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nonparametric statistical inference

Probability And Statistical Inference 2nd Edition Robert Bartoszynski, Magdalena Niewiadomska-Bugaj - Solutions

13.5.2 Let X be a single observation from a distribution with density f(z0,) =1 - 02(z - 0.5)3 for 0 < z < 1 and zero otherwise, -1 < 8 < 1. Find a UMP unbiased test of NO : 8 = 0 against HI : 8 # 0.
13.5.1 Suppose that XI, . . . , X, is a random sample from the U[O, 61 distribution.Test hypothesis HO : 8 = 80 against the two-sided alternative H I : O # 80 using an unbiased test that rejects HO if X,:, < c1 or X,:, > c2. Find c1 and c2 if B0 = 5 , n = 10, and Q = 0.05.
13.4.10 A reaction time to a certain stimulus (e.g., time until solving some problem)is modeled as a time of completion of T processes, running one after another in a specified order. The times TI, . . . , 7,. of completion of these processes are assumed to be iid exponential with mean 1 / X . If T
13.4.9 Let XI,. . . , X, be a random sample from the distribution f(2;8 ) = C[8/(8+1)Iz, where z = 1 , 2 , . . . and C is the normalizing constant. Determine a UMP test of the hypothesis HO : 8 = 00 against the alternative H1 : 8 > 00.
13.4.8 Recall Problem 13.3.8. Assume that R = 50 and that a student with a score at most 30 will fail. Does there exist a UMP test for the hypothesis HO : 0 5 30? If yes, find the test; if no, justify your answer.
13.4.7 Let X I , . . . , X, be a random sample from the GAM(a, A) distribution. (i)Derive a UMP test for the hypothesis HO : a 5 QO against the alternative HI : Q >QO if X is known. (ii) Derive a UMP test for the hypothesis HO : X 5 XO against the alternative H1 : X > XO if a is known.
13.4.6 Suppose that XI,. . . , X, is a random sample from the U[O, 81 distribution.(i) Hypothesis HO : 8 5 80 is to be tested against the alternative H I : 8 > 80.Argue that the UMP test rejects HO if X,:, >c. Find c for 80 = 5 ; n = 10, and Q = 0.05. (ii) If the hypothesis HO : 6 2 00 is tested
13.4.5 The effectiveness of a standard drug in treating specific illness is 60%. A new drug was tested and found to be effective in 48 out of 70 cases when it was used. Specify an appropriate alternative hypothesis and perform the test at the 0.01 level of significance. Find the p-value.
13.4.4 Suppose that the number of defects in magnetic tape of length t (yards) has POI(Xt) distribution. (i) Assume that 2 defects were found in a piece of tape of length 500 yards. Test the hypothesis HO : X 2 0.02 against the alternative H1 :X < 0.02. Use a UMP test at the level Q 5 0.01. (ii)
13.4.3 Let XI,. . . , X , be a random sample from a folded normal distribution with density f(s;6 ) = m 6 e x p {- O2s2/2), for 2 > 0,6 > 0. (i) Derive the UMP test for Ho : 6 = 60 against H I : 6’ > 60. (ii) Show that the power function is increasing.
13.4.2 Let X I , . . . , X , be a random sample from Laplace distribution with density f(z; A) = (X/2) exp{-X/z(}. Find a UMP test for testing Ho : X = 1 against H1 : X < 1 at the significance level 0.01.
13.4.1 Check whether the following families of distributions have a monotone likelihood ratio in the parameter specified: (i) Poisson. (ii) Exponential. (iii) Gamma, for each parameter separately. (iv) Beta, for each parameter separately.
13.3.9 Let X1 , . . . , X, have a joint density f(x;O) , and let U be a sufficient statistic for 0. Show that the most powerful test of HO : 0 = Oo against H1 : 6 = O1 can be expressed in terms of U .
13.3.8 A multiple-choice exam gives five answers to each of its n questions, only one being correct. Assume that a student who does not know the answer chooses randomly and is correct with probability 0.2. Let 8 be the number of questions to which the student knows the answers, and let X be the
13.3.7 An urn contains six balls, T red and 6 - T blue. Two balls are chosen without replacement. Find the most powerful test of Ho : T = 3 against the alternative H1 : T = 5 , with a size as close to Q = 0.05 as possible. Find the probability of a type I1 error for all T # 3.
13.3.6 The sample space of a test statistic X has five values:a, b,c, d,e. Test the HO : f = fo against Ha : f = fl, where distributions fo and f1 are given by the table X a b C d e fo 0.2 0.2 0.0 0.1 0.5 f i 0.2 0.4 0.3 0.0 0.1
13.3.5 Let X I ,. . . , X , be a random sample from EXP(X) distribution. Null hypothesis Ho : X = XO is tested against the alternative H1 : X = XI, where A1 > XO.Compare the power functions of the two tests: (a) the most powerful test, and (b)the most powerful test based on the statistic XI:^.
