Question 3 Suppose that we have the random sample $X_1$, $X_2$ from the distribution $f(x) = \begin{cases} \frac{1}{\theta} & 0 \le x \le \theta \\ 0 & \text{otherwise} \end{cases}$ Of these estimators: 1) $\hat{\theta}_1 = \frac{X_1+X_2}{2}$ 2) $\hat{\theta}_2 = \frac{3X_1+X_2}{2}$ 3) $\hat{\theta}_3 = X_2/4$ Which is unbiased? (hint: compute E(X) first) Only estimators 1 and 3 There is no enough information None All the estimators Only estimator 2 Question 4
Added by Anna A.
Close
Step 1
The probability density function (PDF) is given by $f(x) = \frac{1}{\theta}$ for $0 \le x \le \theta$ and $0$ otherwise. The expected value of a continuous random variable X is given by the integral of $x \cdot f(x)$ over its range: $E(X) = \int_{-\infty}^{\infty} Show more…
Show all steps
Your feedback will help us improve your experience
Victor Salazar and 99 other Intro Stats / AP Statistics educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Suppose that $Y_{1}, Y_{2}, Y_{3}$ denote a random sample from an exponential distribution with density function $$f(y)=\left\{\begin{array}{ll} \left(\frac{1}{\theta}\right) e^{-y / \theta}, & y>0 \\ 0, & \text { elsewhere } \end{array}\right.$$ Consider the following five estimators of $\theta$ : $$\hat{\theta}=Y_{1}, \quad \hat{\theta}_{2}=\frac{Y_{1}+Y_{2}}{2}, \quad \hat{\theta}_{3}=\frac{Y_{1}+2 Y_{2}}{3}, \quad \hat{\theta}_{4}=\min \left(Y_{1}, Y_{2}, Y_{3}\right), \quad \hat{\theta}_{5}=\bar{Y}$$ a. Which of these estimators are unbiased? b. Among the unbiased estimators, which has the smallest variance?
Estimation
The Bias and Mean Square Error of Point Estimators
Dominador T.
(15 points). Suppose that X1 and X2 are independent random samples from a population with mean θ, and variance σ 2 . Also, it is the case that ˆθ1, ˆθ2, and ˆθ3 are all point estimators for a parameter θ, the mean of our population X. a. (5 points) Suppose further, that E[ ˆθ1] = aθ + b, where a and b are nonzero constants. What is the bias of estimator ˆθ1? (You can leave your answer in terms of a, b & θ.) b. (4 points) Show that ˆθ2 = 0.25X1 + 0.75X2 and ˆθ3 = 1.25X1 − 0.25X2 are both unbiased. c. (6 points) Find the relative efficiency of ˆθ3 with respect to ˆθ2.
Sri K.
Recommended Textbooks
Elementary Statistics a Step by Step Approach
The Practice of Statistics for AP
Introductory Statistics
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD