  We consider the extreme eigenvalues of the sample covariance matrix $Q=YY^*$
under the generalized elliptical model that $Y=\Sigma^{1/2}XD.$ Here $\Sigma$
is a bounded $p \times p$ positive definite deterministic matrix representing
the population covariance structure, $X$ is a $p \times n$ random matrix
containing either independent columns sampled from the unit sphere in
$\mathbb{R}^p$ or i.i.d. centered entries with variance $n^{-1},$ and $D$ is a
diagonal random matrix containing i.i.d. entries and independent of $X.$ Such a
model finds important applications in statistics and machine learning.
  In this paper, assuming that $p$ and $n$ are comparably large, we prove that
the extreme edge eigenvalues of $Q$ can have several types of distributions
depending on $\Sigma$ and $D$ asymptotically. These distributions include:
Gumbel, Fr\'echet, Weibull, Tracy-Widom, Gaussian and their mixtures. On the
one hand, when the random variables in $D$ have unbounded support, the edge
eigenvalues of $Q$ can have either Gumbel or Fr\'echet distribution depending
on the tail decay property of $D.$ On the other hand, when the random variables
in $D$ have bounded support, under some mild regularity assumptions on
$\Sigma,$ the edge eigenvalues of $Q$ can exhibit Weibull, Tracy-Widom,
Gaussian or their mixtures. Based on our theoretical results, we consider two
important applications. First, we propose some statistics and procedure to
detect and estimate the possible spikes for elliptically distributed data.
Second, in the context of a factor model, by using the multiplier bootstrap
procedure via selecting the weights in $D,$ we propose a new algorithm to infer
and estimate the number of factors in the factor model. Numerical simulations
also confirm the accuracy and powerfulness of our proposed methods and
illustrate better performance compared to some existing methods in the
literature.
