Point Estimates

Question 1

Define a point estimate

Answer 1

A point estimates of a parameter, theta, is a SINGLE number that can be regarded as a SENSIBLE value for theta.

A pt estimate is obtained by SELECTING a suitable STATISTIC and COMPUTING its value from a given SAMPLE data.

The selected statistic is called the POINT ESTIMATOR of theta.

Question 2

Given 20 random sample obs which point estimate i best?

Answer 2

(a) Estimator Xbar, estimate xbar = Sigma xi/n = 27.791
(b) Estimator Xtilde, estimate xtilde = median = 27.96

(c) Estimator = [min(Xi)+max(Xi)]/2
= [min(xi)+max(xi)]/2 = 27.67

(d) Estimator = Xbar_tr(10) trimmed mean = 27.838

Ea. of estimators (a)-(d) uses a different measure of the center of the sample to estimate mu.. Which estimate is closest to the true value? We cannot answer this without knowing the true value.

A question that can be answered is, “which estimator, when used on other samples of Xis will tend to produce estimates closest to the true values?”

Question 3

What is an unbiased estimator?

Answer 3

A point estimator theta_hat is said to be an UNBIASED ESTIMATOR of theta if E(theta_hat) = theta for every possible value of theta.

If theta is NOT unbiased, the difference E(theta_hat) - theta is called the BIAS of theta_hat.

i.e. theta_hat is UNBIASED if its probability distribution is always “centered” at the true value of theta.

Question 4

What is an unbiased estimator of a binomial rv?

Answer 4

When X is a BINOMIAL rv with parameters n and p, the sample proportion phat = X/n is an UNBIASED estimator of p.

E(phat)
= E(x/n)
= 1/n E(X)
= 1/n (n*p) binomial EV = np

= p

Question 5

What is an unbiased estimator of variance?

Answer 5

Let X1,X2,…,Xn be a random SAMPLE from a distribution with mean mu and variance sigma^2.

Then the estimator if sigma^2

sigma^2_hat
= S^2
=Sigma(Xi - Xbar)^2 /(n-1)

is an UNBIASED estimator of sigma^2.

Question 6

What is an estimator with minimum variance?

Answer 6

Suppose theta_hat_1 and theta_hat_2 are BOTH unbiased. Then although the distribution of ea estimator is centered at the true value of theta, the spread of the distributions about the true value may be DIFFERENT.

Principal of Minimum Variance Unbiased Estimation:

Among all estimators of theta that are unbiased, hoose the one that has MINIMUM VARIANCE. The resulting theta_hat is called the MINIMUM VARIANCE UNBIASED ESTIMATOR (MVUE) of theta.

e.g. given two normal pdfs theta_hat_1 and theta_hat_2 which BOTH have the same center sample unbiased estimator, the MVUE (better estimator) is the estimator which has a SMALLER variance (smaller spread).

Question 7

What is a standard error?

Answer 7

Besides reporting the value of a point estimate, some INDICATION of its PRECISION should be given. The usual measure of precision is the STANDARD ERROR of the estimator used.

The STANDARD ERROR of an estimator theta_hat is its std dev sigma_theta = sqrt(V(theta_hat)). If the std error itself involves UNKNOWN parameters whose values can be estimated, substitution of these estimates into sigma_theta yields the ESTIMATED STANDARD ERROR (estimated standard dev). The estimated std error cab be denoted by s_theta.

Question 8

What the standard error of a normally distributed sample mean Xbar?

Answer 8

If n=20 and the value of sigma is KNOWN to be 1.5, the standard error of Xbar is sigma_Xbar = sigma/sqrt(n) = 1.5/sqrt(20) = .335.

If population sigma is UNKNOWN, but the SAMPLE std dev sigma_hat is computed as 1.462, the estimate sigma_hat_Xbar
= s /sqrt(n)
= 1.462 / sqrt(20)
= .327

Question 9

What is a standard error of a sample proportion?

Answer 9

The standard error of phat = X/n is:

sigma_phat 
= sqrt(V(X/n)
= sqrt(V(X) /n^2)
= sqrt(npq /n^2)
= sqrt(pq /n)