Maksimiosuusmenetelmällä

Maksimiosuusmenetelmällä, often translated as the "maximum likelihood method" or "maximum likelihood estimation," is a fundamental statistical technique used for estimating the parameters of a probability distribution. The core idea is to find the parameter values that maximize the probability of observing the actual data. In essence, it asks: given the data we have, which set of parameters makes this data most likely to have occurred?

The process involves defining a likelihood function, which quantifies the probability of the observed data as a function of the unknown parameters. This function is typically derived from the probability mass function or probability density function of the assumed statistical model. Once the likelihood function is established, statistical methods are employed to find the parameter values that yield the highest possible value for this function. This is often achieved through calculus by taking derivatives and setting them to zero, or through numerical optimization algorithms.

Maximum likelihood estimation has several desirable properties. It is consistent, meaning that as the sample size increases, the estimates converge to the true parameter values. It is also asymptotically efficient, implying that for large sample sizes, it achieves the lowest possible variance among all unbiased estimators. Furthermore, it possesses the property of invariance, meaning that the maximum likelihood estimate of a function of a parameter is the function of the maximum likelihood estimate of the parameter itself. This makes it a versatile and widely used tool in various fields, including econometrics, genetics, machine learning, and signal processing.