A reason for electing a biased estimation is the result of minimizing the Mean Square Error of the estimator, at least approximately. However, it is possible also minimizing the Variance of a unbiased estimator, solution which resolves simultaneously two problems: centered estimation and minimum mean square error.

may be if the biased one is more efficient.

Thank you, Subrata, but the higher efficiency is usually approximate when it is compared with other unbiased estimator which does not exploit the same used available information with the biased estimator.

Some cases biased estimator  variance will be smaller than the unbiased one.

For example: ridge regression, Hill estimators ( extreme value theory)

Yes, it could occur, but its variance is smaller for estimating other different parametric function than the interest one.

Pierre, the intention with the estimator is to approximate efficiently a parametric function which depends of the values of the interest or study variable, but from a sample of units of the population with determinated sample size.

Yes, sometimes we choose a biased estimation over an unbiased one if the mean squared error (MSE) of the biased estimation is less than (MSE) for unbiased estimator, that is mean the variance of unbiased estimation is higher than the variance of biased estimation such that MSE for biased estimation is less than (MSE) for unbiased estimator where MSE is a summation of variance and biased squared.

Thank you, Huda. These facts are certain:

1. There is not a uniformly minimum MSE estimator for a parametric function (Ruiz Espejo, 1987).

2. It is elected a biased estimator which uses auxiliary information, over a unbiased one which does not use auxiliary information. The biased estimator has approximately lesser MSE (or MAE, mean absolute error) than the respective MSE (or MAE) of the unbiased one (Cochran, 1977).

But it is possible too that a unbiased estimator which uses auxiliary information has got less MSE (or MAE) for some region of the parameter in RN than the MSE (or MAE) of the biased estimator. This would make admissible the unbiased estimator vs. the biased estimator.

Yes, We look at mean square error rather than variance 

As many commentators state, the MSE has the advantage of taking into account both bias and variability. There is often a trade-off between these (see for example, http://statweb.stanford.edu/~tibs/ElemStatLearn/, which can be downloaded).

However, there are situations where you would choose a biased estimator over an unbiased one even if they have the same variability.  It depends on how the estimator is used and what the costs/values of the different types of errors are. For example, suppose after extensive research I find that it takes on average 12 minutes to bake the perfect chocolate chip cookies. But this time varies. If they are over-cooked your dessert is ruined. If they are under-cooked you just close the oven door and check again in a minute or so. Therefore, the estimator that would be listed in the cookbook (if you were only allowed to use one number) would be lower than 12 minutes because there is little cost if you under-estimate the time but high cost (if you like cookies) if you over-estimate the time.

Daniel, do you understand "same variability" as "same MSE"? In this case both estimators are equally efficient, and they are equivalent in MSE. But, in this case, I would prefer the unbiased one because it is centered also.

I'm even saying the MSE of the biased estimator will be less, but I would still prefer to use that as my estimator.

Mariano, if you choose the unbiased estimator (12 minutes) for setting the oven alarm, this would mean that your cookies will often be ruined (burnt) more often than mine. True, I will have to get up from the couch more often (I'm assuming cookies are baked often) and wait a little, but I am assuming the cost of that is less than cookies being burnt. For me, the value of good cookies is large!

Similarly, if the unbiased estimator to drive to the train station is 1 hour, if it is important to get on that train I would leave more than an hour before departure time. But if trains left every five minutes so there was little cost for missing a train, I might leave just an hour before the train I wanted.

The following is an older paper, but it is useful for describing how the cost of different outcomes can inform what you report.

http://www.psychologicalscience.org/journals/pspi/pdf/pspi001.pdf

Hi Mariano,

As I indicated earlier, choice of the estimator depends on MSE or MPE for that estimator

Please clarify your question by example or references to get more accurate answer

Huda, could you explain or clarify us what do you understand for MPE?

