Statistics Paper III 2021 IES/ISS – Section A & B Questions

Q0. Q1. (a) and Simple Random Sampling Without Replacement (SRSWOR) and find the value of n such that variance of the sample mean in SRSWOR is exactly half of the variance of the sample mean in SRSWR of the same size. (10 marks)

Q0. Q1. (b) Consider the simple linear regression model: y_i = beta0 + beta1 x_i + epsilon_i, i = 1, 2, ..., n; epsilon_i are iid with mean zero and var sigma². Show that least squares estimates beta0_hat and beta1_hat are linear functions of y1, y2, ..., yn and also compute the variance-covariance matrix of [beta0_hat; beta1_hat] and its determinant. (15 marks)

Q0. Q1.
(c) Let Y_t = Z_t + theta Z_{t-1}, where {Z_t} is a sequence of iid random variables with mean zero and variance sigma_Z^2. Show that a real-valued function on Z defined by gamma(h) = { 1 if h=0; rho if h=±1; 0 otherwise } is an autocovariance function if |rho| < 1/2. (15 marks)

Q0. Q1. (d) Find the spectral density function f(t) of a continuous parameter process. (15 marks)

Q0. Q2. (a) Having correlation function ρ(t) = e^{-t²}, -∞ < t < ∞. Also find the value of f(√log 16). (10 marks)

Q0. Q2. (b) From bivariate population of N units, a simple random sample (x_i, y_i); i = 1, 2, ..., n is drawn without replacement with corresponding means (x̄_n, ȳ_n). Show that Cov(x̄_n, ȳ_n) = (1/n - 1/N) S_xy. (15 marks)

Q0. Q2.
(c) Explain the term exponential smoothing. When is exponential smoothing most useful? Interpret the smoothing constant alpha, what is its range? How is alpha related to degree of smoothing? (15 marks)

Q1.

(a) SECTION A Both the questions are compulsory. Compare Simple Random Sampling Without Replacement (SRSWOR) Q1. and Simple Random Sampling With Replacement (SRSWR) and find the value of n such that variance of the sample mean in SRSWOR is exactly half of the variance of the sample mean in SRSWR of the same size.

(b) Consider the simple linear regression model : y_i = β_0 + β_1 x_i + \varepsilon_i, i = 1, 2, ..., n; \varepsilon₁, \varepsilon₂, ..., \varepsilon_n are independent and identically distributed with mean zero and constant variance σ^

Q2.

(a) Show that least square estimates \hatβ_0 and \hatβ_1 are linear functions of y₁, y₂, ..., y_n and also compute variance-covariance matrix of \beginpmatrix \ddotβ_0 \\ \hatβ_1 \endpmatrix and its determinant. Let Y_t = Z_t + θ Z_t-1, where \Z_t\ be a sequence of iid random variables
(c) with mean zero and variance σ_Z². Show that a real valued function on Z, defined as : γ(h) = \beginarraycc 1 & \quad h = 0, \\[1ex] \rho & \quad h = ±\,1, \endarray is an autocovariance function if |\rho| < 1/2. Find the spectral density function f(t) of a continuous parameter process, Q2. having correlation function \rho(t) = e^-t2, -∞ < t < ∞. Also find the value of f(√(log 16)).

(b) From bivariate population of N units, a simple random sample (x_i, y_i); i = 1, 2, ..., n is drawn without replacement with corresponding means (\barx_n, \bary_n). Show that Cov(\barx_n, \bary_n) = (1)/(n) - (1)/(N)) S_xy.

(c) Explain the term exponential smoothing. When is exponential smoothing most useful? Interpret the smoothing constant α, what is its range? How is α related to degree of smoothing? 2