What is the best most efficient unbiased estimator of a population mean given it is a normally distributed population?

What is the best most efficient unbiased estimator of a population mean given it is a normally distributed population?

Efficiency: The most efficient estimator among a group of unbiased estimators is the one with the smallest variance. For example, both the sample mean and the sample median are unbiased estimators of the mean of a normally distributed variable.

Why is mean the best estimator?

We know the standard error of the mean is . Therefore in a normal distribution, the SE(median) is about 1.25 times . This is why the mean is a better estimator than the median when the data is normal (or approximately normal).

What is the best estimation of mean?

The best estimate of a population mean is the sample mean. star_plus. The most fundamental point and interval estimation process involves the estimation of a population mean. Suppose it is of interest to estimate the population mean, μ, for a quantitative variable.

Which is the best estimator of population mean?

The most fundamental point and interval estimation process involves the estimation of a population mean. Suppose it is of interest to estimate the population mean, μ, for a quantitative variable.

What are the 3 properties of good estimator?

Properties of Good Estimator

• Unbiasedness. An estimator is said to be unbiased if its expected value is identical with the population parameter being estimated.
• Consistency.
• Efficiency.
• Sufficiency.

What is the best estimate in statistics?

Point estimation involves the use of sample data to calculate a single value or point (known as a statistic) which serves as the “best estimate” of an unknown population parameter. The point estimate of the mean is a single value estimate for a population parameter.

How to approximate the distribution of the mean?

We can approximate the distribution of the sample mean with its asymptotic distribution. So the distribution of the sample mean can be approximated by a normal distribution with mean and variance Taboga, Marco (2017). “Point estimation of the mean”, Lectures on probability theory and mathematical statistics, Third edition.

Why does the mean estimator have a normal distribution?

The estimator has a normal distribution: Note that the sample mean is a linear combination of the normal and independent random variables (all the coefficients of the linear combination are equal to ). Therefore, is normal because a linear combination of independent normal random variables is normal.

Is the estimator of the mean the sample mean?

Again, the estimator of the mean is the sample mean: The expected value of the estimator is equal to the true mean and is therefore unbiased: The proof is the same found in the previous example. Also in this case the proof is the same found in the previous example.

How is the mean and variance of the distribution derived?

The mean and the variance of the distribution have already been derived above. The sequence satisfies the conditions of Kolmogorov’s Strong Law of Large Numbers ( is an IID sequence with finite mean). Therefore, the sample mean converges almost surely to the true mean : that is, the estimator is strongly consistent.