The references I used:

Ruiz Espejo, M. (1987). Sobre estimadores UMV y UMECM en poblaciones finitas. Estadística Española 29 (115), 105-111.

Cochran, W. G. (1977). Sampling Techniques (3rd edition). Wiley. New York.

Daniel, in my research I do not think on cooking but in estimating a defined parametric function for approximating its exact value with the data of a sample of the finite population.

Jochen, the aim is not "to determine the exact value" but it is "to estimate the exact value" of the parametric function for a concrete given finite population. I agree with your idea.

Jochen, but the bias of the estimator is usually other known or unknown parametric function to be estimated too. This bias is not known before sampling the population.

The bias is a parametric function, and its exact value is unknown before sampling.

MLE needs a supposed model. I think your model is normal, but there is not any normal fixed finite population.

to compare the efficiency of  estimators you can use mean square Errors (MSE’s) or  mean absolute percentage error (MPE), also known as mean absolute percentage deviation (MAPD)

Dear Huda, and how do you estimate unbiasedly the MPE or MAPD?

I really think that unbiasedness is not very important in some occasions.

Even unbiased variance is not the best estimator when the purpose of

the estimation is prediction. "Third variance", which I found a few

years ago, is better than unbiased variance in terms of prediction.

The detail of the “third variance” is described in my paper:

A Revision of AIC for Normal Error Models

Open Journal of Statistics Vol.2 No.3, July 6, 2012

http://www.scirp.org/journal/PaperInformation.aspx?PaperID=20651

I hope that the concept of "third variance" will help you

consider biased estimation.

In other sense, we should also look at robustness of an estimator.  Since, efficiency and robustness are two opposite poles, when efficiency increases robustness decreases and vice-versa.  Do  biased estimator also trade off between efficiency and robustness ?

Dear Kunio and B. G.,

Prediction and robustness are based on supposed models. They are subjective forms of statistical treatment. When the model is objective and with sufficient resources we would be able of measure all the units of the finite population, prediction and robustness are substituted by estimation and efficiency based on realities which could be objectively measured with sufficient resources.

Thank you very much for Prof. Espejo's reply to my previous message.

I meant in my previous message:

(1) Predictability should be measured by log-likelihood in the

light of future data rather than MSE.

(2) Unbiased variance is not the best estimator in terms of

log-likelihood in the light of future data. This is a good example

to show that unbiased estimators are not always optimal

estimators for prediction.

Dear Kunio,

My surnames are Ruiz Espejo (from father and mother). I suggest this article on optimal unbiased estimation for free-distribution settings. In it, the theorem 4.1 was corrected in a future publication (in Spanish):

Ruiz Espejo, M. (2015). Sobre estimación insesgada óptima del cuarto momento central poblacional. Estadística Española 57 (188), 287-290.

Dear Prof. Ruiz Espejo,

The importance of unbiased estimators are emphasized in statistics.

One of the reasons is, I think, that unbiased estimators are often expressed in

elegant mathematics. Your paper here is a prominent example.

However, even if "a" is an unbiased estimator, the reciprocal of "a" may

not be unbiased. Hence, estimates given by unbiased estimators should be

treated very carefully; but most people do not. On the other hand, a "predictive

estimator" (such as "third variance") yielded by maximizing log-likelihood

in the light of future data (i.e. maximizing predictability) is invariant because it

inherits invariant nature from ordinary log-likelihood (in the light of data at hand).

In my opinion, when we have good predictive estimators, we should

use them in stead of unbiased estimator. A "good predictive estimator"

means here that its characteristics such as variance are favorable.

Dear Prof. Ruiz Espejo,

If your purpose is prediction and

predictability is measured using log-likelihood

in the light of future data, an unbiased estimator

is not always the best choice. For example, "third

variance" is better than unbiased estimator

from the perspective of prediction.

I am wondering if you agree to this idea.

