Teaching CSIR NET Mock Test Series Mathematical Science Statistics & Exploratory Data Analysis Methods of Estimation
Let Y1....Yn (n ≥ 2) be independent observations; Yi ∼ N(βxi, σ2), i = 1,...,n; where x1,...,xn and σ2(> 0) are known constants and β ∈ ℝ is an unknown parameter. Consider N(β0, τ2) prior for the parameter β, where β0 and τ2(> 0) are known constants, and N(μ, λ2) denotes a normal distribution with mean μ and variance λ2, Suppose \(\rm \bar y=\frac{1}{n}\Sigma_{i=1}^ny_i\ and \ \bar x=\frac{1}{n}\Sigma_{i=1}^nx_i\); are observed sample means. Under squared error loss function, which of the following statements are true?
1
Bayes estimate of β tends to β0 as τ2 → 0
2
Bayes estimate of β tends to \(\rm \frac{\bar y}{x}\) as τ2 → 0
3
Bayes estimate of β tends to the BLUE of β as τ2 → ∞
4
Bayes estimate of β tends to MLE of β as τ2 → ∞