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Stochastics Introduction To Probability And Statistics 2nd Edition Hansotto Georgii - Solutions

The influence of vitamin C on the frequency of a cold is to be tested. Therefore, 200 test subjects are randomly split into two groups, of which one receives a certain dose of vitamin C and the other a placebo. The resulting data are the following:For the level 0.05, test the null hypothesis that
It is suspected that in a horse race on a circular racing track, the starting positions affect the chance of winning. The following table classifies 144 winners according to their starting positions (where the starting positions are numbered from the inside to the outside).Test the null hypothesis
S Test against decreasing trend. Consider the infinite product model for the 2-test of goodness of fit, and let be the uniform distribution on E D ¹1; : : : ; sº. Suppose the null hypothesis H0 W # D is not to be tested against the full H1 W # 6D , but only against H0 1W #1 > #2 > >
Tendency towards the mean. Teachers often face the suspicion that when awarding grades they tend to avoid the extremes. In a certain course, 17 students obtain the following average grades (on a scale from 1 to 6, where 1 is best):For simplicity, assume that these average grades are obtained from a
S In the setting of Section 11.2, let c > 0 be a fixed threshold, a probability density on E, and ¹Dn; cº the acceptance region of the associated 2-test of goodness of fit after n observations. Show thatfor all # ¤ , which means that the power tends to 1 at an exponential rate. Hint: Combine
An algorithm for generating pseudo-random numbers is to be tested. So one performs an experiment, in which the algorithm produces, say, n D 10 000 digits in ¹0; : : : ; 9º. Suppose the following frequencies are observed:Choose an appropriate level and perform a 2-test of goodness of fit for a
Let E D ¹1; 2; 3º and be the density of the uniform distribution on E. Determine, for c1; c2 2 R and large n, a normal approximation of the multinomial probability Mn;` 2 ZE C W `.1/ `.2/ c1 pn=3; `.1/ C `.2/ 2 `.3/ c2 pn=3:
The ‘portmanteau theorem’ about convergence in distribution. Let .E; d/ be a metric space equipped with the -algebra E generated by the open sets, .Yn/n1 a sequence of random variables on a probability space . ;F; P/ taking values in .E; E /, and Q a probability measure on .E; E /. Prove the
S p-value and combination of tests. Consider all tests with a rejection region of the form¹T > cº for a given real-valued statistic T . Suppose that the distribution function F of T does not depend on # on the null hypothesis ‚0, in that P#.T c/ D F.c/ for all # 2 ‚0 and c 2 R. In
S Two-sample problem in the Gaussian product model. Let X1; : : : ; Xk, X0 1; : : : ; X0 l be independent random variables with distribution Nm;v and Nm0;v , respectively; the parameters m;m0 and v are assumed to be unknown. Show that each likelihood ratio test for the test problem H0 W m m0
A supplier of teaching materials delivers a set of electric resistors and claims that their resistances are normally distributed with mean 55 and standard deviation 5 (each measured in Ohm). Specify a test of level ˛ D 0:05 for each of the two test problems(a) H0 W m 55 against H1 W m > 55,(b)
S Approximate power function of the t -test. Let 'n be the t -test for the one-sided test problem H0 W m 0 against H1 W m > 0 at a given level ˛ in the two-parameter n-fold Gaussian product model. Show the following: For large n, the power function of 'n admits the normal approximation G'n.m; v/
Let ' be a one-sided or two-sided t -test for the expectation in the two-parameter Gaussian product model. Express the power function G'.m; v/ of ' in terms of the non-central t -distributions defined in Problem 9.15.
S Let ' be the one-sided t -test of Theorem (10.18). Show that its power function is monotone in m, so that ' is indeed an unbiased test of size ˛. Hint: Problem 2.15.
Consider the two-parameter Gaussian product model and show that a uniformly most powerful test for the one-sided test problem H0 W m 0 against H1 W m > 0 does not exist.
