Question
Consider a random sample of size n from a gamma distribution, Xi~GAM(theta, kappa).
a) Assuming kappa is known, derive a 100(1-alpha)% equal tailed confidence interval for theta based on the sufficient statistic
b) Assuming theta=1, and for n=1, find an equal tailed 90% confidence interval for kappa if x1=10 is observed
HINT: note that 2X1~ChiSquare(2kappa)
Explanation / Answer
Abstract: Circulant matrices with general shift by k places have been studied in the literature and formula for their eigenvalues is known. We first reestablish this formula and some further properties of these eigenvalues in a manner suitable for our use. We then consider random k=k(n) circulants A_{k,n} with $n o infty$ and whose input sequence {a_i} is independent with mean zero and variance one and $sup_n n^{-1}sum_{i=1}^n E|a_i|^{2+delta}< infty$ for some $delta > 0$. Under suitable restrictions on {k(n)},we show that the limiting spectral distribution (LSD) of the empirical distribution of suitably scaled eigenvalues exists and identify the limits. As examples, (i) if k^g = -1+ s n where $g ge 1 $ fixed and $s=o(n^{1/3})$, then the LSD is $U_1(prod_{i=1}^g E_i)^{1/2g}$ where E_i are i.i.d. Exp(1) and U_1 is uniformly distributed over the (2g)th roots of unity, independent of the {E_i}, and (ii) if k^g = 1+ sn where $g ge 2$ is fixed and $s=o(n^{ rac{g+1}{g-1}})$ or $s=o(n)$ according as $g ge 2$ is odd or even, then the LSD is $U_2(prod_{i=1}^g E_i)^{1/2g}$ where {E_i} are i.i.d. Exp(1) and U_2 is uniformly distributed over the unit circle, independent of the {E_i}. We then consider the limit distribution of the spectral norm of A_{k,n}. We show that when $n=k^2+1 o infty$, the spectral norm, with appropriate scaling and centering, which we provide explicitly, converges to the Gumbel distribution.