Dear collegues

Thank you Ette and Joachim

In order to complete the idea, which is given in my question, the following remarks may be required:

In fact, in many practical problems for estimating the unknown parameter we have a number of experiments which contain useful real data. It is well known that performing any experiment and collecting data cost in terms of effort and money.

In addition, in many real life problems, and due to huge experiments in our life, the experimenter has prior information about many real problems, i.e. regarding the value of due to past experiences. For example, in lung cancer problems, there are a lot of patients in society who suffer from this disease, and in places where the treatment is mainly based on using chemical therapy. Assume that in Hospital A, and based on a previous experiment, the doctor believes that the patient will survive six more months at most. There is another experiment for studying the mean life of another group of lung cancer patients in Hospital B. The information gained from Hospital A can be utilized in the new estimation problem, i.e. the 180 days can be considered as a prior value of experiment B. Similarly, in an experiment for testing the mean life of an electronic batch, as an outcome of an experiment, the experimenter obtains that the mean life equals to 2000 hours. For testing the coming batches, the prior information =2000 hours can be taken.

Regards.

At best the parameter is unknown. Why then are we talking of the true value of the parameter? How do we know its true value?It's true value cannot be known. There can be a prior probability distribution for the parameter which can be updated with the availability of data to obtain its posterior distribution for statistical inference.

I would like to complete Ette's answer with a remark, that beside the bayesian way of estimating (which works with a introductory distribution of the unknown parameter(s)) it is also possible to involve a single value stated e.g. by experts. Simply one can treat it as result of measurement (observtion) by taking this value with a suitable weight,

Example. The data are 5 numbers x1, x2, x3, x4, x5 close to the real parameter say a. And an expert clims that (probably) a is equal to a0. The proposed algorithm requires to dicide about the weight of the expert's value. If we think that it value the same as a mean of 3 additional measurement, then the simplest and even though reasonble way for point-estimation of a would be (x1 + x2 + x3 + x4 + x5 + 3 a0 ) /8. Obviously, more detailed analysis of the "mixed" estimator (e.g. on its dispersion) would require additional info from the expert about say some a'priori estimates of the dispersion of the value of a0.

Regards, Joachim

Dear collegues

Thank you Ette and Joachim

In order to complete the idea, which is given in my question, the following remarks may be required:

In fact, in many practical problems for estimating the unknown parameter we have a number of experiments which contain useful real data. It is well known that performing any experiment and collecting data cost in terms of effort and money.

In addition, in many real life problems, and due to huge experiments in our life, the experimenter has prior information about many real problems, i.e. regarding the value of due to past experiences. For example, in lung cancer problems, there are a lot of patients in society who suffer from this disease, and in places where the treatment is mainly based on using chemical therapy. Assume that in Hospital A, and based on a previous experiment, the doctor believes that the patient will survive six more months at most. There is another experiment for studying the mean life of another group of lung cancer patients in Hospital B. The information gained from Hospital A can be utilized in the new estimation problem, i.e. the 180 days can be considered as a prior value of experiment B. Similarly, in an experiment for testing the mean life of an electronic batch, as an outcome of an experiment, the experimenter obtains that the mean life equals to 2000 hours. For testing the coming batches, the prior information =2000 hours can be taken.

Regards.

@ Z. A. Al-Hemyari

Dear Sir, your examples are very important since they are related to real problems, including carrying of the human's life and taking into account very practical problem of dimnishing costs of the needed research. Basically these are problems of the relativity theory which covers also the ways of modelling aging of people, devices or other objects. In such a concrete setting as it is given in your response the very simple algorithm suggested in my answer is neither sufficient nor consistent. Let me just mention that in many cases the real data are much more detailed and choosing a suitable mathematical model requires advanced methods of estimation. One of the ways is taking into account that the dat are censored e.g. by the fact that some histories show exact time to death, and some indicate only that a patied was livilg at least such and such time period (without any further history). All this can be and should be taken into account. However, there are bad customs that not always the full data are released/exchanged between say hospitals, research centers, etc. But even then any partial nformation can be taken into account, though with less confidence then (all can be modelled with taking suitable weights for ll auxiliary data).

Regards, Joachim

Dear All

It may be useful to add the following:

In many real world situations, the experimenter may have some prior information regarding the unknown parameters due to past experiences. However, in certain situations the prior information is available only in the form of an initial guess value (natural origin)

θo of θ, then this guess may be utilised in the estimation of θ to improve the estimation procedure, i.e., reducing the MSE of the new estimators or it may give a saving in sample size. Indeed, the prior information costs time and money; and incorporating such prior information in the estimation of the unknown parameters is also utilises the past cost of sampling units.

For example, an experiment may study the relation between the highway gasoline mileage performance (Y) and engine displacement for a type of vehicles for model year 2008 (X). At the same time, another experiment studying the same problem for the year (2007) and for the same vehicle where 3Y 2007 = 28.114 (in ), and 2007 βˆ = −0.028. Here we may take μ0 = 28.114 (in3), and β0 = –0.028 (i.e., θ0 = (μ0, β0)T).

Regards,

Zuhair