13.3.4 Assume that X has a N(2, a2) distribution. Find the best critical region for testing Ho : o2 = 2 against: (i) : o2 = 4. (ii) H I : a2 = 1.
13.3.3 Let X have a negative binomial distribution with parameters r and p . Find the most powerful test of HO : r = 2, p = 1/2 against H I : T = 4,p = 1/2 at significance level Q = 0.05. Find probability of type I1 error. Use randomized test if necessary.
13.3.2 A single observation X is taken from a BETA(a,b) distribution. Find the most powerful test ofthe null hypothesis Ho:a = b = 1, against the alternative H I :(i) a = b = 5. (ii) a = 2, b = 3 (iii) a = b = 1/2. Use significance level Q = 0.05.
13.3.1 Let XI, . . . , Xl o be a random sample from a POI(@ distribution. (i) Find the best critical region for testing Ho : 6 = 0.2 against H1 : 6 = 0.8 at the significance level Q = 0.05. (ii) Determine the size of the test in (i).
13.2.6 Let X I , . . . , Xg be a random sample from the N(p, 1) distribution. To test the hypothesis HO : p 5 0 against H1 : p > 0, one uses the test “reject HO if 3 5 x 5 5.” Find the power function and show that this is a bad test.
13.2.5 An urn contains five balls, T red and 5 - T white. The null hypothesis states that all balls are of the same color (i.e., Ho : T = 0 or T = 5 ) . Suppose that we take a sample of size 2 and reject Ho if the balls are of different colors. Find the power of this test for T = 0, . . . , 5 if
13.2.4 Let X I , X2 be a random sample of size 2 from the U[O, 6’1 distribution. We want to test HO : 6’ = 3 against H1 : 6’ = 2 (observe that Ho and H1 do not exhaust all possibilities). (i) HO will be rejected if x
13.2.3 Consider three tests C1, C2 , and C3 of the same hypothesis, performed independently(e.g., for each of these tests the decision is based on a different sample).Consider now the following three procedures:A: Reject HO only if all three tests reject it; otherwise, accept Ho, B: Reject Ho only
13.2.2 Consider the following procedure for testing the hypothesis HO : p 2 0.5 against the alternative H1 : p < 0.5 in BM(10, p ) distribution. We take observation X I , and reject HO if X1 = 0 or accept HO if X1 2 9; otherwise, we take another observation X2 (with the same distribution as X1 and
13.2.1 Let X 1 , X2 be a random sample from the U[O, 6’ + 11 distribution. In the test of No : 6’ = 0 against H1 : 6’ > 0, NO is rejected when X I + X2 > Ic. Find the power function of the test that has probability of the type I error equal 0.05
12.7.16 A sample of 200 trees in a forest has been inspected for a presence of some bugs, out of which 37 trees were found to be infested. (i) Assuming a binomial model, give a 90% confidence interval for the probability p of a tree being infested.Use the exact and the approximated formulas. (ii)
12.7.15 Suppose that the arrivals at a checkout counter in a supermarket (i.e., times of arriving at the counter or joining the queue, whichever is earlier) form a Poisson process with arrival rate A. Counting from noon, the thirteenth customer arrived at 12: 18 p.m. Find a 90% CI for: (i) A. (ii)
12.7.14 Based on a random sample X I , . . . , X, from the U[O, 81 distribution find:(i) P{X,:, < 0 < 2X,,,}. (ii) Such k that the interval (X,,,, kX,:,) is a (1 -a) 100% CI for 8.
12.7.13 Suppose that the lifetime T of a certain kind of device (e.g., a fuel pump in a car) has an EXP(X) distribution. The observed lifetimes of a sample of the devices are 350,727,615,155,962 (in days). Find a 95% CI’s for: (i) A. (ii) E(T). (iii) The standard deviation of the lifetime of the
12.7.12 Based on a random sample of size n selected from the WEI(0,4) distribution, derive a 95% confidence interval for 0 based on XI:,.
12.7.11 (i) Use the large sample distribution of MLE of mean A in Poisson distribution to construct an approximate (1 - a)-level CI for A. (ii) Assuming that the numbers of new cars of a given make sold per week in 15 consecutive weeks-5, 5, 6 , 3 , 5,8, 1,4, 7, 7, 5,4,3, 0,9-form a random sample
12.7.10 Suppose that the largest observation recorded in a sample of size n = 35 from a distributionuniform on [0,8] is 5.17. Find a 90% CI for 8.
