim Pr(X > 1).
d→∞
2.28. a) Show that the joint density of X, Y1, . . ., Yn given in equations (2.31),
(2.35), and (2.36) is proportional to
exp −n(¯x − µ) 2 )/2σ 2 exp −(y 2 1 + · · · + y 2 n)/2σ 2 .
b) Show that the joint density of W1, . . ., Wn−1 given in equation (2.39)
is the same as n − 1 iid N(0, σ2 ) random variables.
2.29. Higher dimensions. Let X ∼ Nd(0, I) and consider
1
E .
X 2 2
Theory of Statistics c○2000–2013 James E. Gentle