4.4 Maximum likelihood estimation
5 min read•august 13, 2024
is a powerful statistical method for finding the best-fit parameters of a probability distribution. It works by maximizing the , which measures how probable the observed data is given different parameter values.
MLE is widely used in engineering and science due to its versatility and desirable properties. It provides consistent, efficient estimates that improve with larger sample sizes, making it valuable for parameter estimation across various statistical models and applications.
Maximum Likelihood Estimation
Concept and Principles
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- Maximum likelihood estimation is a statistical method used to estimate the parameters of a probability distribution by maximizing the likelihood function
- The likelihood function measures the probability of observing the data given the parameters of a statistical model
- The maximum likelihood estimate is the set of parameter values that maximize the likelihood function, making the observed data the most probable outcome
- MLE is based on the principle that the best estimate of the parameters is the one that makes the observed data the most likely to occur
- MLE is a versatile and widely used method for parameter estimation in various fields (engineering, economics, social sciences)
Applications and Advantages
- MLE is used to estimate parameters in a wide range of statistical models and probability distributions
- It provides a consistent framework for parameter estimation across different models and data types
- MLE estimators possess desirable statistical properties (consistency, asymptotic normality, efficiency) under certain regularity conditions
- MLE can be used with both small and large sample sizes, although its performance improves as the sample size increases
- MLE is often the foundation for other statistical inference techniques (hypothesis testing, confidence intervals)
Likelihood Functions for Distributions
Formulation and Solving
- The likelihood function is formulated by expressing the joint probability density function (for continuous random variables) or the joint probability mass function (for discrete random variables) of the observed data in terms of the unknown parameters
- For independent and identically distributed (i.i.d.) samples, the likelihood function is the product of the individual probability density or mass functions
- Common probability distributions used in MLE include normal, binomial, Poisson, exponential, and gamma distributions
- To find the maximum likelihood estimate, the likelihood function is often transformed into the function, which simplifies the optimization process
- The maximum likelihood estimate is obtained by setting the first derivative of the log-likelihood function with respect to each parameter equal to zero and solving the resulting system of equations
Analytical and Numerical Solutions
- In some cases, closed-form solutions for the maximum likelihood estimates can be derived analytically ( with unknown and known )
- When analytical solutions are not available, numerical optimization techniques, such as gradient descent or Newton's method, may be required to find the maximum likelihood estimates
- Numerical optimization techniques iteratively search for the parameter values that maximize the log-likelihood function
- The choice of optimization technique depends on the complexity of the likelihood function, the number of parameters, and the desired accuracy of the estimates
- Software packages and libraries (R, Python, MATLAB) often provide built-in functions for MLE, making the implementation more convenient
Properties of MLE Estimators
Consistency and Asymptotic Normality
- Maximum likelihood estimators possess several desirable statistical properties, which make them attractive for parameter estimation
- Consistency: As the sample size increases, the maximum likelihood estimator converges in probability to the true parameter value, meaning the estimator becomes more accurate as more data is collected
- Asymptotic normality: Under certain regularity conditions, the maximum likelihood estimator is asymptotically normally distributed, allowing for the construction of confidence intervals and hypothesis tests using the normal distribution
- The asymptotic normality property enables the use of standard statistical inference techniques based on the normal distribution, even when the true distribution of the estimator is unknown
Efficiency and Invariance
- Efficiency: Maximum likelihood estimators are often the most efficient estimators, meaning they have the smallest variance among all unbiased estimators
- The efficiency property implies that MLE makes the most use of the information contained in the data, resulting in more precise estimates compared to other estimation methods
- Invariance: The maximum likelihood estimator is invariant under parameter transformations, meaning that if a function of the parameter is estimated using MLE, the result is the same as applying the function to the MLE of the original parameter
- The invariance property allows for the estimation of transformed parameters without the need to rederive the likelihood function or recalculate the estimates
MLE in Engineering Applications
Implementation Steps
- Identify the relevant probability distribution that best describes the data or the underlying process in the engineering application (normal distribution for measurement errors, Poisson distribution for rare events)
- Formulate the likelihood function by expressing the joint probability density or mass function of the observed data in terms of the unknown parameters
- If necessary, transform the likelihood function into the log-likelihood function to simplify the optimization process
- Determine the maximum likelihood estimates by setting the first derivative of the log-likelihood function with respect to each parameter equal to zero and solving the resulting system of equations
- If closed-form solutions are not available, employ numerical optimization techniques, such as gradient descent or Newton's method, to find the maximum likelihood estimates
Interpretation and Usage
- Assess the quality of the estimates by examining their statistical properties, such as consistency and asymptotic normality, and by calculating confidence intervals or performing hypothesis tests
- Interpret the results in the context of the engineering application (estimated mean and variance of a process, estimated failure rate of a component)
- Use the estimated parameters for further analysis, prediction, or decision-making in the engineering application (process control, reliability analysis, system optimization)
- Consider the assumptions and limitations of the MLE approach, such as the correctness of the assumed probability distribution and the sample size, when interpreting and using the results
- Validate the MLE results using other estimation methods or domain knowledge to ensure the robustness and reliability of the estimates in the engineering context
Key Terms to Review (17)
Akaike Information Criterion: The Akaike Information Criterion (AIC) is a measure used to compare different statistical models, providing a way to evaluate their goodness of fit while penalizing for complexity. It helps in selecting the model that best explains the data without being overly complex, thus avoiding overfitting. The AIC is calculated using the likelihood of the model and the number of parameters, making it an essential tool in maximum likelihood estimation.
