A consistent estimator is one that uniformly converges to the true value of a population distribution as the sample size increases. A population value is a characteristic of the population that describes its actual distribution, while a distribution parameter is a characteristic of the distribution used to estimate the population. The parametric description of the population is consistent if the parametric distribution converges to the population distribution.

When the population is fixed and each member of the population is sampled, then the frequency of occurrence of a characteristic is the probability of occurrence. No parametric description is necessary, but a simpler distribution than the actual distribution may be assumed for ease of calculation. If the parametric distribution is sufficiently accurate for the purpose of the investigation, then it is also consistent.

Actual data do not follow the assumed parametric distribution. If deviations from the parametric distribution are important to the outcome for a region of the distribution, then stratification should be considered.

Consistency is always a good qualification. Is not it we all look for consistency in our acts.

In 'scale-free' power-law distributions you never reach consistency, because the more numerous the sampling the more frequent the exteme positive values become and the mean progressively increaes without reaching a stable value.

Respected Alessandro Giuliani, can you explain me a little bit whatever you have written "In 'scale-free' power-law distributions you never reach consistency",( with example) so that I could understand your standpoint better.

check out this article, it may be useful for you with an example of scale-free networks.

The large-scale organization

of metabolic networks

Respected Subrata Sir, Can you help me by giving two real life examples, where consistency do not exist.

1. a logical system that is so constituted that none of the propositions deducible from the axioms contradict one another is consistent, otherwise inconsistent.

2. Complete lack of consistency can be seen some times political leader where there is hardly any uniformity or agreement among what they say, claim,do!

these comments are not to be generalized though.

Actually Sir, my question was, When we go for data analysis(say quantitative data), then what type of data not satisfied the criterion of consistency.

In other words, you are asking why at all we need to go for consistency! Good question!

In case of Bayesian estimation, how consistency can be define.

Consistency is generally a weak test of a sampling distribution. It is expected that an increase in sample size leads to smaller standard deviation. A contrary result occurs if uniformity is localized or selection criteria are relaxed. Localized uniformity may result in preliminary data with unintended sampling from a narrow distribution. Expansion of the sampling space may then lead to a larger standard deviation. Relaxing of selection criteria is another way of expanding the sampling space. Consistency cannot be required upon expansion of the sampling space, but is expected in the expanded space.

Consistency can have a different meaning in Bayesian analysis. Bayesian analysis expects consistency in the updating of the posterior distribution. The posterior is expected to converge to the true value. Caution must be observed in Bayesian updating. Convergence to the mean is the central limit case. One must be certain that the central limit is the desired measure. The probability distribution of the mean is not the same as the sample distribution. A heavy-tailed sampling distribution says that the mean is part of the information and the sampling distribution may indicate important differences from the mean.

The consistency test concerns the sampling distribution, not the posterior distribution. There are no barriers to using consistency in Bayesian analysis, but with the caution that consistency may have different meanings for the sampling and posterior distributions.

Respected Joseph Alvarez, please if it is possible give some real life examples where consistency do not exist.

Non-consistency that follows an increase in sample size is usually caused by a more heterogeneous sample than indicated by initial results. For example, a questionnaire may be given to university students for initial evaluation, but when given to the general population there is a larger standard deviation.

Health studies may be performed on a group of volunteers that is well-controlled by the investigator. Expansion to the general population invariably increases the standard deviation over the volunteers.

Plant and animal studies encounter temporal and spatial variation. A small study area over a short time may show small variation in a characteristic of the population. There may be several controlling factors that affect the characteristic, temperature, nutrients, length of day, and predation. Expansion of the study area and/or increasing the time of the study may show a significant change in the controlling factors. Consistency will be found only by compensating for the controlling factors.

It is often necessary to eliminate or account for covariance.

Respected Joseph Alvarez,in my view, actually before going to sampling, to check the heterogeneity and the geographical location we first need to go for pilot survey, and if it is heterogeneous then there are different sampling techniques, then why we why at all we need to go for consistency.

Exactly. It is more important to identify the heterogeneity and confounding factors. Then devise an appropriate sampling plan.

Yes, Respected Joseph. But after doing all these why we should have to go for consistency. I think consistency will automatically exist if we devise the appropriate sampling plane after pilot survey.

There is no need. Consistency is a weak test.

Then you mean to say that it is not a very good property.

Consistency is a qualitative rather than a quantitative property. We expect consistency, but it is not necessary nor can we propose a measure for consistency. It is one of many evaluation methods when collecting data. If we observe an increasing spread in the standard deviation instead of decreasing with increasing data, it is to be noted and if possible explained. Data analysis cannot follow a cook book, because every study is different. We may follow the same methods of analysis, but no step confirms or denies a result.

Most important is clarity in the intent of the analysis. Clear logic syllogisms should state the investigation at hand. It is equally important at the outset to determine what is necessary to reach a conclusion. What are we trying to show? What evidence and strength of evidence meets the criterion of demonstrating the effect?

The data collected and the conclusions drawn should allow one to predict the results of a replicating experiment.

@Joseph Alvarez

thanks sir, for your kind response.