代做MATH0099 Statistical Methods and Data Analytics 2018 Problem Sheet 1代写留学生Matlab语言程序

MSc Financial Mathematics

Statistical Methods and Data Analytics 2018

MATH0099

Problem Sheet 1

Let X0, X1, . . . , Xn be i.i.d. copies of a random variable X ∼ N(0, 1). The following two distributions that arise from transformations of standard normal random variables play a fundamental role in statistics.

• the Chi-squared distribution with n degrees of freedom is the distribution of X1 2 + . . . + X2n . Its density is

2 −n/2Γ(n/2)−1 e −z/2 z n/2−1 , z ≥ 0.

• the t-distribution with n degrees of freedom is the distribution of  Its density is

Problem 1. (t-statistics) Let X1, . . . , Xn be i.i.d. copies of X ∼ N(µ, σ2 ). Show that the arithmetic mean ¯X and the sample variance S2n , defined by

are independent random variables. Moreover, show that X ∼ N(µ, σ2/n) and (n − 1)S2 n /σ2 ∼ χ 2 n−1 . Deduce that the t-statistic

is distributed according to the t-distribution with n − 1 degrees of freedom.

The median of a cumulative distribution function F is defined by m := F −1 (1/2). Let X1, . . . , Xn be i.i.d. with cumulative distribution F and median m = 0 and suppose that F 0 (0) exists and is strictly greater than zero. Let Zn be the sample median, i.e. Zn := Xk where k = [n/2 + 1] and X(1) ≤ . . . ≤ X(n) is an increasing ordering of the random variables X1, . . . , Xn. To solve the below problem you may assume the result that

P( √ nZn ≤ x) → Φ(2F 0 (0)x), n → ∞.

Problem 2. (Comparison between arithmetic and sample mean) Let X1, . . . , Xn be i.i.d. copies of a random variable X ∼ N(µ, σ2 ). The parameters µ and σ are unknown and ought to be estimated. Two possible estimators for µ are

where the random variables X(1) < X(2) < . . . < X(2n+1) are arranged in increasing order.

(a) Compute c (1)n and c (2)n such that

(b) Compute q ∈ R+, such that

How can q be interpreted? (In words.)

Problem 3. This exercise proves the Neyman-Pearson Lemma from lectures. The general problem of determining a most powerful statistical test with sufficiency level α is equivalent to determining an optimal 0 ≤ ϕ ≤ 1 such that

I(ϕp0) ≤ α, I(ϕp1) → max,                        (1)

where I denotes the integral with respect to the measure given by P0 + P1. Now follow the following steps:

(a) using the technique of Lagrange multipliers reformulate the problem (1) into an optimisation problem involving an appropriately chosen Lagrangian and Lagrange multiplier c;

(b) by considering the cumulative distribution function F of the random variable Q := p1(X)/p0(X) for X ∼ P0, show that there exists a Lagrange multiplier for the optimisation problem (1).

The above steps prove 1. in the statement of the Neyman-Pearson Lemma. The claims 2. and 3. are now easily shown by once again considering the optimisation problem.