12.7.9 Find the probability that the length of a 95% confidence interval for the mean of normal distribution with unknown u is less than u, n = 25.
12.7.8 Obtain a (1 - a)1 00% CI for 8 if XI,. . . , X , is a random sample from a:(i) N(p, 8) distribution with p and 8 unknown. (ii) BETA(1, 8) distribution.
12.7.7 Let XI , . . . , X, be a random sample from N(p, a’) distribution with both parameters unknown. Let L, be the length of the shortest confidence interval for p on confidence level 1 -a. (i) Find E ( L t ) as a function of n, u’ anda. (ii) Find the smallest n such that E(L:) 5 u2/2 for a
12.7.6 A large company wants to estimate the fraction p of its employees who participate in a certain health program. It has been decided that if p is below 25%, a special promotion campaign will be launched. In a random sample of 85 employees the number of those who participated in the program was
12.7.5 Seven measurements of the concentration of some chemical in cans of tomato juice are 1.12, 1.18,1.08,1 .13,1.14,1.10,1.07A. ssume that these numbers represent a random sample from the distribution N(0, a’). (i) Find the shortest 95% and 99% CI’s for 8, if u’ is unknown. (ii) Answer
12.7.4 Based on a random sample 1.23, 0.36, 2.13, 0.91, 0.16, 0.12 selected from the GAM(2.5,6) distribution, find an exact 95% CI for parameter 6.
12.7.3 Let X I , . . . , X , be a random sample selected from the Pareto distribution with density f(s;6 ) = 62es-(e+1), Find: (i) The sufficient statistic for 6. (ii) A 95% CI for 6.
12.7.2 Continuing Problem 12.5.10, obtain the 90% Bayesian interval for 6 and for/I = 1 / 6 i f n = 6 , 2 = 4,andp = 2.
12.7.1 Six randomly selected adults are asked if they favor additional taxes to help fund more affordable health care. Four of them respond “yes.” Assuming that the prior distribution is BETA( 1, l), determine the posterior density and obtain the probability that in the population the
12.6.11 Let XI , . . . , X, be a random sample from the U[0, 0+l] distribution. Show that statistic T = X,:, - XI:i,s ancillary for 0.
12.6.10 Let X1 , Xz be a random sample of size n = 2 from the EXP(X) distribution.Show that statistic T = X1/X2 is ancillary for A.
12.6.9 Let XI, . . . , X, be a random sample from the EXP(X) distributions. Suppose that only first k order statistics XI:,,. . . , XkZn are observed. Find a minimal sufficient statistic for A.
12.6.8 Suppose that a random sample is taken from a distribution with density f(z;0 ) = 2z/02 for 0 5 5 5 0 and f(z;0 ) = 0 otherwise. Find the MLE of the median of this distribution, and show that this estimator is a minimal sufficient statistic.
12.6.7 Show that the family of GAM(a, A) distributions is in an exponential class, and find the minimal jointly sufficient statistics.
12.6.6 Show that the following families of distributions are in the exponential class:(i) POI(X). (ii) EXP(X). (iii) NBIN(T, 0), T known. (iv) BETA(&, 02). (v) WEI(0, A), X known.
12.6.5 Let X I , . . . , X , be a random sample from the distribution with a density f(z;A, 0) = Ae-'("-@) for z 2 0 and 0 otherwise. Determine a pair of jointly sufficient statistics for parameters X and 0.
12.6.4 Show that the N(O,0) family is not complete.
12.6.3 Find a sufficient statistic for 0 if observations are uniformly distributed on the set of integers 0, 1, . . . , 0.
12.6.2 Generalizing Example 12.53, let X1 , . . . , X, be n independent Bernoulli trials. Show that T = xyZXl i is sufficient for probability of success p by finding the joint distribution of (XI , . . . X,) given T = t . Find the marginal distribution
12.6.1 Find sufficient statistic(s) for parameter 0 in the following distributions: (i)f(z,0) = (z/02)e-s2/2e2 for z > 0 (Rayleigh). (ii) f(z,0) = (1/20).~-1~1'(double exponential). (iii) BETA(0,20). (iv) U[0, 201.
12.5.24 Let XI,. . . , X, be a random sample from U[O, 8s 13 distribution. (i) Show that T = c( X,:, - 1) + (1 - c)X1,, 0 < c < 1, is the MLE of 0, and find the value of c that minimizes its MSE. (ii) Determine the asymptotic distribution of T .