Bayesian Information Criterion: The Bayesian Information Criterion (BIC) is a statistical tool used to evaluate the fit of a model while penalizing for the number of parameters in the model. This helps to prevent overfitting, as it balances model complexity with goodness of fit. The BIC is particularly useful when comparing different models, guiding the choice of a model that best explains the data without being overly complicated.
David R. Cox: David R. Cox is a prominent statistician known for his significant contributions to the field of statistics, particularly in the area of maximum likelihood estimation and survival analysis. His work has laid the groundwork for many modern statistical methods and theories, influencing how data is interpreted and understood in various applications. One of his most notable contributions is the Cox proportional hazards model, which has become a standard tool in survival analysis.
Exponential Distribution: The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process. It is characterized by its memoryless property, meaning that the probability of an event occurring in the future is independent of any past events. This distribution is significant in various fields, including reliability engineering and queuing theory, making it essential for understanding system behavior and performance.
Identifiability: Identifiability refers to the ability to uniquely estimate model parameters based on observed data. In the context of statistical modeling, a parameter is identifiable if it can be determined without ambiguity from the likelihood function. This concept is crucial because when parameters are identifiable, it ensures that maximum likelihood estimation can yield meaningful and reliable estimates, allowing statisticians to infer conclusions from their models.
Independence of Observations: Independence of observations refers to the statistical assumption that the observations in a dataset are not influenced by each other. This means that each data point is collected in such a way that it does not affect or is not affected by other data points, which is crucial for ensuring the validity of many statistical methods. When this assumption is met, it allows for accurate estimation and inference from data, including parameter estimation and hypothesis testing.
Likelihood Function: The likelihood function is a mathematical function that represents the probability of obtaining observed data given specific values of model parameters. It plays a crucial role in statistical inference, particularly in maximum likelihood estimation, where the aim is to find the parameter values that maximize this function. The likelihood function connects observed data with the underlying statistical model, allowing for the estimation of parameters and comparison between different models based on how well they explain the data.
Log-likelihood: Log-likelihood is a measure used in statistics that assesses the probability of observing the given data under a specific statistical model. It is the natural logarithm of the likelihood function, which expresses how well a particular model explains the observed data. Log-likelihood is crucial in maximum likelihood estimation, as it transforms the multiplication of probabilities into a summation, making calculations more manageable and stable.
Maximum Likelihood Estimation: Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a probability distribution by maximizing the likelihood function, which measures how well a statistical model explains the observed data. This technique connects with various probability distributions, both discrete and continuous, as it can be applied to find the most likely parameters for these distributions given observed data. MLE is especially relevant in engineering models, failure time analysis, and even factor analysis, providing a robust framework for estimating unknown parameters based on empirical evidence.
Mean: The mean is a measure of central tendency that represents the average value of a dataset, calculated by summing all the values and dividing by the total number of values. It provides a simple way to summarize a set of data points, making it easier to understand trends and patterns in various fields, including engineering.
Model Fitting in Quality Control: Model fitting in quality control refers to the process of selecting and applying statistical models to analyze and predict the behavior of manufacturing processes, ensuring that products meet specified quality standards. This technique involves adjusting the parameters of the model to best represent the data collected from quality measurements, helping in identifying variations and controlling the production process. Effective model fitting aids in decision-making by providing insights into product quality, process capability, and potential improvements.
Normal Distribution: Normal distribution is a continuous probability distribution characterized by its symmetric bell-shaped curve, where most of the observations cluster around the central peak, and probabilities for values further away from the mean taper off equally in both directions. This distribution is crucial because it serves as a foundation for many statistical methods, including those that estimate parameters and test hypotheses.
Parameter Estimation in Reliability Engineering: Parameter estimation in reliability engineering refers to the process of using statistical methods to estimate the parameters of a probability distribution that models the time until failure or the reliability of a system. This process is essential for understanding how long a product will last, predicting failures, and improving designs. The accuracy of these estimates directly affects decision-making in design and maintenance, and helps in ensuring safety and performance standards.
Regression models: Regression models are statistical tools used to understand the relationship between a dependent variable and one or more independent variables. They aim to estimate the expected value of the dependent variable based on the values of the independent variables, helping to identify trends, make predictions, and analyze the strength of relationships. In the context of maximum likelihood estimation, these models can be optimized to find the best-fitting parameters that maximize the likelihood of observing the given data.
Ronald A. Fisher: Ronald A. Fisher was a prominent statistician, geneticist, and evolutionary biologist known for his foundational contributions to the field of statistics and experimental design. His work laid the groundwork for many modern statistical methods, including point estimation, maximum likelihood estimation, analysis of variance, and experimental designs that remain vital in research today.
Survival Analysis Models: Survival analysis models are statistical techniques used to analyze time-to-event data, often focusing on the time until an event of interest occurs, such as death or failure of a system. These models take into account censoring, which occurs when the event of interest has not been observed for all subjects during the study period. They are particularly useful in fields like medicine and engineering, where understanding the duration until an event can inform risk assessment and decision-making.
Variance: Variance is a statistical measure that represents the degree of spread or dispersion in a set of data points. It indicates how much the values in a dataset differ from the mean, providing insight into the variability of the data, which is crucial for understanding the distribution and behavior of different types of data and random variables.