S Two-sided chi-square variance test. In the two-parameter Gaussian product model, consider the two-sided test problem H0 W v D v0 against H1 W v ¤ v0 for the variance. It is natural to consider a decision rule that accepts H0 if c1 n1 v0 V c2 for suitably chosen c1 and c2.(a) Calculate the
Two-sided binomial test. Construct a two-sided binomial test of level ˛, that is, a test in the binomial model of the null hypothesis H0 W # D #0 against the alternative H1 W # ¤ #0, where 0 < #0 < 1. Also deduce an asymptotic version of this test by using the de Moivre–Laplace theorem (5.22).
A newspaper chooses the winner of its weekly competition by drawing (with replacement)from the post cards received, until a correct answer is found. Last week, 7 cards had to be checked; so the responsible editor suspects that the proportion p of correct solutions was less than 50%, implying that
Testing the life span of appliances. Consider the n-fold product of the statistical model.0;1OE;B0;1OE;Q# W # > 0/, where Q# is the Weibull distribution with known exponentˇ > 0 and unknown scale parameter # > 0; recall Problem 3.27. That is, Q# has the Lebesgue density#.x/ D #ˇ xˇ1 exp OE#
S Optimality of the power function on the null hypothesis. Let ‚ R and .X;F; P# W# 2 ‚/ be a statistical model with increasing likelihood ratios relative to a statistic T .For given #0 2 ‚, let ' be a uniformly most powerful level-˛ test of the null hypothesis H0 W # #0 against the
In the situation of Problem 7.1, determine a uniformly most powerful test of level ˛ D 0:05 for the null hypothesis that the radiation burden is at most 1, based on n D 20 independent observations. Plot the power function (using a suitable program).
S Normal approximation for Neyman–Pearson tests. Let .E; E IQ0;Q1/ be a statistical standard model with simple null hypothesis and alternative and strictly positive densities 0, 1. Consider the negative log-likelihood ratio h D log. 0= 1/ and assume that its variance v0 D V0.h/ exists. In the
Consider the situation of Example (10.5), where the proper functioning of a satellite was tested. The satellite manufacturer has the choice between two systems A and B. The signal-to-noise ratio of system A is m.A/1 =p v D 2, at a price of e 105. System B, with a ratio of m.B/1 =p v D 1, costs only
Among 3000 children born in a hospital, 1578 were boys. On the basis of this result, would you, with 95% certainty, insist on the null hypothesis that newborns are male with probability 1/2?
S Minimax-Tests. Consider a binary standard model .X;FIP0; P1/. A test ' of P0 against P1 is called a minimax test, if the maximum of the type I and type II error probabilities is minimal. Establish the existence of a Neyman–Pearson test ' with E0.'/ D E1.1'/, and show that it is a minimax test.
Bayes tests. Let ' be a test of P0 against P1 in a binary standard model .X;FIP0; P1/, and let ˛0; ˛1 > 0. Show that ' minimises the weighted error probability ˛0 E0.'/ C˛1 E1.1'/ if and only if ' is a Neyman–Pearson test with threshold value c D ˛0=˛1.Such a ' is called a Bayes test with
S Neyman–Pearson geometry. In the situation of the Neyman–Pearson lemma (10.3), let G.˛/ WD sup®E1. / W a test with E0. / ˛¯be the greatest possible power that can be achieved for the level 0 < ˛ < 1. Show the following:(a) G is increasing and concave.(b) If ' is a Neyman–Pearson
On the basis of n draws in the lottery ‘6 out of 49’, it is to be tested whether ‘13’ is an unlucky number, because it appears less frequently than expected. Formulate the test problem and specify a most powerful test of approximate level ˛ D 0:1 (use the normal approximation of binomial
During a raid, the police searches a gambler and discovers a coin, of which another player claims that ‘heads’ appears with a probability of p D 0:75 instead of p D 0:5. Because of time restrictions, the coin can only be checked n D 10 times. Choose a null hypothesis and an alternative
S Specify a most powerful test of H0 W P D P0 against H1 W P D P1 of level ˛ 2 0; 1=2OE in each of the following cases:(a) P0 D U0;2OE, P1 D U1;3OE.(b) P0 D U0;2OE, P1 has the density 1.x/ D x 10;1.x/ C 1 2 1OE1;2OE.x/.
