Statistics for Finance
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1 Statistics for Fiace. Lecture 3:Estimatio ad Likelihood. Oe of the cetral themes i mathematical statistics is the theme of parameter estimatio. This relates to the fittig of probability laws to data. May families of probability laws deped o a small umber of parameters. For example, the ormal distributios are determied by the mea µ ad the stadard deviatio σ. Eve though, oe may make a reasoable assumptio o the type of the distributio, e.g. ormal, oe usually does ot kow the parameters of the distributio, e.g. mea ad stadard deviatio, ad oe eeds to determie these from the available data. The philosophical foudatio of our approach is that sample data, say, X, X,..., X, of a sample of size, are thought of as a ( subset of a ifiite) collectio of idepedet, idetically distributed (i.i.d.) radom variables, followig the probability distributio i questio. A bit of explaatio is required at this poit. We are used of sample data haveig the form of real umbers. Whe for example we measure the heights of a sample of 5 studets i Warwick, we may record heights 78, 89, 70, 60, 64. So what does X, X, X 3, X 4, X 5 stad for? The aswer is that, although, we may ed up with cocrete real umbers, a priori these umbers are ukow ad could be aythig. That is why we treat them as radom ad ame them X, X,..., X 5... Sample Mea ad Variace. The method of momets. We have already itroduced the sample mea ad variace, but let us view the relatio of these quatities to the parameters of the uderlyig distributio. Let us remid that the sample mea is defied as X = X i ad the sample variace as s = (X i X). Defiitio. A estimator ˆθ of a parameter θ of a distributio is called ubiased estimator if E[ˆθ] = θ A few words of explaatio. The estimator ˆθ will be a fuctio of the measuremets (X,..., X ) o the sample, i.e. ˆθ = ˆθ(X,..., X ). As we discussed before
2 the measuremets (X,..., X ), are cosidered as i.i.d radom variable havig the uderlyig distributio. If f(x; θ) deotes the pdf of the uderlyig distributio, with parameter θ, the the expectatio i the above defiitio should be iterpreted as E[ˆθ] = E[ˆθ(X,..., X )] = ˆθ(x,..., x )f(x ; θ) f(x ; θ) dx dx ad the defiitio of ubiased estimator correspods to the fact that the above itegral should be equal to the parameter θ of the uderlyig distributio. Defiitio. Let ˆθ = ˆθ (X,..., X ) a estimator of a parameter θ based o a sample (X,..., X ) of size. The ˆθ is called cosistet if ˆθ coverges to θ i probability, that is P ( ˆθ θ ɛ) 0, as Here, agai, as i the previous defiitio, the meaig of the probability P is idetified with the uderlyig distributio with parameter θ. Propositio. The sample mea ad variace are cosistet ad ubiased estimators of the mea ad variace of the uderlyig distributio. Proof. It is easy to compute that [ ] X + + X E = µ ad [ E ] (X i X) = = = ad ow expadig the E[X ] as E [ (X X) ] ( ) E[X ] E[X X] + E[X ] ( E[X ] ) ( ) E[X ] E[X X ] + E[X ] E[X ] = ( E[X ] + ( )E[X X ] ) ad also usig the idepedece, e.g. E[X X ] = E[X ]E[X ] = µ we get that the above equals to E[X ] µ = σ. We, therefore, obtai that the sample mea ad sample variace are ubiased estimators.
