Is MLE of exponential distribution biased?

Is MLE of exponential distribution biased?

In this case, the MLE estimate of the rate parameter λ of an exponential distribution Exp(λ) is biased, however, the MLE estimate for the mean parameter µ = 1/λ is unbiased. Thus, the exponential distribution makes a good case study for understanding the MLE bias.

Can MLE be biased?

It is well known that maximum likelihood estimators are often biased, and it is of use to estimate the expected bias so that we can reduce the mean square errors of our parameter estimates.

Are MLE estimators unbiased?

MLE is a biased estimator (Equation 12).

How do you find the maximum likelihood estimator of an exponential distribution?

The maximum likelihood estimate (MLE) is the value $ \hat{\theta} $ which maximizes the function L(θ) given by L(θ) = f (X1,X2,…,Xn | θ) where ‘f’ is the probability density function in case of continuous random variables and probability mass function in case of discrete random variables and ‘θ’ is the parameter …

How do you calculate an estimator bias?

1 Biasedness – The bias of on estimator is defined as: Bias( ˆθ) = E( ˆ θ ) – θ, where ˆ θ is an estimator of θ, an unknown population parameter. If E( ˆ θ ) = θ, then the estimator is unbiased.

Is MLE of Poisson distribution unbiased?

Conclude that the MLE is unbiased.

What are the properties of maximum likelihood?

In large samples, the maximum likelihood estimator is consistent, efficient and normally distributed. In small samples, it satisfies an invariance property, is a function of sufficient statistics and in some, but not all, cases, is unbiased and unique.

How do you prove an estimator is unbiased?

An estimator is said to be unbiased if its bias is equal to zero for all values of parameter θ, or equivalently, if the expected value of the estimator matches that of the parameter.

What is the maximum likelihood estimator of λ?

STEP 1 Calculate the likelihood function L(λ). log(xi!) STEP 3 Differentiate logL(λ) with respect to λ, and equate the derivative to zero to find the m.l.e.. Thus the maximum likelihood estimate of λ is ̂λ = ¯x STEP 4 Check that the second derivative of log L(λ) with respect to λ is negative at λ = ̂λ.

What is the difference between bias and MSE?

MSE can be high even if bias is 0, because positive and negative deviations of the estimates from the true mean do not cancel out. MSE is the sum of the variance of an estimate plus the square of its bias.