A delivery of 10 appliances contains an unknown number of defective items. Unfortunately, a defect can only be discovered by a very expensive quality check. A buyer who is only interested in a completely flawless delivery performs the following inspection upon delivery.He checks 5 appliances. If
Testing the range of a uniform distribution. In the model .Rn;Bn; UOE0;#˝n W # > 0/of Example (7.3), consider the test problem H0 W # D 1 against H1 W # ¤ 1. Determine the power function of the test with acceptance region® 1 2 < max¹X1; : : : ; Xnº 1¯.
Relation between confidence regions and tests. Let .X;F; P# W # 2 ‚/ be a statistical model. Show the following:(a) If C W X ! P.‚/ is a confidence region with error level ˛ and #0 2 ‚ is chosen arbitrarily, then ¹#0 62 C./º is the rejection region of a test of H0 W # D #0 against H1 W #
S Sequential confidence intervals of given maximal length, Charles M. Stein 1945. Let X1;X2; : : : be independent, Nm;v-distributed random variables with unknown parameters m and v. Furthermore, let n 2 and 0 1I1˛=2 the ˛=2-fractile of the tn1-distribution. Consider the unbiased estimatorsof m
(a) Let X1; : : : ; Xn be independent, Nm;v-distributed random variables with known m and unknown v. Determine a confidence interval for v with error level ˛.(b) Two-sample problem with known expectations. Let X1; : : : ; Xk, Y1; : : : ; Yl be independent random variables, where the Xi are
S Two-sample problem in Gaussian product models with known variance. Consider independent random variables X1; : : : ; Xn, Y1; : : : ; Yn, where each Xi has distribution Nm;v and each Yj distribution Nm0;v . The expectations m;m0 are assumed to be unknown, but the common variance v > 0 is known.
Confidence region in the Gaussian product model. Consider the n-fold Gaussian product model with unknown parameter # D .m; v/ 2 R 0;1OE . Let ˛ 2 0; 1OE be given and define ˇ˙ D .1 ˙p 1 ˛/=2. Also, let u D ˆ1.ˇC/ be the ˇC-quantile of N0;1 and c˙ D 2 n1Iˇ˙the ˇ˙-quantile of 2n1. Show
Non-central Student distributions. The non-central t -distribution tn; with n degrees of freedom and non-centrality parameter > 0 is defined as the distribution of the random variables T D Z=p S=n for independent random variables Z and S with distribution N;1 and 2n, respectively. Show that tn;
Non-central F-distributions. Let m; n 2 N, a1; : : : ; am 2 R, D Pm iD1 a2 i , and X1; : : : ; Xm, Y1; : : : ; Yn be i.i.d. standard normal random variables. Show that the random variablehas the distribution densityfor y 0. The corresponding probability measure is called the non-central
Approximation of t - and F-distributions. Let c 2 R, 0 < ˛ < 1 and m 2 N be given.Show that(a) tn.1; c/ ! ˆ.c/ and tnI˛ ! ˆ1.˛/,(b) Fm;n.OE0; c/ ! 2m.OE0; mc/ and fm;nI˛ ! 2 mI˛=m as n ! 1. Here, tnI˛, fm;nI˛, 2 mI˛, and ˆ1.˛/ are the ˛-quantiles of tn, Fm;n, 2m, and N0;1,
S Fisher’s approximation. Suppose that Sn has the chi-square distribution 2n. Use Problem 5.21 to show that p2Sn p 2n !d N0;1. Deduce that p22 nI˛p 2n ! ˆ1.˛/ as n!1. Use the tables in the appendix to compare this approximation with the one from the previous problem for the levels ˛ D
Approximation of chi-square quantiles. For 0 < ˛ < 1 and n 2 N, let 2 nI˛ be the˛-quantile of the 2 n-distribution and ˆ1.˛/ the ˛-quantile of the standard normal distribution.Show that .2 nI˛n/=p 2n ! ˆ1.˛/ as n ! 1. Hint: Use Problem 4.17 or Example (7.27b).
