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)  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)  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)  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)  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)  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)  Describe what happens to the ﬁrst moment when t ! 0 and t ! 1. EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

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