12.5.23 For independent variables Y1, . . . , Y, with distribution N( a + psi, a2), where zl1 . . . , 5, are fixed, show that the LS-estimator and ML-estimator of 0 =(a,p ) coincide.
12.5.22 Y 1 , . . . , Y, are independent variables. Assuming that 2 1 , . . . ,s, are such that c(zi - T ) 2 > 0, compare the MSE's of the MLE and LS-estimators ofparameter 8, if: (i) Y , - EXP(6'zi). (ii) yi - POI(6'si).
12.5.21 Let XI,. . . , X, be a random sample from the N(p1,uT) distribution and let Y1, . . . , Y, be a random sample from the N(p2,og) distribution, with Xi's being independent from 5 ' s . Find the MLE of: (i) p1,p2,u2 if 01 = 02 = u. (ii)p , u:, ug where p1 = p2 = p.
12.5.20 Let X I , . . . , X, be a random sample from a log-normal distribution with a parameters p and u2 [this means that log X i " ( p , a2)]. Find the MLE of p and 6 2 .
12.5.19 Suppose that the median of 20 observations, taken from a normal distribution with an unknown mean and variance, is 5 and that only one observation differs from the median by more than 3. Suggest an estimate of the probability that the next two observations will both be between 4 and 5.
12.5.18 For R. observations taken from the U[O, 6'1 distribution, let U, be the number of the ones that are less than 3. Find the MLE of 0.
12.5.17 Find the MLE of the mean of a U[&, 6'21 distribution based on a random sample of size n.
12.5.16 Let XI,X2 be a random sample from a N( p ,a 2)d istribution with p and a 'unknown. Find the MLE of a2 if the only available information is that the difference between observations equals 3.
12.5.15 Let XI,. . . , X, be a random sample from N( p , a') distribution, p is known.Find the MLE of a: (i) Directly. (ii) First finding the MLE of variance a2 and then using the invariance property.
12.5.14 Find the MME and MLE of the standard deviation of a Poisson distribution.
12.5.13 Let XI,. . . , X, be a random sample from POI(X) distribution. Find the MLE of X assuming that: (i) XI + . . + X, > 0. (ii) X1 + . . . + X, = 0.
12.5.12 Two independent Bernoulli trials resulted in one failure and one success.What is the MLE of the probability of success 0 if it is known that: (i) 6' is at most 1/4. (ii) 0 exceeds 1/4.
12.5.11 Suppose that there were 15 successes in 24 trials. Find the MLE of the probability of success 0 if it is known that 6' 5 1/2.
12.5.10 Let XI,. . . , X, be a random sample from the EXP(6') distribution, and let the prior distribution of 6' be EXP(,B). Find Bayes estimator of 6' and p = l/6' using:(i) Squared error loss. (ii) Absolute error loss.
12.5.9 Show that the family of gamma distributions provides conjugate priors for the exponential distribution. Determine the the posterior distribution.
12.5.8 Find the distributionof the MLE ofthe probability of success 6' based on two Bernoulli trials.
12.5.7 A single observation of a random variable X with a geometric distribution results in X = I;. Find the MLE of the probability of success 0 if: (i) X is the number of failures preceding the first success. (ii) X is the number of trials up to and including the first success.
12.5.6 Some phenomena (e.g., headway in traffic) are modeled to be a distribution of a sum of a constant and an exponential random variable. Then the density of X where a > 0 and b > 0 are two parameters. Find: (i) The MME of 6' = (a; b). (ii)The MLE of 0.
12.5.5 (Bragging Tennis Player) As in Example 12.34, consider tennis players A and B who from time to time play matches against each other. The probability that A wins a set against B is p .Suppose now that we do not have complete data on all matches between A and B; we learn only of A's victories,
12.5.4 Let X I , . . . , X, be a random sample from Poisson distribution with mean A.Find the MLE of P(X = 0 ) .
12.5.3 Let XI., . . , X, be a random sample from the distribution uniform on the union of the two intervals: [ -2, -11 and [0, 01. Find: (i) The MME of 0. (ii) The MLE of 0. (iii) The MLE of 0 if positive Xi’s are recorded exactly, and negative Xi’s can only be counted. (iv) The MLE of 6 if
12.5.2 Find the MME of parameter 0 in the distribution with density f(z,0) =(e + i ) ~ - ( ~for+ 5 ~> )1,,e > 0.
12.5.1 Let X1 , . . . , X , be a random sample from GAM(a, A) distribution. Find: (i)The MME of 0 = (a,A) , using the first two moments. (ii) The MME of a when A is known, and the MME of X when a is known.
12.4.7 Show that the estimator T = x satisfies relation (12.46) and determine functions y1 and 7 2 if a random sample XI,Xz ,. . . , X, is selected from: (i) N(0, a')distribution with D known. (ii) BIN(1,O) distribution.