3 The fact that the sample mea is a cosistet estimator follows immediately from the weak Law of Large Number (assumig of course that the variace σ is fiite). The fact that the sample variace is also a cosistet estimator follows easily. First, we have by a easy computatio that s = ( ) Xi X. The result ow follows from the Law of Large Numbers sice X i s ad hece Xi s are iepedet ad therefore Xi E[X ] ad X = X + + X E[X ]. The above cosideratios itroduce us to the Method of Momets. recall that the k th momet of a distributio is defied as µ k = x k f(x)dx 3 Let us If X, X,... are sample data draw from a give distributio the the k th sample momet si defied as ˆµ k = Xi k ad by the Law of Large Numbers (uder the appropriate coditio) we have that ˆµ k approximates µ k, as the sample size gets larger. The idea behid the Method of Momets is the followig: Assume that we wat to estimate a parameter θ of the distributio. The we try to express this parameter i terms of momets of the distributio. Example. Cosider the Poisso distributio with parameter λ, i.e. λ λk P (X = k) = e k! It is easy to check (check it!) that λ = E[X]. Therefore, the parameter λ ca be estimated by the sample mea of a large sample. Example. Cosider a ormal distributio N(µ, σ ). Of course, we kow that µ is the first momet ad that σ = Var(X) = E[X ] E[X] = µ µ. So estimatig the first two momets, gives us a estimatio of the parameters of the ormal distributio.
4 4.. Maximum Likelihood. Maximum likelihood is aother importat method of estimatio. May wellkow estimators, such as the sample mea ad the least squares estimatio i regressio are maximum likelihood estimators. Maximum likelihood estimatio teds to give more efficiet estimates tha other methods. Parameters used i ARIMA time series models are usually estimated by maximum likelihood. Let us start describig the method. Suppose that we have a distributio, with a parameter θ = (θ,..., θ k ) R k, that we wish to estimate. Let, also, X = (X,..., X ) a set of sample data. Viewed as a collectio of i.i.d variables, the sample data will have a probability desity fuctio f(x,..., X ; θ) = f(x i ; θ). This fuctio is viewed as a fuctio of the parameter θ, we will deote it by L(θ) ad call it the likelihood fuctio. The product structure is due to the assumptio of idepedece. The maximum likelihood estimator (MLE) is the value of the parameter θ, that maximises the likelihood fuctio, give the observed sample data, (X,..., X ). It is ofte mathematically more tractable to maximise a sum of fuctios, tha a product of fuctio. Therefore, istead of tryig to maximise the likelihood fuctio we prefer to maximise the log-likelihood fuctio log L(θ) = log f(x i ; θ). Example 3. Suppose that the uderlyig distributio is a ormal N(µ, σ ) ad we wat to estimate the mea µ ad variace σ from sample data (X,..., X ), usig the maximum likelihood estimator. First, we start with the log-likelihood fuctio, which i this case is log L(µ, σ) = log σ log(π) σ (X i µ). To maximise the log-likelyhood fuctio we differetiate with respect to µ, σ ad obtai L µ = (X σ i µ) L σ = σ + σ 3 (X i µ)
5 the partials eed to be equal to zero ad therefore solvig the first equatio we get that ˆµ = X i := X. Settig the secod partial equal to zero ad substitutig µ = ˆµ we obtai the maximum likelyhood estimator for the stadard deviatio as ˆσ = (X i X) Remark: Notice that the MLE is biased sice E[ ˆσ ML] = σ Example 4. Suppose we wat to estimate the parameters of a Gamma(α, θ) distributio The maximum likelihood equatios are f(x; α, θ) = θ α Γ(α) xα e x/θ 0 = log θ + 0 = αθ log X i Γ (α) Γ(α) Solvig these equatios i terms of the parameters we get X i 5 ˆθ = Xˆα 0 = log ˆα log X + log X i Γ (ˆα) Γ(ˆα). Notice that the secod equatio is a oliear equatio which caot be solved explicitly!i order to solve it we eed to resort to umerical iteratio scheme. To start the iterative umerical procedure we may use the iitial value obtaied from the method of momets. Propositio. Uder appropriate smoothess coditios o the pdf f, the maximum likelihood estimator is cosistet.
