ECON 4130 Supplementary Exercises 1-4

Transcription

1 HG Set. 0 ECON 430 Sulementary Exercises - 4 Exercise Quantiles (ercentiles). Let X be a continuous random variable (rv.) with df f( x ) and cdf F( x ). For 0< < we define -th quantile (or 00-th ercentile), x, as the solution of PX ( x) = or, with other words, the solution of F( x ) x 0.5 F (0.5) =. Hence x = F ( ). Then, e.g., = is the median and x 0.5 the lower quartile. ( If the equation F( x ) = has several solutions, then x is by convention chosen as the smallest one. This may occur when F( x ) is flat in some regions as will be the case when X is discrete.) a. Suose X is exonentially distributed with arameter λ ( df f( x) = λe λx, x > 0, λ > 0 ). Find x as a function of λ. This distribution is right skewed (i.e. has a heavy tail to the right). For such distributions usually the exected value is larger than the median. Illustrate this by calculating the median and the exectation for X when λ = 3. b. Based on a samle of (total) incomes from n = 4867 men and n = 486 women drawn from the Norwegian oulation in 998, a number of quantiles of the income distributions for men and women have been estimated. The results are given in the table. Table. Estimated quantiles of income distributions for men and women. Norway 998 x Men (NOK 000) x Women (NOK 000) Based on the quantiles, draw a histogram for the incomes in the samle both for men and for women. [Hint: The class limits are given by the x s. Therefore the length of the intervals will vary. The relative frequency of the intervals must then be equal to the area

2 of the corresonding boxes of the histogram. Note that the relative frequency of, e.g., the interval ( x0., x 0.4) is 0. etc.] Exercise (Pareto Distribution ) (Note: Section a and d are based on Problem in the exam aer 003 H ) Vilfredo Pareto (848-93) claimed that incomes often are distributed according to a articular form, at least incomes above a certain limit ( the uer tail ). This tye of distribution is called the Pareto distribution. Let b denote the lower limit of incomes considered. The density of the Pareto distribution is defined by f( x) = θb ( x/ b) θ for x> b, and f( x ) = 0 for x b. Here b is a chosen cut-off oint (usually known) and θ > 0 is a arameter that determines the form of the distribution (i.e. the length of the uer tail before it gets negligible). a. Let X be Pareto distributed as described. Show that that the cdf is Fx ( ) = ( bx) θ for x> b. What is the value of F( x ) for x b? Find E(X). Note that E(X) does not exist if θ. Why? Illustrate by considering the case θ =. b. Show that the -th quantile is given by ln( x ) ln( b) θ ln( ) = ). x b θ = ( ) (or c. Let 0., 0.78, 0.54, 0.98 be 4 observations drawn (i.e. simulated by a comuter) from the uniform distribution over [0, ]. Transform these observations to 4 observations drawn (i.e. simulated) from the Pareto distribution with b = NOK and θ =.8. y d. Show that Y = ln( X b) is exonentially distributed with df fy ( y) = θe θ for y > 0. Exercise 3 (Lorenz Curve) Income distributions are sometimes studied by the so called Lorenz curve, L( ) 0, that says something about the income inequality in the oulation. The

3 3 (emirical) Lorenz curve based on a samle of incomes drawn from the oulation tells us, for given, the ercentage that the total sum of incomes from the 00% in the samle with lowest income constitutes of the total sum of incomes from the comlete samle. See figure. Fig. Lorentz Curve, L() For a theoretical definition consider an income distribution given by the df, f( x ), concentrated on the ositive axis, and let the rv X be distributed according to f. Then X reresents the income of a randomly drawn member from the oulation. Let x be the -th quantile of X, and ut µ = E( X ) = xf ( x) dx. The Lorenz curve is defined by 0 () x L( ) = xf ( x) dx µ 0 [Rationale: Consider a finite oulation with N members and an income distribution close to f. Then the mean income in the oulation is (close to) µ and the total sum of incomes is Nµ. We now need an exression for the sum of incomes that are lower than a cut-off oint, c. Let Y be X

4 4 truncated at c, in the sense that Y = X when X c and Y = 0. when X > c. Then the total sum of incomes in the oulation that are less or equal to c is N E( Y). Now Y is a mixed rv, i.e. artly discrete and artly continuous (note that PY ( = 0) = P( X> c) > 0 ). The theory for finding the exectation of such a rv is not covered in this course, but it can be shown that E( Y ) = 0 P( Y = 0) + xf ( x) dx = xf ( x) dx c 0 0 c. Hence, the sum of incomes that are at most c, divided by the total sum of incomes in the oulation is NE( Y) E( Y) = = xf ( x) dx Nµ µ µ ] c 0 a. Suose that X is Pareto distributed as defined in exercise where we assume that θ > so that E(X) exists. Show that the (theoretical) Lorenz curve in this case is given by L( ) = ( ) θ b. One ossible measure of the degree of income inequality in the oulation is the so called Gini coefficient, G, which is defined as the shaded area in figure divided by the area of the triangle ABC, i.e. G is equal to times the shaded area. Hence 0 G. Under what circumstances is G = 0? Show that for the Pareto distribution G = θ c. For the Norwegian income data from 998, referred to in exercise, it turns out that the Pareto distribution fits satisfactorily well for incomes above NOK , both for women and for men. Assume b = Then (maximum likelihood) estimates for θ based on these data are: Women: ˆ θ = 3.83 Men: ˆ θ =.83 Calculate corresonding estimates for the Gini coefficient and comment shortly on the results. Exercise 4 (Chi-square distribution) One class of distributions that occurs very often in alied statistics is the chi-square distribution with d degrees of freedom (d =,, 3,..), (written in short χ d ). The df is given by:

5 5 f( z) = z e d d Γ( ) d z for z > 0 f( z ) = 0 for z 0 a. Suose the rv Z is distributed according to this distribution (written Show that i) E( Z) = d ii) var( Z) = d Z ~ χ d ). d r r Γ ( + r) iii) E( Z ) = d Γ( ) for any real r > 0 (even non-integers) d (Hint: Identify Z as a gamma distributed rv (i.e. Z ~ Γ (, ) ), and use the formula, derived in the lectures, r Γ ( α + r) E( X ) = r λ Γ( α) for any real r > 0, when X ~ Γ ( αλ, ). ) b. Suose that X, X,, Xn are indeendent and identically distributed (X ~ iid) with exectation, E( X i ) = µ and variance, var( X i ) = σ. From the introductory statistics course we know that an unbiased estimator of σ is S X X (i.e. n = ( i ) n i= E( S ) = σ ) where X is the samle mean. Show that the samle standard deviation, S = S, under-estimates σ, i.e. E( S) < σ. (Hint: Use the result that for any rv, Y, that is not a constant, we have 0 < var( Y) = E( Y ) E( Y). Use this with Y = S ). [ ] c. Hence S is a biased estimator for σ (the bias can be shown to be negligible for large n in most cases). We will calculate the bias in a secial case. Write E( S) = cσ where c is a constant with 0< c <. Calculate c for the secial case that n = 0 and the X s are normally distributed (i.e. X ~ N ( µσ, )). i i

6 6 (Hint: Use the famous theorem (see e.g. rule 58 in Løvås, Statistikk for universiteter ( n ) S og høgskoler ) that says that, under these conditions, V = is chi-square σ σ distributed with n- (= 9 in this case) degrees of freedom. Note that S = V. ) n