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Wikipedia / Point estimation

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 rather than an interval), which serves as a "best guess" or "best estimate" of an unknown quantity, for example, the population mean, the variance of a distribution, or a model parameter (in a parametric model).

Point estimation can be contrasted with interval estimation: 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.

Properties of point estimators

Biasedness

The bias is defined as the difference between the expected value of the estimator and the true value of the population parameter being estimated. It can also be described that the closer the expected value of a parameter is to the measured parameter, the lesser the bias. When the estimated number and the true value is equal, the estimator is considered unbiased. This is called an unbiased estimator. The estimator will become a best unbiased estimator if it has minimum variance. However, a biased estimator with a small variance may be more useful than an unbiased estimator with a large variance.

. is called biased. Examples of unbiased estimators are the sample mean and the unbiased sample variance.

fulfillsThus, a more general condition for unbiasedness can be defined by, then the estimator is called median-unbiased, since the median is a minimiser of the mean absolute error.

Consistency

A point estimator is called consistent, if the probability that the estimate is close to the true value tends to 1 as the sample size grows to infinity. If the estimate (almost) surely gets arbitrarily close to the true value, eventually, as the sample size grows to infinity, then the estimator is even called strongly consistent. Intuitively, a consistent estimator will be "probably approximately correct", and a strongly consistent estimator even "surely approximately correct", if the sample size is large enough. In the special case, where an estimator is unbiased, it is already consistent, if its variance decreases to zero as the sample size grows to infinity.

Efficiency

Efficiency is a property used to investigate the variance of unbiased estimators. According to the bias-variance decomposition, the variance of an unbiased estimator is equal to its mean squared error (that is, the Cramér–Rao bound is then called efficient.

(efficiency can also be defined in a similar manner for parameter vectors) fulfilling the Cramér–Rao regularity conditions fulfills the Cramér–Rao bound

is the Fisher-information is an exponential family) is the natural sufficient statistic.

would be called more efficient..

Uniform minimum variance and unbiasedness (UMVU)

The concept of efficiency is restricted to estimators and distributions, which fulfill the Cramér–Rao regularity conditions. Such estimators are then called (uniformly) minimum-variance unbiased estimators (UMVUE/MVUE).

it holds

.

Sufficiency

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 is said to be sufficient.

Asymptotic properties

are:

  • asymptotic unbiasedness: Roughly speaking, the distribution converges in distribution converges to zero (though it might be non-zero for finite sample sizes). Every consistent estimator is asymptotically unbiased.

  • asymptotic normality: The distribution, converges in distribution to a standard normal distribution.

  • asymptotic efficiency is the parameter of a parametric model tends to infinity, to a distribution whose variance is the inverse of the Fisher information, that is, the smallest possible variance for a regular unbiased estimator based on one data point.

Estimation methods

Below are some commonly used methods for estimating unknown parameters. The methods vary in their domain of applicability and on the underlying statistical paradigm (frequentist or Bayesian).

Frequentist methods

Maximum likelihood estimation (MLE)

The method of maximum likelihood of a parametric model. According to this method, the best estimate for the unknown model parameter is the one that maximizes the probability (or probability density) of observing the data that has been observed. This probability as a function of the model parameters is called the likelihood, giving the method its name.

fulfills

In practice, the likelihood function is often differentiable is a solution of the equations

Under certain assumptions on the likelihood, the MLE is strongly consistent, asymptotically efficient and asymptotically normal.

Method of moments (MoM)

The method of moments is one of the oldest methods of estimation, which was first introduced by Karl Pearson in 1895. This method uses the functional relationship between quantities of interest of a distribution (e.g. the variance or model parameters) and the moments of that distribution, leading to a set of equations that are solved for the aforementioned quantities. Motivated by the law of large numbers, the moments of the distribution are estimated by the sample moments, that is, the arithmetic averages of powers of the data values. The method stands out for its simplicity and wide domain of applicatibility, but often do not yield the best estimators and sometimes even no estimator at all. There also exists generalizations of the method of moments.

are established:

such that the parameters can be expressed by the moments via

, yielding the moment estimators:

, the moment estimators are strongly consistent and asymptotically normal.