6 6 Proof. We will oly give a outlie of the proof, which, evertheless, presets the ideas. We begi by observig that by the Law of Large Numbers, as teds to ifiity, we have that L(θ) = log f(x i ; θ) E log f(x; θ) = log f(x; θ)f(x; θ 0 ) dx I the above θ 0 is meat to be the real value of the parameter θ of the distributio. The MLE will ow try to fid the ˆθ that maximises L(θ)/. By the above covergece, we have that this should the be approximately the value of θ that maximises E log f(x; θ). To maximise this differetiate with respect to θ to get that θ f(x; θ)/ θ log(f(x; θ)) f(x; θ 0 )dx = f(x; θ 0 )dx. f(x; θ) Settig θ = θ 0 i the above we get that is is equal to θ f(x; θ 0)dx = f(x; θ 0 )dx = 0. θ Therefore θ 0 maximisises the E[log f(x; θ)] ad therefore the maximiser of the loglikelihood fuctio will approach, as grows, to the value θ Comparisos. We itroduced two methods of estimatio: the method of momets ad the maximum likelihood estimatio. We eed some way to compare the two methods. Which oe is more likely to give better results? There are several measures of the efficiecy of the estimator. Oe of the most commoly used is the mea square error (MSE). This is defied as follows. Suppose, that we wat to estimate a parameter θ, ad we use a estimator ˆθ = ˆθ(X,..., X ). The the mea square error is defied as ] E [(ˆθ θ). Therefore, oe seeks, estimators that miimise the MSE. Notice that it holds ] ( E [(ˆθ θ) = E[ˆθ] θ) + Var(ˆθ). If the estimator ˆθ is ubiased, the the MSE equals the Var(ˆθ). So havig a ubiased estimator may reduce the MSE. However this is ot ecessary ad oe should be willig to accept a (small) bias, as log as the MSE becomes smaller. The sample mea is a ubiased estimator. Moreover it is immediate (why?) that Var(ˆµ) = σ where σ is the variace of the distributio. Therefore, the MSE of the sample mea is σ /.
7 I the case of a maximum likelihood estimator of a parameter θ we have the followig theorem Theorem. Uder smoothess coditios o f, the probability distributio of I(θ0 )(ˆθ θ 0 ) teds to stadard ormal. Here [ ( ) ] I(θ) = E log f(x; θ) θ [ ] = E log f(x; θ) θ We will skip the proof of this importat theorem. The reader is refered to the book of Rice. This theorem tells us that the maximum likelihood estimator is approximately ubiased ad that the mea square error is approximately /I(θ 0 ). A way to compare the efficiecy of two estimators, say ˆθ ad θ we itroduce the efficiecy of ˆθi terms of θ as eff (ˆθ, θ) = Var( θ) Var(ˆθ). Notice that the above defiitio makes sese as a compariso measure betwee estimators that are ubiased or that have the same bias..4. Exercises.. Cosider the Pareto distributio with pdf f(x) = aca, x > c. xa+ Compute the maximum likelihood estimator for the parameters a, c.. Cosider the Gamma distributio Gamma(α, θ). Write the equatios for the maximum liekliehood estimators for the parameters α, θ. Ca you solve them? If you caot solve them directly, how would you proceede to solve them? 3. Compute the mea of a Poisso distributio with parameter λ. 4 Cosider a Gamma distributio Gamma(α, θ). Use the method of momets to estimate the parameters α, θ of the Gamma distributio. 5. Cosider the distribtuio f(x; α) = + αx, < x <. The parameter α lies i betwee ±. A. Use the method of momets to estimate the parameter α. 7
8 8 B. Use the maximum likelihood method to estimate α. If you caot solve the equatios explai why is this ad describe what would you do i order to fid the MLE. C. Compare the efficiecy betwee the two estimators. 6. Cosider the problem of estimatig the variace of a ormal distributio, with ukow mea from a sample X, X,..., X, of i.i.d ormal radom variables. I aswerig the followig questio use the fact ( see Rice, Sectio 6.3 ) that ( )s σ χ ad that the mea ad the variace of a chi-square radom variable with r degrees of freedom is r ad r, respectively. A. Fid the MLE ad the momet-method estimators of the variace. Which oe is ubiased? B. Which oe of the two has smaller MSE? C. For what values of ρ does the estimator ρ (X i X) has the miimal MSE?
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