Point estimation

Biasedness “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. Most importantly, we prefer point estimators that have the smallest mean square errors. If we let T = h(X1,X2, . . . , Xn) be an estimator based on a random sample X1,X2, . . . , Xn, the estimator T is called an unbiased estimator for the parameter θ if E[T] = θ, irrespective of the value of θ. == Types of point estimation ==

Bayesian point estimation Bayesian inference is typically based on the posterior distribution. Many Bayesian point estimators are the posterior distribution's statistics of central tendency, e.g., its mean, median, or mode: • Posterior mean, which minimizes the (posterior) risk (expected loss) for a squared-error loss function; in Bayesian estimation, the risk is defined in terms of the posterior distribution, as observed by Gauss. • Posterior median, which minimizes the posterior risk for the absolute-value loss function, as observed by Laplace. • maximum a posteriori (MAP), which finds a maximum of the posterior distribution; for a uniform prior probability, the MAP estimator coincides with the maximum-likelihood estimator; 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. Bayesian estimators are admissible, by Wald's theorem. 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: • Kalman filter • Wiener filter Several methods of computational statistics have close connections with Bayesian analysis: • particle filter • Markov chain Monte Carlo (MCMC) == Methods of finding point estimates ==

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. Method of maximum likelihood (MLE) 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. 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 dlogL(θ)/dθi=0, i = 1, 2, …, k. If θ is a vector, then partial derivatives are considered to get the likelihood equations. However, due to the simplicity, this method is not always accurate and can be biased easily. Let (X1, X2,…Xn) be a random sample from a population having p.d.f. (or p.m.f) f(x,θ), θ = (θ1, θ2, …, θk). The objective is to estimate the parameters θ1, θ2, ..., θk. Further, let the first k population moments about zero exist as explicit function of θ, i.e. μr = μr(θ1, θ2,…, θk), r = 1, 2, …, k. In the method of moments, we equate k sample moments with the corresponding population moments. Generally, the first k moments are taken because the errors due to sampling increase with the order of the moment. Thus, we get k equations μr(θ1, θ2,…, θk) = mr, r = 1, 2, …, k. Solving these equations we get the method of moment estimators (or estimates) as mr = 1/n ΣXir. == 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 n) = 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. 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 ==