#1 STA355H1F due Wednesay, October 5, 2016 Problems to hand in: 1. Suppose that X1 , , Xn are independent Exponential random variables with density f (x; ) = exp(x) for x 0 where > 0 is an unknown parameter. (a) Show that the quantile of the Exponential distribution is F 1 ( ) = 1 ln(1 ). (b) The form of the quantile function in part (a) can be used to give a quantile-quantile (QQ) plot to graphically assess whether the Exponential model is reasonable for a given data set. Specifically, if x(1) x(2) x(n) are the ordered data, we can plot x(k) versus ln(1 k ) (for k = 1, , n) where k = (k 3/8)/(n + 1/4) (or something similar); if the data are Exponential then the points should fall close to a straight line (with slope 1 and y-intercept 0). A simple R function to implement this Exponential QQ plot is given on Blackboard feel free to modify it as you wish. Using both Exponential (which can be generated in R by x <- rgamma(n,1) where n is the sample size) and non-Exponential data with different sample sizes, get a feel for how this QQ plot works. (Some suggestions are given on Blackboard.) Comment on the qualitative differences in the QQ plot between Exponential and non-Exponential data. (Later in the course, we will discuss some more \"objective\" approaches.) (c) Data on intervals (in hours) between failures of air conditioning units on ten Boeing aircraft are given (and analyzed) in Example T of Cox & Snell1 (1981). These data (198 observations) are provided in a file on Blackboard. Use the QQ plot from part (b) to assess whether an Exponential distribution is a reasonable model for these data. 2. (a) Suppose that X has a Gamma distribution with shape parameter and scale parameter ; the density of X is f (x) = x1 exp(x) for x > 0 () Find expressions for the skewness and kurtosis of X in terms of and . (Do these depend on ?) What happens to the skewness and kurtosis as ? (b) Suppose that X1 , , Xn are independent and define Sn = X1 + + Xn . Assuming that E(Xi3 ) is well-defined for all i, show that the skewness of Sn is given by skew(Sn ) = n X i=1 1 i2 !3/2 n X i3 skew(Xi ) i=1 Cox, D.R. and Snell, E.J. (1981) Applied Statistics: Principles and Examples. Chapman and Hall, New York. where i2 = Var(Xi ). (Hint: Follow the proof given for the kurtosis identity assuming for simplificity that E(Xi ) = 0; this is more simple since E(Sn ) involves a triple summation, most of whose terms are 0.) Supplemental problems (not to be handed in): 3. Suppose that X1 , , Xn are independent random variables with distribution function F where = E(Xi ) and 2 = Var(Xi ). For some families of distributions, the variance is a function of the mean so that 2 = 2 (). A function g is said to be a variance stabilizing transformation for the family of distributions if d n ) g()) N (0, 1) n(g(X (a) Show that g defined above must satisfy the differential equation g 0 () = 1 . () (Note that g is not unique.) (b) Find variance stabilizing transformations for (i) Poisson distributions; (ii) Exponential distributions; (iii) Bernoulli distributions. 4. Suppose that X1 , , Xn are independent random variables with some continuous distribution function F . Given data x1 , , xn (outcomes of X1 , , Xn ), we can make a boxplot to graphically represent the data observations beyond the \"whiskers\" (which extend to at most 1.5 interquartile range from the upper and lower quartiles) are flagged as possible outliers. When n is large enough, we can obtain a crude estimate for the expected number of outliers as follows: (i) Compute the lower and upper quartiles of F , F 1 (1/4) and F 1 (3/4) and define IQR = F 1 (3/4) F 1 (1/4). (ii) Compute the probability of an outlier by F (F 1 (1/4) 1.5 IQR) + 1 F (F 1 (3/4) + 1.5 IQR) (iii) The expected number of outliers is simply n times the probability in part (ii). Compute the expected number of outliers for the following distributions. (a) Normal distribution - note that the probability in (ii) will not depend on the mean and variance so you can assume a standard normal distribution. (The R functions pnorm and qnorm can be used to compute the distribution function and quantiles, respectively, for the normal distribution.) (b) Laplace distribution with density f (x) = 1 exp(|x|). 2 (No R functions for the distribution functions and quantiles seem to exist for the Laplace distribution. However, both are easy to evaluate analytically.) (c) Cauchy distribution with density f (x) = 1 (1 + x2 ) (The R functions pcauchy and qcauchy can be used to compute the distribution function and quantiles, respectively, for the Cauchy distribution.) (d) Comment on the differences between the 3 distributions considered in parts (a)-(c). In particular, how does the proportion of outliers change as the \"tails\" (i.e. the rate at which f (x) goes to 0 as |x| ) of the distributions change? 5. Suppose that X1 , X2 , is a sequence of independent random variables with mean and n = n1 (X1 + + Xn ). Describe the limiting behaviour (that variance 2 < ; define X is, either convergence in probability or convergence in distribution as well as the limit as n ) of the following random variables. n 1 X n )2 . (a) Sn2 = (Xi X n 1 i=1 n )/Sn . (b) n(X n ) exp())/Sn . (c) n(exp(X n X 1 n |. (The limit here should be intuitively clear; however, proving it is not (d) |Xi X n i=1 easy!) d 6. Suppose that an (Xn ) Z (where an ) and that g(x) is an infinitely differentiable function (that is, it has derivatives of all orders). The Delta Method says that d an (g(Xn ) g()) g 0 ()Z; p if g 0 () = 0 then the right hand side above is 0 and so an (g(Xn ) g()) 0. (a) Suppose that g 0 () = 0 and g 00 () 6= 0. Use the Taylor series expansion 1 g(x) = g() + g 0 ()(x ) + g 00 ()(x )2 + rn 2 (where rn /(x )2 0 as x ) to find the limiting distribution of a2n (g(Xn ) g()). (b) Extend the result of part (a) to the case where g 0 () = g 00 () = = g (k1) () = 0 but g (k) () 6= 0 (g (k) denotes the k-th derivative of g)