Final answer:
The maximum likelihood estimator for the parameter β of an exponential distribution is found by maximizing the likelihood function and is the reciprocal of the sample mean. This estimator coincides with the method of moments estimator in this case. Bayesian approaches and central limit theorem are related but distinct concepts that involve different assumptions and implications for parameter estimation.
Step-by-step explanation:
Maximum Likelihood Estimator for β in an Exponential Distribution
To find the maximum likelihood estimator (MLE) for the parameter β of an exponential distribution based on a sample of size n, we use the likelihood function for the exponential distribution. The likelihood function L(β) for a sample x_1, x_2, …, x_n is the product of the individual probabilities:
L(β) = β^n e^{-β(x_1 + x_2 + … + x_n)}
The log-likelihood function, which is the natural logarithm of the likelihood function, is then maximized to find the estimator for β. Taking the derivative of the log-likelihood with respect to β, setting it equal to zero, and solving for β gives us the MLE:
β_hat = n / (x_1 + x_2 + … + x_n)
The method of moments estimator for β would be the reciprocal of the sample mean, which in this case coincides with the MLE for β.
There are differences between the MLE and other estimation methods like the Bayesian approaches, which take into account prior information and assume that model parameters are variable, not fixed point values. The Bayesian approach often provides higher certainty in parameter estimates and is less prone to certain errors, but it requires more complex calculations, often involving Markov Chain Monte Carlo (MCMC) methods.
Central Limit Theorem
The central limit theorem states that the sum (or the mean) of a sufficiently large number of independent random variables (each with a finite mean and variance) will be approximately normally distributed. For a sample of sizes sufficiently large, the sample sum becomes more normally distributed, regardless of the shape of the original distribution. This theorem is fundamental for inferential statistics and serves as the basis for constructing confidence intervals and hypothesis testing.