The plug-in principle

The plug-in principle is a technique used in probability theory and statistics to approximately compute or to estimate a feature of a probability distribution (e.g., the expected value, the variance, a quantile) that cannot be computed exactly. It is widely used in the theories of Monte Carlo simulation and bootstrapping.

Roughly speaking, the plug-in principle says that a feature of a given distribution can be approximated by the same feature of the empirical distribution of a sample of observations drawn from the given distribution. The feature of the empirical distribution is called a plug-in estimate of the feature of the given distribution. For example, a quantile of a given distribution can be approximated by the analogous quantile of the empirical distribution of a sample of draws from the given distribution.

https://www.statlect.com/asymptotic-theory/plug-in-principle

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quadratic forms

Basic matrix operations

BASIC FACTS ABOUT MATRICES

[1] A + B = B + A

[2] x(A + B) = xA + xB,           where x is any number

[3] (x+y)A = xA + yB

[4] AB does not always equal BA

[5] A(BC) = (AB)C

[6] A(BC) does not always equal (AC)B (for example, consider A = I)

[7] AA^-1 = I, the Identity matrix

[8] (A^T)^T = A

[9] (A + B)^T  = A^T + B^T

[10] (xA)^T = x A^T

[11] (AB)^-1 = B^-1 A^-1

[12] (AB)^T = B^T A^T

[13] (A^-1)^T = (A^T)^-1

symmetric matrix and positive definite

A real symmetric matrix  is positive definite iff there exists a real nonsingular matrix  such that

(5)

where  is the transpose (Ayres 1962, p. 134). 

Why is the largest element of symmetric, positive semidefinite matrix on the diagonal

http://math.stackexchange.com/questions/1382940/why-is-the-largest-element-of-symmetric-positive-semidefinite-matrix-on-the-dia/1382954

http://math.stackexchange.com/questions/1286813/largest-entry-in-symmetric-positive-definite-matrix

Eigenvalues of Positive Semi-definite Matrices

Suppose that A is a Hermitian matrix. Then A is positive semi-definite matrix if and only if whenever λ is an eigenvalue of A, then λ≥0.

http://linear.ups.edu/jsmath/0212/fcla-jsmath-2.12li103.html