Suppose you have N items with values xi from which we want to sample n items...

Suppose you have N items with values x_i from which we want to sample n items with replacement. Each item has its own probability p_i of being selected across all times we pick items.

Let T = > 1, 1=1

What's the bias for the following estimator of T: $\hat{T} = \sum_{i \in S} \frac{x_i}{p_i}$

Let T = > 1, 1=1

Solutions

Expert Solution

The bias of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated.

Given that

So, the bias is given by:

orchestra answered 1 year ago

Next > < Previous

Related Solutions

Suppose you have N items with values xi from which we want to sample n items...

Suppose you have N items with values xi from which we want to sample n items with replacement. Each item has its own probability pi of being selected across all times we pick items. What's the estimated variance of the Horvitz-Thompson estimator of T? Let T = > 1, 1=1

Suppose we have N = 6 values from a population. These values are 4, 8, 0,...

Suppose we have N = 6 values from a population. These values are 4, 8, 0, 10, 14 and 6. Let μ and σ denote the population mean and population standard deviation of these six values, respectively. (a) What are the values of μ and σ, respectively? μ = 4.8580; σ = 7 μ = 7; σ = 4.8580 μ = 7; σ = 4.4347 μ = 7; σ = 19.6667 μ = 7; σ = 23.6 (b) What percentage...

Suppose, for a random sample selected from a normal population, we have the values of the...

Suppose, for a random sample selected from a normal population, we have the values of the sample mean x ̄ = 67.95 and the standard deviation s = 9. a. Construct a 95% confidence interval for population mean μ assuming the sample size n = 16. b. Construct a 90% confidence interval for population mean μ assuming n = 16. c. Obtain the width of the confidence intervals calculated in a and b. Is the width of 90% confidence interval...

Suppose we have 6 distinct items and we want to place 4 in one bin and...

Suppose we have 6 distinct items and we want to place 4 in one bin and 2 in the other. How many ways can this be done if order does not matter? (answer is 15)

Suppose we have a random sample of size 50 from a N(μ,σ2) PDF. We wish to...

Suppose we have a random sample of size 50 from a N(μ,σ2) PDF. We wish to test H0: μ=10 versus H1: μ=10. The sample moments are x ̄ = 13.4508 and s2 = 65.8016. (a) Find the critical region C and test the null hypothesis at the 5% level. What is your decision? (b) What is the p-value for your decision? (c) What is a 95% confidence interval for μ?

We observe a random sample of n values from the beta distribution with parameters c and...

We observe a random sample of n values from the beta distribution with parameters c and 2. (a) Find a general expression for the method of moments (MOM) estimate of c, and calculate this MOM estimate for the case where we observe two values, 0.96 and 0.84. (b) Find the bias of the MOM estimator of c for the case where c = 1 and n = 1. (c) Find a general expression for the maximum likelihood estimate (MLE) of...

We want to study the zinc concentration from a river. We have a sample of measurements...

We want to study the zinc concentration from a river. We have a sample of measurements taken in 25 diﬀerent locations in a river with sample mean x = 3 and population standard deviation σ = 0.3. The population is normally distributed. 1. Find the 95% and 99% conﬁdence intervals for the mean zinc concentration in the river. 2. Is the following statement correct? “If we repeat the same experiment multiple times and each time calculate the two conﬁdence intervals...

Suppose that you have a random sample of size n from a population with Gamma density...

Suppose that you have a random sample of size n from a population with Gamma density with α= 3 but unknown β. Write down the likelihood function, and find a sufficient statistic. Find the MLE and the MOM estimators forβ. (Hint: They should be equal.) Then find the MSE for this estimator by finding the bias and the variance. Is it consistent? Is it MVUE? Explain why or why not.

Suppose the possible values of X are {xi}, the possible values of Y are {yj}, and...

Suppose the possible values of X are {xi}, the possible values of Y are {yj}, and the possible values of X + Y are {zk}. Let Ak denote the set of all pairs of indices (i,j) such that xi+yj =zk;thatisAk ={(i,j):xi+yj =zk} a. Argue that b. Argue that P{X+Y=zk}= ? P{X=xi,Y=yj} (i,j )∈Ak E[X+Y]=? ? (xi+yj)P{X=xi,Y =yj} k (i,j )∈Ak c. Using the formula in b, argue that E[X+Y]=??(xi+yj)P{X=xi,Y =yj} ij d. Show that P(X =xi)=?P(X =xi,Y =yj),P(Y =yj)=?P(X =xi,Y...

We have a random sample of size n (which is large), and we wish to test...

We have a random sample of size n (which is large), and we wish to test - Ho: X~UNIF(0,1) vs Ha: X~exp(1). How would you conduct hypothesis testing? Describe procedure.

Subjects