Dear Kunio,

I am usually interested in estimation because in such estimation is possible to study objectivity and objective properties of estimators. In prediction, all depends of the assumed model, which could be true with a probability zero almost sure, or false. This occurs also in statistical modelization, when the researcher assigns a supposed model to an ideal population. For example, in the "a priori" distribution of a parameter in Bayesian statistics. In my opinion these are subjective studies.

Thank you for your participation in my Question.

In this article, I show that the usual linear regression estimator (which is not unbiased) can be improved by other unbiased linear regression estimator.

The traditional criterium for optimization of estimators is "unbiasedness and uniformly of minimum variance".

The criterium "uniformly minimum mean square error" is also "unbiasedness and uniformly of minimum variance" for the parametric function "expectation of the estimator".

Other article on "unbiased multivariate regression estimation" (in Spanish) has been published recently (October 2016). You can see it in the Contributions of my profile in RG.

Thank you for the invitation. I am not professional in this field and, thus, I will not risk an  answer to the question. Nevertheless, the discussion thread will be a good opportunity for me to start learning about this interesting subject.

Regards

My last book Sampling Science is other good resource for unbiased estimation.

http://www.bubok.es/libros/251372/CIENCIA-DEL-MUESTREO

In some circumstances, when there is not a known unbiased estimator for the parametric function, it would be possible to use biased estimation with good accuracy properties.

It is not statistically proper to use biased estimator.

The estimator to be used should have the minimum four qualities namely consistency, unbiasedness, efficiency and sufficiency.

However, if in a situation there does not exist unbiased estimator then in such situation a biased estimator can be used provided that it satisfies the other qualities.

Dear Mariano Ruiz Espejo ,

Your recommendation has made me more encouraged in innovative works.

The age of unbiased estimator is over. "Predictive estimator" is in. Please refer to

http://cse.naro.affrc.go.jp/takezawa/simu_e-1.html

Dear Kunio,

For economic variables, unbiased estimation is the nearest to just estimation. The justice never passes.

Kunio Takezawa is correct.

I recommend his answer.

Dear Dhritikesh Chakrabarty,

Thank you very much for your agreement.

But I would like to know the details about your opinion.

Do you acknowledge the value of the concepts of

"third variance" and "predictive estimator"?

The major purpose to introduce new estimators is to minimize variability in estimates. If MSE of a biased estimator is less than the variance of an unbiased estimator, we may prefer to use biased estimator for better estimation.

Hello

The mean squared error (MSE) can be the selection criterion

An estimator to be used should satisfy basically four properties/criteria. These four criteria are consistency, unbiasness, efficiency and sufficiency. Some other properties are completeness, compactness etc.

However, consistency is an essential property while unbiasness is a desirable property.

Consistency can not be sacrificed. On the other hand, unbiasness is also not sacrificed usually. It may be sacrificed only in unavoidable situation. However, in that situation consistency must be remained intact.

Thus, it is scientifically not correct to use a biased estimator for a parameter when there is an unbiased estimator for it.

Only in unavoidable situation, it is suggested to use a biased estimator (which must be consistent) for a parameter when there is no unbiased estimator for it.

Here you have an article which presents an optimal unbiased estimator for distribution free setting which is better than linear regression estimator, ratio estimator and product estimator among all others based on an auxiliary variable: "Recientes frutos en bioestadística" of 2018, in my Research, Articles.

"Recientes frutos en bioestadística" does it for a finite number of auxiliary variables too (year 2018).

"Sobre la optimización del estimador de regresión lineal" provides an unbiased estimator based on the same conditions of linear regression estimator and it is optimal (uniformly of minimum variance) for estimating the finite population mean of an interest variable. You can see it in my profile too (year 2022).

@ Mariano Ruiz Espejo

I have not found any reason for choosing a biased estimator over an unbiased estimator when the later exists. Only in unavoidable situation, it is suggested to use a biased estimator (which must be consistent) for a parameter when there is no unbiased estimator for it.