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
For each set of probabilities, determine whether the events A and B are independent or dependent.
lozanna [386]

Answers:

  • (a) Independent
  • (b) Dependent
  • (c) Dependent
  • (d) Independent

========================================================

Explanation:

If events A and B are independent, then the two following equations must both be true

  • P(A | B) = P(A)
  • P(B | A) = P(B)

This is because the conditional probability P(A|B) means "P(A) when B has happened". If B were to happen, then P(A) must be the same as before. In other words, event B does not affect A, and vice versa.

For part (a), we have P(B) = 1/4 and P(B|A) = 1/4 showing that P(B|A) = P(B) is true, and therefore we can say the events are independent. We don't need the info that P(A) = 1/8.

------------------------

Unlike part (a), part (b) has the answer "dependent" because P(A) = 1/8 and P(A | B) = 1/3 differ in value. Event A starts off at probability 1/8, but then event B occurring means P(A) gets increased to 1/3. The prior knowledge about B changes the chances of A. The P(B) = 1/5 is unneeded.

------------------------

If A and B were independent, then,

P(A and B) = P(A)*P(B)

However,

P(A)*P(B) = (1/4)*(1/5) = 1/20

which is not the same as P(A and B) = 1/6. Therefore the two events are dependent.

------------------------

Refer back to part (a)

P(A) = 1/4 and P(A|B) = 1/4 are identical in value, so P(A|B) = P(A) which leads to the events being independent. Whether we know event B happened or not, it does not affect the outcome of event A. P(B) = 1/9 is unneeded.

7 0
2 years ago
Solve the equation and check your solution. (If an answer does not exist, enter DNE. If all real numbers are solutions, enter RE
r-ruslan [8.4K]
5x-4=4x+8
-4x. -4x
X-4=8
+4. +4
X=12
5(12)-4=4(12)+8
60-4=48+8
56=56
5 0
3 years ago
Can somebody please help my with this question​
ki77a [65]

Answer:

All the possible values of a-b lies in the interval 3.

Step-by-step explanation:

As the range of a is

3

As the range of b is

4

As a contains the set of number between 3 to 4, and b contains the values of from 4 to 5.

a-b can be obtained by subtracting the values of b from a.

So,

a-b

⇒ (3

⇒ 3

Therefore, all the possible values of a-b lies in the interval 3.

Keywords: number, value, interval

Learn more about number interval from brainly.com/question/13048073

#learnwithBrainly

8 0
3 years ago
Best anwser gets brainly no work needed just correct letter best gets brain ​
umka2103 [35]

Answer:

True

Step-by-step explanation:

i used photomath on the problem and this is the simplified version.

8 0
2 years ago
What is 1/4 divided by 2/4
Alisiya [41]
\dfrac{1}{4} \div  \dfrac{2}{4}

------------------------------------------------------------------------
Change the divide fraction to multiplication fraction :
------------------------------------------------------------------------

\dfrac{1}{4} \times  \dfrac{4}{2}

------------------------------------------------------------------------
Combine into single fraction :
------------------------------------------------------------------------

\dfrac{4}{8}

------------------------------------------------------------------------
Reduce to simplest form :
------------------------------------------------------------------------

\dfrac{4 \div 4}{8 \div 4} =  \dfrac{1}{2}

8 0
3 years ago
Other questions:
  • Peter bought 2 1/2 pounds of hamburger meat.He is planning to make 1/4 pound hamburgers.How many 1/4 pound hamburgers can he mak
    12·1 answer
  • Use the drop-down menus to choose the correct locations on the number line.
    6·1 answer
  • Jackson is able to craft in Minecraft A cake is crafted from a recipe that requires 3 milk, 2sugar, 1egg and 3wheat. If Jackson
    9·2 answers
  • Find the missing value.<br> 2 cookies<br> $3.00<br> 4 cookies<br> X
    13·2 answers
  • Where is Point B on the number line? ?? im Stuck on this ????? ​
    9·1 answer
  • Yall i need help please yuhhh
    8·1 answer
  • Horizon Cell Company charges $15 fee plus 10 cents per text, t, and unlimited phone calls.
    15·1 answer
  • A Ferris wheel at an amusement park is modeled by (x – 100)2 + (y – 75)2 = 4,900, where the measurements are in feet. A slingsho
    8·1 answer
  • Helppppppppppppp i need this homework done by today
    10·1 answer
  • Ann studied the effects of watching television on time spent exercising
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!