What is the criteria of a good estimator, explain the points in your own word

There are two number of estimates about a population:

1. A point estimate and

2. An interval estimate:
Estimator:

In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter; an estimate is the result from the actual application of the function to a particular set of data. Many different estimators are possible for any given parameter. Some criterion is used to choose between the estimators, although it is often the case that a criterion cannot be used to clearly pick one estimator over another. To estimate a parameter of interest (e.g., a population mean, a binomial proportion, a difference between two population means, or a ratio of two population standard deviation), the usual procedure is as follows:

1. Select a random sample from the population of interest.
2. Calculate the point estimate of the parameter.
3. Calculate a measure of its variability, often a confidence interval.
4. Associate with this estimate a measure of variability.

There are two types of estimators: Point Estimators and Interval Estimators.

1. A point Estimate: is a single number that is used to estimate an unknown population parameter. A point estimate is often insufficient, because it is either right or wrong. If you are told only that her point estimate of enrollment is wrong, you do not know how wrong it is, and you cannot be certain of the estimate’s reliability. If you learn that is off by only 10 students, you would accept 350 students as a good estimate of future enrollment.

Point estimation:

In statistics, point estimation involves the use of sample data to calculate a single value (known as a statistic) which is to serve as a "best guess" for an unknown (fixed or random) population parameter.

More formally, it is the application of a point estimator to the data.

Point estimation should be contrasted with Bayesian methods of estimation, where the goal is usually to compute (perhaps to an approximation) the posterior distributions of parameters and other quantities of interest. The contrast here is between estimating a single point (point estimation), versus estimating a weighted set of points (a probability density function).

2. An interval estimator: is a range of values used to estimate a population parameter. It indicates the error in two ways: by the parameter lying within that range.

Interval estimation:

In statistics, interval estimation is the use of sample data to calculate an interval of possible (or probable) values of an unknown population parameter. The most prevalent forms of interval estimation are confidence intervals (a frequentist method) and credible intervals (a Bayesian method).

* Behrens-Fisher problem
* fiducial inference

Other common interval estimation are

* Tolerance interval
* Prediction interval - used mainly in Regression Analysis

Criteria of a good estimator:

1. Unbiasedness: This is a desirable property for a good estimator to have. The term unbiasedness refers to the fact that a sample mean is an unbiased estimator of a population mean because the mean of the sampling distribution of sample means taken from the same population is equal to the population mean itself. We can say that a statistic is an unbiased estimator if, on average, it tends to assume values that ate the same extent as it tends to assume values that are below the population parameter being estimated.

2. Efficiency: Another desirable property of a good estimator is that it be efficient. Efficiency refers to the size of the standard error of the statistic. If we compare two statistics from a sample of the same size and try to decide which one is the more efficient estimator, we would pick the statistic that has the smaller standard error, or standard deviation of the sampling distribution. Suppose we choose a sample of a given size and must decide whether to use the sample mean or the sample median to estimate the population mean. If we calculate the standard error of the sample mean and find it to be 1.05 and then calculate the standard error of the sample median and find it to be 1.6, we would say that the sample mean is a more efficient estimator of the population mean be cause its standard error is smaller. It makes sense that an estimator with a smaller standard error (with less variation) will have more chance of producing an estimate nearer to the population parameter under consideration.

3. Consistency: A statistic is a consistent estimator of a population parameter if as the sample size increases, it becomes almost certain that the value of the statistic comes very close to the value of the population parameter. If an estimator is consistent, it becomes more reliable with large samples. Thus, if you are wondering whether to increase the sample size to get more information about a population parameter, find out first whether your statistic is a consistent estimator. If it is not, you will waste time and money by taking larger samples.
Consistency:

A Consistent Estimator is an estimator that converges in probability to the quantity being estimated as the sample size grows without bound.

An estimator tn (where n is the sample size) is a consistent estimator for parameter θ if and only if, for all ε > 0, no matter how small, we have

\lim_{n\to\infty}\Pr\left\{ \left| t_n-\theta\right|<\epsilon \right\}=1.

It is called strongly consistent, if it converges almost surely to the true value.

4. Sufficiency: An estimator is sufficient if it makes so much use of the information in the sample that no other estimator could extract from the sample additional information about the population parameter being estimated.
Efficiency:

Main article: Efficiency (statistics)

The quality of an estimator is generally judged by its mean squared error.

However, occasionally one chooses the unbiased estimator with the lowest variance. Efficient estimators are those that have the lowest possible variance among all unbiased estimators. In some cases, a biased estimator may have a uniformly smaller mean squared error than does any unbiased estimator, so one should not make too much of this concept. For that and other reasons, it is sometimes preferable not to limit oneself to unbiased estimators; see estimator bias. Concerning such "best unbiased estimators", see also Cramér-Rao bound, Gauss-Markov theorem, Lehmann-Scheffé theorem, Rao-Blackwell theorem.