S Representation of non-central chi-square distributions. Let X;X1;X2; : : : be i.i.d. random variables with standard normal distribution N0;1, and let a; a1; a2; : : : 2 R. Show the following.(a) Y D .XCa/2 has the distribution density a.y/ D .2y/1=2 e.yCa2/=2 cosh.a py/ ; y > 0 :(b) The
Non-central chi-square distributions. Let a 2 Rn and X be an Nn.0; E/-distributed ndimensional random vector. Show that the distribution of jXCaj2 depends only on (n and)the Euclidean norm jaj ofa. This distribution is called the non-central 2 n-distribution with non-centrality parameter D jaj2.
Gamma and Dirichlet distributions. Let s 2, ˛ > 0, 2 0;1OEs and suppose that X1; : : : ; Xs are independent random variables with gamma distributions ˛;.1/; : : : ; ˛;.s/.Also, let X D Ps iD1 Xi . Show that the random vector .Xi=X/1is is independent of X and has the Dirichlet
Moments of the chi-square and t -distributions. Let Y and Z be real random variables with distributions 2n and tn, respectively.(a) Show that E.Y k=2/ D ..n k/=2/=OE.n=2/ 2k=2 for k < n.(b) Determine the moments of Z up to order n1 and show that the nth moment of Z does not exist. Hint: Use
S Balls of maximal Nn.0; E/-probability. Show that Nn.0; E/jX mj < r Nn.0; E/jXj < rfor all n 2 N, r > 0 and m 2 Rn; here, X stands for the identity mapping on Rn. That is, balls have maximal Nn.0; E/-probability when they are centred. Hint: You can use induction on n and Fubini’s theorem
S Normal distributions as maximum-entropy distributions. Let C be a positive definite symmetric nn matrix, and consider the class WC of all probability measures P on .Rn;Bn/with the properties F P is centred with covariance matrix C, that is, the projections Xi W Rn ! R satisfy E.Xi / D 0 and
Let X be an Nn.0; E/-distributed n-dimensional random vector, and let A and B be k n and l n matrices of rank k and l , respectively. Show that AX and BX are independent if and only if AB>D 0. Hint: Assume without loss of generality that kCl n. In the proof of the ‘only if’ direction, verify
Geometry of the bivariate normal distribution. Suppose C D v1 c c v2is positive definite, and let 0;C be the density of the corresponding bivariate centred normal distribution. Show the following.(a) The contour lines ¹x 2 R2 W 0;C.x/ D hº for 0 < h < .2p det C /1 are ellipses.Determine the
S Best linear prediction. Suppose that the joint distribution of the random variables X1; : : : ; Xn is an n-dimensional normal distribution. Show the following.(a) X1; : : : ; Xn are independent if and only if they are pairwise uncorrelated.(b) There exists a linear combination OXn WD Pn1 iD0
Find the distribution function and the distribution density of Y D eX when X has distribution(a) Nm;v, or (b) E˛. In case (a), the distribution of Y is called the lognormal distribution for m and v, and in case (b) the Pareto distribution for ˛.
