The Fisher limiting spectral distribution (LSD), denoted Fs;t , has the density function

Question 1 [20 marks]

Suppose we had two independent p-dimensional vector samples X := fx1; : : : ; xn1g and
Y := fy1; : : : ; yn2g where p  n2. We assume that each sample comes from a (possibly
diﬀerent) population distribution with i.i.d. components and ﬁnite second moment.

(a) [3] How is the Fisher LSD related to the two vector samples X and Y ? What is the
relationship between the two parameters (s; t) of Fs;t and the three values (p; n1; n2)
describing the dimensionality and sizes of X and Y ?

(b) [3] Now that you explained the relationship between X, Y , the values (p; n1; n2) and the
parameters (s; t) of Fs;t in part (a), what would you expect the empirical density of
eigenvalues be for the following three choices of triplets (p; n1; n2):
(50; 100; 100); (75; 100; 200); (25; 100; 200):

Plot the three densities on the same ﬁgure with an appropriate legend.

(c) [4] Perform a simulation study for the same choices of the triplets (p; n1; n2) that are
given in part (b). Setup your experiment correctly to demonstrate a histogram
of eigenvalues and compare them to the appropriately parametrised densities from
part (b). For each triplet (p; n1; n2), plot the histogram of eigenvalues, overlay the
appropriate density, and ensure the plot is appropriately titled. Show the code for

(d) [4] Derive a formula for the ﬁrst moment of the Fisher LSD in terms of its parameters
s and t. You can assume that h > t as this always holds by the deﬁnition of h.

(e) [4] Perform a numerical experiment to conﬁrm the formula you derived in part (d). That
is, choose 5 values of (s; t) then use your simulation study code (from part c) to
generate empirical eigenvalues for those values and calculate their sample mean.
Compare the sample means to your formula.

(f) [2] Describe what happens to the ﬁrst moment when t ! 0 and t ! 1.

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