Estimation Point
Zee Palmer
In statistics, point estimation involves the use of sample data to calculate a single value (known as a point estimate since it identifies a point in some parameter space) which is to serve as a "best guess" or "best estimate" of an unknown population parameter (for example, the population mean). More formally, it is the application of a point estimator to the data to obtain a point estimate.
Point estimation can be contrasted with interval estimation: such interval estimates are typically either confidence intervals, in the case of frequentist inference, or credible intervals, in the case of Bayesian inference. More generally, a point estimator can be contrasted with a set estimator. Examples are given by confidence sets or credible sets. A point estimator can also be contrasted with a distribution estimator. Examples are given by confidence distributions, randomized estimators, and Bayesian posteriors.
Consistency is about whether the point estimate stays close to the value when the parameter increases its size. The larger the sample size, the more accurate the estimate is. If a point estimator is consistent, its expected value and variance should be close to the true value of the parameter. An unbiased estimator is consistent if the limit of the variance of estimator T equals zero.
Generally, we must consider the distribution of the population when determining the efficiency of estimators. For example, in a normal distribution, the mean is considered more efficient than the median, but the same does not apply in asymmetrical, or skewed, distributions.
In statistics, the job of a statistician is to interpret the data that they have collected and to draw statistically valid conclusion about the population under investigation. But in many cases the raw data, which are too numerous and too costly to store, are not suitable for this purpose. Therefore, the statistician would like to condense the data by computing some statistics and to base their analysis on these statistics so that there is no loss of relevant information in doing so, that is the statistician would like to choose those statistics which exhaust all information about the parameter, which is contained in the sample. We define sufficient statistics as follows: Let X =( X1, X2, ... ,Xn) be a random sample. A statistic T(X) is said to be sufficient for θ (or for the family of distribution) if the conditional distribution of X given T is free from θ.[2]
The MAP estimator has good asymptotic properties, even for many difficult problems, on which the maximum-likelihood estimator has difficulties.For regular problems, where the maximum-likelihood estimator is consistent, the maximum-likelihood estimator ultimately agrees with the MAP estimator.[5][6][7]Bayesian estimators are admissible, by Wald's theorem.[6][8]
Below are some commonly used methods of estimating unknown parameters which are expected to provide estimators having some of these important properties. In general, depending on the situation and the purpose of our study we apply any one of the methods that may be suitable among the methods of point estimation.
The method of maximum likelihood, due to R.A. Fisher, is the most important general method of estimation. This estimator method attempts to acquire unknown parameters that maximize the likelihood function. It uses a known model (ex. the normal distribution) and uses the values of parameters in the model that maximize a likelihood function to find the most suitable match for the data.[9]
Let X = (X1, X2, ... ,Xn) denote a random sample with joint p.d.f or p.m.f. f(x, θ) (θ may be a vector). The function f(x, θ), considered as a function of θ, is called the likelihood function. In this case, it is denoted by L(θ). The principle of maximum likelihood consists of choosing an estimate within the admissible range of θ, that maximizes the likelihood. This estimator is called the maximum likelihood estimate (MLE) of θ. In order to obtain the MLE of θ, we use the equation
The method of moments was introduced by K. Pearson and P. Chebyshev in 1887, and it is one of the oldest methods of estimation. This method is based on law of large numbers, which uses all the known facts about a population and apply those facts to a sample of the population by deriving equations that relate the population moments to the unknown parameters. We can then solve with the sample mean of the population moments.[10] However, due to the simplicity, this method is not always accurate and can be biased easily.
When f(x, β0, β1, ,,,, βp) is a linear function of the parameters and the x-values are known, least square estimators will be best linear unbiased estimator (BLUE). Again, if we assume that the least square estimates are independently and identically normally distributed, then a linear estimator will be minimum-variance unbiased estimator (MVUE) for the entire class of unbiased estimators. See also minimum mean squared error (MMSE).[2]
There are two major types of estimates: point estimate and confidence interval estimate. In the point estimate we try to choose a unique point in the parameter space which can reasonably be considered as the true value of the parameter. On the other hand, instead of unique estimate of the parameter, we are interested in constructing a family of sets that contain the true (unknown) parameter value with a specified probability. In many problems of statistical inference we are not interested only in estimating the parameter or testing some hypothesis concerning the parameter, we also want to get a lower or an upper bound or both, for the real-valued parameter. To do this, we need to construct a confidence interval.
Here two limits are computed from the set of observations, say ln and un and it is claimed with a certain degree of confidence (measured in probabilistic terms) that the true value of γ lies between ln and un. Thus we get an interval (ln and un) which we expect would include the true value of γ(θ). So this type of estimation is called confidence interval estimation.[2] This estimation provides a range of values which the parameter is expected to lie. It generally gives more information than point estimates and are preferred when making inferences. In some way, we can say that point estimation is the opposite of interval estimation.
In either case, we can't possibly survey the entire population. That is, we can't survey all American college students between the ages of 18 and 24. Nor can we survey all patients with Alzheimer's disease. So, of course, we do what comes naturally and take a random sample from the population, and use the resulting data to estimate the value of the population parameter. Of course, we want the estimate to be "good" in some way.
In this lesson, we'll learn two methods, namely the method of maximum likelihood and the method of moments, for deriving formulas for "good" point estimates for population parameters. We'll also learn one way of assessing whether a point estimate is "good." We'll do that by defining what a means for an estimate to be unbiased.
Point estimators are functions that are used to find an approximate value of a population parameter from random samples of the population. They use the sample data of a population to calculate a point estimate or a statistic that serves as the best estimate of an unknown parameter of a population.
Most often, the existing methods of finding the parameters of large populations are unrealistic. For example, when finding the average age of kids attending kindergarten, it will be impossible to collect the exact age of every kindergarten kid in the world. Instead, a statistician can use the point estimator to make an estimate of the population parameter.
The bias of a point estimator is defined as the difference between the expected value of the estimator and the value of the parameter being estimated. When the estimated value of the parameter and the value of the parameter being estimated are equal, the estimator is considered unbiased.
Consistency tells us how close the point estimator stays to the value of the parameter as it increases in size. The point estimator requires a large sample size for it to be more consistent and accurate.
You can also check if a point estimator is consistent by looking at its corresponding expected value and variance. For the point estimator to be consistent, the expected value should move toward the true value of the parameter.
The most efficient point estimator is the one with the smallest variance of all the unbiased and consistent estimators. The variance measures the level of dispersion from the estimate, and the smallest variance should vary the least from one sample to the other.
Generally, the efficiency of the estimator depends on the distribution of the population. For example, in a normal distribution, the mean is considered more efficient than the median, but the same does not apply in asymmetrical distributions.
The two main types of estimators in statistics are point estimators and interval estimators. Point estimation is the opposite of interval estimation. It produces a single value while the latter produces a range of values.
A point estimator is a statistic used to estimate the value of an unknown parameter of a population. It uses sample data when calculating a single statistic that will be the best estimate of the unknown parameter of the population.
On the other hand, interval estimation uses sample data to calculate the interval of the possible values of an unknown parameter of a population. The interval of the parameter is selected in a way that it falls within a 95% or higher probability, also known as the confidence interval.
The confidence interval is used to indicate how reliable an estimate is, and it is calculated from the observed data. The endpoints of the intervals are referred to as the upper and lower confidence limits.
The process of point estimation involves utilizing the value of a statistic that is obtained from sample data to get the best estimate of the corresponding unknown parameter of the population. Several methods can be used to calculate the point estimators, and each method comes with different properties.
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