Determining the speed of light. Consider the data collected by Michelson in order to determine the speed of light, see Problem 8.7. Which confidence intervals for the speed of light can you specify, individually for each series as well as jointly for all measurements, if you want to avoid any
S Normal approximation of order statistics. Let .Xi /i1 be a sequence of i.i.d. random variables with common distribution Q on an interval X. Assume that the distribution function F D FQ is differentiable on X with continuous derivative D F 0 > 0. Let 0 < ˛ < 1, q 2 X the associated ˛-quantile
S Law of large numbers for order statistics. Consider a sequence .Xi /i1 of i.i.d. random variables taking values in an interval X. Let Q be the common distribution of the Xi , and suppose that the distribution function F D FQ of Q is strictly increasing on X. Also, let 0 < ˛ < 1 and .jn/ be any
S Distribution density of order statistics. Let X1; : : : ; Xn be i.i.d. real random variables with continuous distribution density , and suppose that the set X WD ¹ > 0º is an interval.Determine the distribution density of the kth order statistic XkWn. Hint: Use either Problem 1.18 and Section
Sensitivity of sample mean, sample median and trimmed mean. Let Tn W Rn ! R be a statistic that assigns an ‘average’ to n real outcomes. The sensitivity function Sn.x/ D nTn.x1; : : : ; xn1; x/ Tn1.x1; : : : ; xn1/then describes how much Tn depends on a single outcome x 2 R when the
Take the data from Example (8.6) to plot the empirical distribution function, i.e., the distribution function FL of the empirical distribution L. Determine the sample median and the sample quartiles and draw the corresponding box plot.
S Median-unbiased estimators. Consider the non-parametric product model .Rn;Bn;Q˝n W Q continuous/. An estimator T of a real characteristic .Q/ is called median-unbiased if, for every continuous Q, .Q/ is a median of Q˝n ı T 1. Show the following.(a) If T is a median-unbiased estimator of
S Non-uniqueness of quantiles. Let Q be a probability measure on .R;B/, 0 < ˛ < 1, q an ˛-quantile of Q, and q0 > q. Show that q0 is another ˛-quantile of Q if and only if Q.1; q/ D ˛ and Q.q; q0OE/ D 0.
Is the drawing of the lottery numbers fair? Using the method of the previous problem, determine, for some of the lottery numbers between 1 and 49, an approximate confidence interval for the probability that the respective number is drawn. First choose an error level and obtain the up-to-date
Consider the binomial model .¹0; : : : ; nº;P.¹0; : : : ; nº/; Bn;# W 0 < # < 1/: Use the method discussed after Theorem (8.11) to determine an approximate confidence interval for# with error level ˛ D 0:02 and n D 1000, and verify some of the corresponding entries of Table 8.3.
S In a chemical plant, n fish are kept in the sewage system. Their survival probability #serves as an indicator for the degree of pollution. How large must n be to ensure that # can be deduced from the number of dead fish with 95% confidence up to a deviation of ˙0:05? Use(a) Chebyshev’s
History of the European Union. On June 23, 1992, the German newspaper Frankfurter Allgemeine Zeitung (FAZ) reported that 26% of all German people agreed with a single European currency; furthermore, 50% were in favour of an enlargement of the European Union towards the East. The percentages were
S Confidence points. Consider the Gaussian product model .Rn;Bn; P# W # 2 Z/ with P# D N#;v˝n for a known variance v > 0 and unknown integer mean. Let ni W R ! Z be the ‘nearest integer function’, that is, for x 2 R let ni.x/ 2 Z be the integer closest to x, with the convention ni.z 1 2 / D
Beta representation of the binomial distribution. Show by differentiating with respect to p thatfor all 0 1º. Bn,p({k+1,k+2,.....n})= -k) ) So" t* (1-1)-k-dt=Pk+1,nk ([0, p])
Determining the speed of light. In 1879 the American physicist (and Nobel prize laureate of 1907) Albert Abraham Michelson made five series of 20 measurements each to determine the speed of light; you can find his results under http://lib.stat.cmu.edu/DASL/Datafiles/Michelson.html. Suppose that the
Consequences of choosing the wrong model. An experimenter takes n independent and normally distributed measurements with unknown expectation m. He is certain he knows the variance v > 0.(a) Which confidence interval for m will he specify for a given error level ˛?(b) Which error level does this
Let .Rn;Bn;Q˝n#W # 2 ‚/ be a real n-fold product model with continuous distribution functions F# D FQ# . Furthermore, let T# D Pn iD1 log F#.Xi /, # 2 ‚. For which probability measure Q on .R;B/ is .QI T# W # 2 ‚/ a pivot? Hint: Problem 1.18, Corollary(3.36).
