1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Elenna [48]
3 years ago
12

Let $$X_1, X_2, ...X_n$$ be uniformly distributed on the interval 0 to a. Recall that the maximum likelihood estimator of a is $

$a = max(X_i)$$. Argue intuitively why aˆ cannot be an unbiased estimator for a. b. Suppose that E(a) = na/(n + 1). Is it reasonable that aˆ consistently underestimates a? Show that the bias in the estimator approaches zero as n gets large. c. Propose an unbiased estimator for a. d. Let $$Y = max(X_i)$$. Use the fact that Y ≤ y if and only if each $$X_i ≤ y$$ to derive the cumulative distribution function of Y . Then show that the probability density function of Y is. $$f(y) = [ny^n - ^1/a^n 0$$, 0 ≤ y ≤ a otherwise, Use this result to show that the maximum likelihood estimator for a is biased. e. We have two unbiased estimators for a: the moment estimator $$a_1=2\overline{\mbox{X}}$$ and $$a_2 = [(n + 1)/n] max(X_i)$$, where max $$(X_i)$$ is the largest observation in a random sample of size n. It can be shown that $$V(a_1) = a^2/(3n)$$ and that $$V(a_2) = a^2/[n(n + 2)]$$. Show that if n > 1, aˆ2 is a better estimator than aˆ. In what sense is it a better estimator of a?
Mathematics
1 answer:
Solnce55 [7]3 years ago
8 0

Answer:

a) \hat a = max(X_i)  

For this case the value for \hat a is always smaller than the value of a, assuming X_i \sim Unif[0,a] So then for this case it cannot be unbiased because an unbiased estimator satisfy this property:

E(a) - a= 0 and that's not our case.

b) E(\hat a) - a= \frac{na}{n+1} - a = \frac{na -an -a}{n+1}= \frac{-a}{n+1}

Since is a negative value we can conclude that underestimate the real value a.

\lim_{ n \to\infty} -\frac{1}{n+1}= 0

c) P(Y \leq y) = P(max(X_i) \leq y) = P(X_1 \leq y, X_2 \leq y, ..., X_n\leq y)

And assuming independence we have this:

P(Y \leq y) = P(X_1 \leq y) P(X_2 \leq y) .... P(X_n \leq y) = [P(X_1 \leq y)]^n = (\frac{y}{a})^n

f_Y (Y) = n (\frac{y}{a})^{n-1} * \frac{1}{a}= \frac{n}{a^n} y^{n-1} , y \in [0,a]

e) On this case we see that the estimator \hat a_1 is better than \hat a_2 and the reason why is because:

V(\hat a_1) > V(\hat a_2)

\frac{a^2}{3n}> \frac{a^2}{n(n+2)}

n(n+2) = n^2 + 2n > n +2n = 3n and that's satisfied for n>1.

Step-by-step explanation:

Part a

For this case we are assuming X_1, X_2 , ..., X_n \sim U(0,a)

And we are are ssuming the following estimator:

\hat a = max(X_i)  

For this case the value for \hat a is always smaller than the value of a, assuming X_i \sim Unif[0,a] So then for this case it cannot be unbiased because an unbiased estimator satisfy this property:

E(a) - a= 0 and that's not our case.

Part b

For this case we assume that the estimator is given by:

E(\hat a) = \frac{na}{n+1}

And using the definition of bias we have this:

E(\hat a) - a= \frac{na}{n+1} - a = \frac{na -an -a}{n+1}= \frac{-a}{n+1}

Since is a negative value we can conclude that underestimate the real value a.

And when we take the limit when n tend to infinity we got that the bias tend to 0.

\lim_{ n \to\infty} -\frac{1}{n+1}= 0

Part c

For this case we the followng random variable Y = max (X_i) and we can find the cumulative distribution function like this:

P(Y \leq y) = P(max(X_i) \leq y) = P(X_1 \leq y, X_2 \leq y, ..., X_n\leq y)

And assuming independence we have this:

P(Y \leq y) = P(X_1 \leq y) P(X_2 \leq y) .... P(X_n \leq y) = [P(X_1 \leq y)]^n = (\frac{y}{a})^n

Since all the random variables have the same distribution.  

Now we can find the density function derivating the distribution function like this:

f_Y (Y) = n (\frac{y}{a})^{n-1} * \frac{1}{a}= \frac{n}{a^n} y^{n-1} , y \in [0,a]

Now we can find the expected value for the random variable Y and we got this:

E(Y) = \int_{0}^a \frac{n}{a^n} y^n dy = \frac{n}{a^n} \frac{a^{n+1}}{n+1}= \frac{an}{n+1}

And the bias is given by:

E(Y)-a=\frac{an}{n+1} -a=\frac{an-an-a}{n+1}= -\frac{a}{n+1}

And again since the bias is not 0 we have a biased estimator.

Part e

For this case we have two estimators with the following variances:

V(\hat a_1) = \frac{a^2}{3n}

V(\hat a_2) = \frac{a^2}{n(n+2)}

On this case we see that the estimator \hat a_1 is better than \hat a_2 and the reason why is because:

V(\hat a_1) > V(\hat a_2)

\frac{a^2}{3n}> \frac{a^2}{n(n+2)}

n(n+2) = n^2 + 2n > n +2n = 3n and that's satisfied for n>1.

You might be interested in
I need help again please 5/6-1/6
Murljashka [212]
The answer is 2/3 in lowest terms.
6 0
2 years ago
Michael bought 0.44 pound of slice turkey. what is the value of the digit in the hundredths place?
jarptica [38.1K]

Answer:

4

Step-by-step explanation:

You have 0.44

0 is in the ones place

4 is in the tenths place (0.4)

And the other 4 is in the hundredths place (0.04)

8 0
2 years ago
Read 2 more answers
What are the side lengths of the equilateral triangle?
Alchen [17]

5x  +  (6x - 5) + (3x + 10) = 360

5x  + 6x - 5 + 3x + 10 = 360

11x - 5 + 3x + 10 = 360

14x + 5 = 360

14x = 360 - 5

14x = 355

x = 25.36

<h2><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u></h2>

x = 25.36

5x = 5 \times 25.36 = 126.8

6x - 5 = 6 \times 25.36 - 5 = 147.16

3x + 10 = 3 \times 25.36 + 10 = 86.08

4 0
3 years ago
The graph below shows a line of best fit for data collected on the average high temperature in El Calafate as a function of the
Novosadov [1.4K]

Answer:

95xhsshsjsjeeiejjejsjsjsjwnnsnsmsmmwwnsnnsns

4 0
3 years ago
What is the anwers please help me.
GrogVix [38]
2x=18

X=9
2(9)=18
X is the number of glass pieces
3 0
2 years ago
Read 2 more answers
Other questions:
  • Look at attachment for question
    14·1 answer
  • The length of a rectangle is 3 feet more than twice the width. The perimeter is 128 feet. Find the length and width.
    11·2 answers
  • This question is a fair question: “Which do you think is the most common age group of people who like pop music?” true or false
    13·1 answer
  • 39 = 1 3/10 b<br><br> Find out what b is.
    12·1 answer
  • Someone please help me get the answers to these
    8·1 answer
  • If you have 25 cookies and 1/8 are sugar cookies. how many cookies are chocolate chip (not 7/8)
    14·2 answers
  • Will mark brainliest! in kite UVWX, m
    7·1 answer
  • Emilio borrows $1200 from a bank with 8% simple interest per year. How much will he have to pay back total in 2 years?
    6·2 answers
  • 2
    8·1 answer
  • Can someone help me with this question.
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!