12.4.6 Let X I , . . . , X , be a random sample from a Bernoulli distribution with an unknown p . Show that the variance of any unbiased estimator of (1 - p ) 2 must be at least 4p( 1 - ~ ) ~ / n .
12.4.5 Let X I , XZ be a random sample of size 2 from N(p, a2) distribution. Determine the amount of information about p and about c2 contained in: (i) X1 + X2.(ii) X1 - X2.
12.4.4 Find Fisher information I ( 0 ) in a random sample of size n from the Cauchy distributionwithdensity f ( s , 0 ) = {.[I + (z - 0)2]}-1.
12.4.3 Let XI,. . . , X , be a random sample from EXP(X) distribution. Propose an efficient estimator of 1/X and determine its variance.
12.4.2 Let X have EXP(X) distribution. Find the Fisher information I ( X )
12.4.1 Let x 2 k be the sample mean of 2k independent observations from a normal distribution with mean 0 and known variance r2. Find the efficiency of 7, (i.e., of estimator that uses only half of the sample).
12.3.8 Let X I , . . . , X4 be a random sample from U[O, 81 distribution. Compare the mean squared errors of four estimators of 8: TI = 5X1:4, T2 = (5/2)x2,4, T3 = (5/3)X3:4, andT4 = (5/4)X4:4.
12.3.7 Let U = XI:,, and let V = X,:" in a random sample from UIO - 1,B + 11 distribution. (i) Show that x and (V + V)/2 are both unbiased estimators of 8. (ii)Determine the MSE's of estimators in (i).
12.3.6 Let X1 , . . . , X, be a random sample from a N(p, a2) distribution ( p and a2 are unknown), and let be two estimators of a2, (i) Compare the MSE's of S2 and ST. (ii) Consider n i=l as estimators of u2 and find k for which Sz has smallest MSE. Explain why, in practice, the only values of k
12.3.5 Let X - 1, . . . , X, be a random sample of size n from the discrete distribution with probability function f ( z , O ) = O(1 - 0). , z = 0,1, . . .. Compare the MSE's of two estimators of 8: TI = x and T2 = [n/(n + l)]x.
12.3.4 Let XI, . . . , X , be n Bernoulli trials with probability of success 0, and let S = Cy=l Xi. Compare the mean squared errors of two estimators of 8: TI = S/n and T2 = (S + l)/(n + 2).
12.3.3 Let X I , . . . , X, be a random sample from EXP(l/e) distribution. Compare the mean squared errors of two estimators of 8: TI = x and T2 = [n/ (n+ l)]y.
12.3.2 Let XI,. . . , X , be a random sample from a N(B, a2)d istribution with g2 known. Show that the estimator T of 8, defined as T(X1: . . . , X,) = 3 (T = 3 regardless of the observations), is admissible.
12.3.1 Let TI and T2 be two unbiased estimators of 0 with variances of, a;, respectively.Find values a and b such that: (i) Estimator aT1 + bT2 is unbiased. (ii)Unbiased estimator aT1 + bT2 has a minimum variance assuming that TI and T2 are independent. (iii) Unbiased estimator aT1 + bT2 has a
12.2.5 Observations are randomly sampled from the U[O, e] distribution. After the sample size reaches n, the experimenter starts recording the minimum observations XI:,, X1:,+1 , . . . . He will continue until he gets X,+N such that x1:n = Xl:n+l = ' ' ' = Xl:n+N-l > Xl:n+N.Suggest an estimator of
12.2.4 Assume that the observations are taken from the U[O, e] distribution, and let U, be the number of observations (out of first n) that are less than 5. Show that if 6 > 5 , then Tn = 5 n / U , is a consistent estimator of 8.
12.2.3 The density of a Pareto distribution is f(z,a , 0 ) = aB'2z-('2'+f1o)r z 2 6'and equals 0 otherwise. Show that T = X I : , is a consistent estimator of 8.
12.2.2 Let X1 , . . . , X , be a random sample from the distributionwith density f(z;6 ')= e--(3-e) for z 2 0, and f(z, 0) = 0 otherwise. Check if T = X I : , is a consistent estimator of 0.
12.2.1 Show that the estimators 2'1, T2, and T3 in Example 12.4 are strongly consistent.
10.6.11 Let X , be a relative frequency of success in n Bernoulli trials. Use the Delta method to find the limiting distributionof g(X,) = X,(1 - X,).
10.6.10 Let x, be a sample mean in a random sample of size n from POI(X). Use the Delta method to find the limiting distribution of g(x=) & z(X - A).

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