S Scaled uniform distributions. Consider the product model .Rn;Bn; UOE#;2#˝n W # > 0/, where UOE#;2# is the uniform distribution on OE#; 2#. For# > 0, let T# D max1in Xi=# :(a) For which probability measure Q on .R;B/ is .QI T# W # > 0/ a pivot?(b) Use this pivot to construct a confidence
Combination of confidence intervals. Let .X;F; P# W # 2 ‚/ be a statistical model and 1; 2 W ‚ ! R two real characteristics of #. You are given two individual confidence intervals C1./ and C2./ for 1 and 2 with arbitrary error levels ˛1 and ˛2, respectively. Use these to construct a
Return to Problem 7.3 of estimating an unknown number N of lots in a lottery drum, and let T be the maximum likelihood estimator determined there. For a given error level ˛, find the smallest possible confidence region for N that has the form C.x/ D ¹T.x/; : : : ; c.T.x//º.
Shifted exponential distributions. Consider the statistical model .R;B; P# W # 2 R/, where P# is the probability measure with Lebesgue density #.x/ D e.x#/1OE#;1OE.x/, x 2 R. Construct a minimal confidence interval for # with error level ˛.
Gamma and Poisson distribution. Let .Zn C;P.Zn C/;P˝n#W # > 0/ be the n-fold Poisson product model. Suppose the prior distribution is given by ˛ D a;r , the gamma distribution with parameters a; r > 0. Find the posterior density x for each x 2 Zn C, and determine the Bayes estimator of #.
S Asymptotics of the residual uncertainty as the information grows. Consider Example(7.39) in the limit as n ! 1. Let x D .x1; x2; : : : / be a sequence of observed values in R such that the sequence of averages Mn.x/ D 1 nPn iD1 xi remains bounded. Let .n/x be the posterior density corresponding
Dirichlet and multinomial distributions. As a generalisation of Example (7.36), consider an urn model in which each ball has one of a finite number s of colours (instead of only two).Let ‚ be the set of all probability densities on ¹1; : : : ; sº. Suppose that the prior distribution˛ on ‚ is
Verify the consistency statement (7.37) for the posterior distributions in the binomial model of Example (7.36).
S Consider the two-sided exponential model of Problem 7.10. For each n 1, let Tn be a maximum likelihood estimator based on n independent observations. Show that the sequence.Tn/ is consistent.
Estimation by the method of moments. Let .R;B;Q# W # 2 ‚/ be a real-valued statistical model, and let r 2 N be given. Suppose that for each # 2 ‚ and every k 2 ¹1; : : : ; rº, the kth moment mk.#/ WD E#.Idk R/ of Q# exists. Furthermore, let g W Rr ! R be continuous, and consider the real
S Estimation of the mutation rate in the infinite alleles model. For given n 1, consider Ewens’ sampling distribution n;# with unknown mutation rate# > 0, as defined in Problem 6.5a. Show the following.(a) ¹ n;# W # > 0º is an exponential family and Kn.x/ WD Pn iD1 xi (the number of different
S Exponential families and maximisation of entropy. Let ¹P# W # 2 Rº be an exponential family relative to a statistic T on a finite set X. According to physical convention, we assume that a.#/ D # and h 1. That is, the likelihood function takes the form(7.40) #.x/ D expOE# T .x/=Z.#/ ; x 2
Relative entropy and Fisher information. Let .X;F; P# W # 2 ‚/ be a regular statistical model with finite sample space X. Show that lim"!0"2H.P#C"IP#/ D I.#/=2 for all # 2 ‚:
Recall the situation of Problem 7.3 and show that the maximum likelihood estimator T to be determined there is sufficient and complete.
Let .X;F; P# W # 2 ‚/ be an exponential model relative to a statistic T , and suppose for simplicity that T takes values in ˙ WD ZC. Show that T is sufficient and complete.