Lehmann–Scheffé theorem

The Lehmann–Scheffé theorem, although not an estimation method in itself, motivates two methods for constructing UMVU estimators in models, where a complete, sufficient is available. This applies, for example, to exponential families, where the natural sufficient statistic is always complete.

.

Method 1: .

Method 2: that is, the conditional expectation might be easier than solving the equation from method 1.

Score matching estimation

Score matching is a rather new method for parameter estimation that is less intuitive than MLE, but is more advantageous for complicated distributions with a large number of parameters.

. The key quantity, that is needed for the score matching estimation, is the score and a normalization constant does not occur in the score matching estimation procedure, thus eliminating this issue.

, the score matching estimate can be shown to be weakly consistent.

Least square estimation (LSE)

The method of least squares is a parameter estimation procedure for regression need to be expressed as functions of the covariates via

accounting for random noise in the data. The ordinary least squares and estimates the parameter by minimizing of the sum of squared differences, called residuals:

are independent and identically normally distributed with mean zero, then the OLS coincides with the maximum likelihood estimator.

If the rank, then the OLS is uniquely given by

and it is the best linear unbiased estimator (BLUE) by the Gauß–Markov theorem. If, as above, the noise variables are independent and identically normally distributed, then the OLS is even UMVU for the entire class of unbiased estimators.

There are multiple variants of least squares estimation:

Bayesian methods

, called the prior distribution, the prior distribution is updated according to Bayes' rule:

The updated distribution of the parameters given the data is called the posterior distribution. Many Bayesian point estimators are the posterior distribution's statistics of central tendency, e.g., its mean, median, or mode.

Bayes estimator

and a loss function that minimizes the Bayes risk:

Obviously, the risk is minimized when the integral under the expectation is minimized as a function of the observations. Thus, a Bayes estimator is equivalently given as a minimizer of the posterior loss:

are the following:

By Wald's theorem, unique Bayesian estimators are admissible., the posterior mean estimator can be shown to be consistent, asymptotically normal and asymptotically efficient.

Maximum a posteriori (MAP)

The maximum a posteriori is a maximizer of the posterior distribution:

In a sense, the MAP estimator is a Bayesian analog to the maximum likelihood estimator. In fact, for a uniform prior distribution, the MAP estimator coincides with the MLE.

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.

Others

The Minimum Message Length (MML) point estimator is based in Bayesian information theory and is not so directly related to the posterior distribution.

Special cases of Bayesian filters are important:

Several methods of computational statistics have close connections with Bayesian analysis:

Point estimate v.s. confidence interval estimate

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.

Confidence interval describes how reliable an estimate is. We can calculate the upper and lower confidence limits of the intervals from the observed data. Suppose a dataset x1, . . . , xn is given, modeled as realization of random variables X1, . . . , Xn. Let θ be the parameter of interest, and γ a number between 0 and 1. If there exist sample statistics Ln = g(X1, . . . , Xn) and Un = h(X1, . . . , Xn) such that P(Ln < θ < Un) = γ for every value of θ, then (ln, un), where ln = g(x1, . . . , xn) and un = h(x1, . . . , xn), is called a 100γ% confidence interval for θ. The number γ is called the confidence level. In general, with a normally-distributed sample mean, Ẋ, and with a known value for the standard deviation, σ, a 100(1-α)% confidence interval for the true μ is formed by taking Ẋ ± e, with e = z1-α/2(σ/n1/2), where z1-α/2 is the 100(1-α/2)% cumulative value of the standard normal curve, and n is the number of data values in that column. For example, z1-α/2 equals 1.96 for 95% confidence.

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. 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.

See also

Notes

References