1. 5. Estimation 5.3 Estimation of the mean K. Desch – Statistical methods of data analysis SS10 Is an efficient estimator for μ ?  depends on the distribution of x ! there is an absolute lower limit on the variance of an estimator (Likelihood method – later) For a Gauss distribution is the most efficient estimator Alternative estimators: Truncated mean: discard the (1-2r)n/2 largest and smallest elements of the sample when calculating the mean r=0.5: sample mean, r→0: median. Example: Breit-Wigner (“μ“=0, Г=1) truncated mean r=0.2, discard 30% of high and low

2. 5. Estimation 5.3 Estimation of the mean K. Desch – Statistical methods of data analysis SS10 Min-Max estimator most efficient estimator uniformly distributed data but not robust (a single „wrong“ measurement can have huge impact)

3. 5. Estimation 5.4 Estimation of the variance K. Desch – Statistical methods of data analysis SS10 The “sample variance”: is an unbiased estimator of the true variance σ 2 as: expectation value of s 2 : as:

4. 5. Estimation 5.4 Estimation of the variance K. Desch – Statistical methods of data analysis SS10 Why ? Since true mean μ is not known and must be estimated also from sample, one looses one degree of freedom If the true mean would be known, then would be an unbiased estimator of the variance Estimator for Variance of s 2 :

5. 6. Max. Likelihood 6.1 Maximum likelihood method K. Desch – Statistical methods of data analysis SS10 (x 1,…x n ) is a data sample with a probability density f(x i,a). How can one construct an estimator for a when f is known (analytically or numerically)? Likelihood-function: a measure for the probability to obtain (x 1,…x n ) for fixed a: Regard L as a function of a ! (x i are fixed, L is not a p.d.f.) Principle of maximum likelihood (ML) Best estimator for a,, is the one which maximizes the Likelihood function: Important: f(x;a) has to be correctly normalized for all a:

6. 6. Max. Likelihood 6.1 Maximum likelihood method K. Desch – Statistical methods of data analysis SS10 When sample size n is large, product is impractical (numerical problems)  use the logarithm of the Likelihood function intstead  product in L is transformed into a sum:

7. More than one parameter: Often negative log-Likelihood is used  minimize F(a) (or even -2lnL → consistency with χ 2 estimater, later ) Properties of ML estimators 1)ML estimators are usually consistent 2)ML estimators are (only) asymptotically unbiased 3)ML estimators are efficient! (they reach the limit of minimum variance) 4)ML estimators are invariant under parameter transformation estimator for θ(a) ? When is a ML estimator for a, then is the ML estimator for . : Drawback: ML estimators do not test whether the data agrees with f – separate tests are necessary (no “goodness of fit” measure) 6. Max. Likelihood 6.1 Maximum likelihood method K. Desch – Statistical methods of data analysis SS10

8. Example of ML estimators : anglular distribution measurements x i, i=1,n 6. Max. Likelihood 6.1 Maximum likelihood method K. Desch – Statistical methods of data analysis SS10

9. 6. Max. Likelihood 6.1 Maximum likelihood method K. Desch – Statistical methods of data analysis SS10 Example of ML estimators: μ and σ 2 of a Gaussian Estimator for μ : Estimator for σ 2 : ML estimator of σ 2 is biased ! ( Unbiased: )

10. Example of ML estimators : exponential distribution p.d.f. : Construct ML estimator for parameter : Log- Likelihood function: Maximum: 6. Max. Likelihood 6.1 Maximum likelihood method K. Desch – Statistical methods of data analysis SS10 arithmetic mean

11. Expectation of : 6. Max. Likelihood 6.1 Maximum likelihood method K. Desch – Statistical methods of data analysis SS10 is unbiased

12. “Statistical error of parameter “ depends on true value of, which we don’t know. How one can estimate ? → Parameter transformation of ML estimator (, ) Variance of : How efficient is the estimator ? Rarely possible analytically → Monte Carlo method 6. Max. Likelihood 6.1 Maximum likelihood method K. Desch – Statistical methods of data analysis SS10

13. 6. Max. Likelihood 6.1 Maximum likelihood method K. Desch – Statistical methods of data analysis SS10 Measurement : means that an experiment estimates the 0.7 ± 0.1 parameter to be 0.7 and by continuous repetition the shows a standard deviation of 0.1

14. Rao-Cramer-Frechet (RCF) bound of minimum variance (w/o proof) Variance of an estimator of single parameter is limited as: is called “efficient” when the bound exactly archived When an efficient estimator exists, it is a ML estimator All ML estimators are efficient for n → ∞ Example: exponential distribution : as (b=0) : 6. Max. Likelihood 6.1 Maximum likelihood method K. Desch – Statistical methods of data analysis SS10