S Sufficiency and completeness. Let .X;F; P# W # 2 ‚/ be a statistical model and T W X ! ˙ a statistic with (for simplicity) countable range ˙. T is called sufficient if there exists a family ¹Qs W s 2 ˙º of probability measures on .X;F/ that do not depend on #and satisfy P#. jT D s/ D Qs
S Shifted uniform distributions. Consider the situation of Problem 7.2. Compute the variances V#.M/ and V#.T / of the estimators M and V , and decide which of them you would recommend for practical use. Hint: For n 3 and # D 1=2, determine first the joint distribution density of min1in Xi and
S Randomised response. In a survey on a delicate topic (‘Do you take hard drugs?’) it is difficult to protect the privacy of the people questioned and at the same time to get reliable answers. That is why the following ‘unrelated question method’ was suggested. A deck of cards is prepared
Consider the n-fold Gaussian product model .Rn;Bn;Nm;#˝n W # > 0/ with known expectation m 2 R and unknown variance. Show that the statisticon Rn is an unbiased estimator of .#/ D p #, but that there is no # at which V#.T / reaches the Cramér–Rao bound 0.#/2=I.#/. T = 2 n IM i=1 [|X; -m
Consider the negative binomial model .ZC;P.ZC/; Br;# W 0 < # < 1/ for givenr > 0.Determine a best estimator of .#/ D 1=# and determine its variance explicitly for each #.
S Uniqueness of best estimators. In a statistical model .X;F; P# W # 2 ‚/, let S; T be two best unbiased estimators of a real characteristic .#/. Show that P#.S D T / D 1 for all #. Hint: Consider the estimator .S C T /=2.
Unbiased estimators can be bad. Consider the model .N;P.N/;P# W # > 0/ of the conditional Poisson distributionsShow that the only unbiased estimator of .#/ D 1e# is the (useless) estimator T .n/ D 1 C .1, n 2 N. Po({n}) = P({n}|N) = n! (e-1) n N.
Consider the binomial model of Example (7.14). For any given n, find an estimator of #for which the mean squared error does not depend on #.
Estimate of a transition matrix. Let X0; : : : ; Xn be a Markov chain with finite state space E, known initial distribution ˛ and unknown transition matrix …. For a; b 2 E, let L.2/.a; b/ D j¹1i n W Xi1 D a; Xi D bºj=n be the relative frequency of the letter pair.a; b/ in the ‘random
Consider the statistical product model .Rn;Bn;Q˝n#W # 2 R/, where Q# is the socalled two-sided exponential distribution or Laplace distribution centred at #, namely the probability measure on .R;B/ with density#.x/ D 1 2 ejx#j; x2 R:Find a maximum likelihood estimator of # and show that it is
Consider the geometric model .ZC;P.ZC/; G# W # 2 0; 1/. Determine a maximum likelihood estimator of the unknown parameter #. Is it unbiased?
S At the summer party of the rabbit breeders’ association, there is a prize draw for K rabbits. The organisers print N K tickets, of which K are winning, the remaining ones are blanks. Much to his mum’s dismay, little Bill brings x rabbits home, 1 x K. How many tickets did he probably
A certain butterfly species is split into three types 1, 2 and 3, which occur in the genotypical proportions p1.#/ D #2, p2.#/ D 2#.1#/ and p3.#/ D .1#/2, 0 # 1.Among n butterflies of this species you have caught, you find ni specimens of type i . Determine a maximum likelihood estimator T of
S Phylogeny. When did the most recent common ancestor V of two organisms A and B live? In the ‘infinite sites mutation model’, it is assumed that the mutations of a fixed gene occur along the lines of descent from V to A and V to B at the times of independent Poisson processes with known
Determine a maximum likelihood estimator(a) in the situation of Problem 7.1,(b) in the product model .0; 1OEn;B˝n0;1OE;Q˝n#W # > 0/, where Q# D ˇ#;1 is the probability measure on 0; 1OE with density #.x/ D #x#1, and check whether it is unique.
Consider again the setting of Problem 7.3. This time little Bill draws n lots without replacing them. Find the maximum likelihood estimator T of N, calculate EN .T /, and give an unbiased estimator of N.

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