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Estimation of Common Location and Scale Parameters in Nonregular Cases Ahmad Razmpour Iowa State University

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Estimation of Common Location and Scale Parameters in Nonregular Cases Ahmad Razmpour Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1982 Estimation of common location and scale parameters in nonregular cases Ahmad Razmpour Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Statistics and Probability Commons Recommended Citation Razmpour, Ahmad, "Estimation of common location and scale parameters in nonregular cases " (1982). Retrospective Theses and Dissertations. 7528. https://lib.dr.iastate.edu/rtd/7528 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This reproduction was made from a copy of a document sent to us for microfilming. While the most advanced technology has been used to photograph and reproduce this document, the quality of the reproduction is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help clarify markings or notations which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure complete continuity. 2. When an image on the film is obliterated with a round black mark, it is an indication of either blurred copy because of movement during exposure, duplicate copy, or copyrighted materials that should not have been filmed. For blurred pages, a good image of the page can be found in the adjacent frame. If copyrighted materials were deleted, a target note will appear listing the pages in the adjacent frame. 3. When a map, drawing or chart, etc., is part of the material being photographed, a definite method of "sectioning" the material has been followed. It is customary to begin filming at the upper left hand comer of a large sheet and to continue from left to right in equal sections with small overlaps. If necessary, sectioning is continued again—beginning below the first row and continuing on until complete. 4. For illustrations that cannot be satisfactorily reproduced by xerographic means, photographic prints can be purchased at additional cost and inserted into your xerographic copy. These prints are available upon request from the Dissertations Customer Services Department. 5. Some pages in any document may have indistinct print. In all cases the best available copy has been filmed. Universi^ Micronlms International 300 N. Zeeb Road Ann Arbor, Ml 48106 8224246 Razmpour, Ahmad ESTIMATION OF COMMON LOCATION AND SCALE PARAMETERS IN NONREGULAR CASES Iowa State University PHD. 1982 University Microfilms Intsrnâtionâl SOO X. zeeb Rœd. Ann Arbor,MI 48106 Estimation of common location and scale parameters in nonregular cases by Ahmad Razmpour A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Major: Statistics Approved; Signature was redacted for privacy. In Chargeflof jnajor Work Signature was redacted for privacy. For the Major Department Signature was redacted for privacy. Iowa State University Ames, Iowa 1982 ii TABLE OF CONTENTS Page 1. INTRODUCTION 1 1.1. Background 1 1.2. Outline 16 2. COMMON LOCATION PARAMETER OF TWO EXPONENTIAL DISTRIBUTION 19 2.1. The Basic Set Up 19 2.2. Maximum Likelihood Estimators 19 2.3. Distribution Function of Z = min(U(X^),V(X2)) 21 2.4. Joint Distribution of T^^ and T^ 21 2.5. The UMVUE of the Common Location Parameter of Two Exponential Distributions 24 2.6. Basic Set-up for the UMVUE of y When the Ratio = P(>0) is Known 30 ^1 °2 2.7. Distribution of T = Z (X -Z)+p Z (X .-Z) 33 i=l j=l 2.8. The UMVUE of u When the Ratio a^/cT_ = p(>0) is Known 35 2.9. Modified Maximum Likelihood Estimator (MLE*) 37 2.10. Comparison of MLE and MLE* in Terms of Their Distribution as Well as MSE 37 2.11. A Monte Carlo Study of the Bias and the Mean Squared Errors 51 3. THE SEVERAL EXPONENTIAL CASE 85 3.1. Basic Set Up 85 3.2. Maximum Likelihood Estimators 85 3.3. Distribution Function of Z* = min (U.(X.)) 86 ]Xi_^ 1 ~3- 3.4. Joint Distribution Function of T = n a 87 X 1 i Xll Page 3.5. The UMVUE of Common Location Parameter of 90 p-Exponential Distributions 3.6. The UMVDE of Common Location Parameter (y), When _1 the Ratios are Known 94 3.7. Distribution Function of Z* 95 P , °i 3.8. Distribution Function of T*« Z (X..-Z*)+ Z p ^ Z 95 j=i i=2 ^ j=i (Z -Z*) 3.9. The UMVDE of Common Location Parameter When P^(=0^cr^^)'s, (i = 1,...,P) are Known 98 3.10. Modified Maximum Likelihood Estimator for Common Location Parameter (y) in P-Populations 101 3.11. Comparison of MLE and Modified MLE in Terms of Their Distribution and MSE 102 4. ESTIMATING THE LOCATION PARAMETER OF AN EXPONENTIAL DISTRIBUTION WITH KNOWN COEFFICIENT OF VARIATION 116 4.1. The Best Linear Estimator of 0 in Class of Unbiased Estimators 116 4.2. Linear Minimum Mean Square Estimation 118 4.3. The Maximum Likelihood Estimator of 0 119 4.4. Best Scale Invariant Estimates 130 4.5. A Class of Bayes Estimators 133 5. REFERENCES 136 6. ACKNOWLEDGEMENTS 138 1 1. INTRODUCTION 1.1. Background The problem of estimating the common mean of several normal popu­ lations with unknown and unequal variances was essentially initiated in the work of Yates (1940). The model for this problem as described by Meier (1953) is as follows. Let X^, X^, —, X^ be k estimators of a parameter p, being independently and normally distributed with 2 2 mean y and variances o^/n^, ...» a^/n^ respectively. Now if the 2 a^'s are known, then the minimum variance unbiased estimator of y is Ic k = z(n.x,/aj)/ z(D./ab (1.1.1) i=l 1 1 ^ i=l ^ ^ ^ 2 2 and the variance of this estimate is 1/ 2(n./a.). When, a.*s are i=l 1 1 ^ 2 2 unknown, let S^, ..., be independent unbiased estimators of 2 2 2 ..., with distributed as chi-squared with degrees of freedom, then analogous estimator y is proposed as follows. w = Z(n,x./S?)/ UnJsb (1.1.2) i=l ^ 1 ^ i=l ^ 1 Meier in his investigation gave the following basic results k k k k (i) var(y) = [ 2(1/aJ)r^{l+2 E nT^[o:^/ Z aT^]Ll-aT^/ I a'^l + i=l ^ i=l i=l ^ ^ i=l ^ k ~ 0( Z n:^)} (1.1.3) i=l ^ 2 (ii) An approximately unbiased estimate of Var(y) is given by V* = [ z s7^R^{i+4 z nT^EsT^/ I sT^]Li-sT^/ z sT^]} (1.1.4) i=l ^ i=l ^ ^ i=l ^ ^ i=l ^ (iii) V* is distributed approximately as a chi-square with f degrees of freedom where f~^ = Z xT^ia'}/ Z 07^)2. i=l ^ ^ i=l ^ Later, Graybill and Deal (1959) proved the following theorem for combined estimator y when k = 2. 2 2 Theorem 1.1.1 For the random variables X^, and S^, defined as above, a necessary and sufficient condition that U = (n^SgX^jhigS^Zg)/ (ni^z+^S^) (1.1.5) is an unbiased estimator of U which is uniformly better than either or Xg is that m^ and m^ are both larger than nine. Proof The proof given here is different from proofs given by Graybill and Deal, Norwood and Hinkelmann (1977). 2 2 Using the independence of (X^,X^) with (S^^Sg), it follows that 2 2 - 2 2 n TiS +n uS E(y) = EE(ii|sf,Sp = E[ . n 3 = y (1.1.6) In order to prove the rest of the theorem, first note that Var(y) = Var[E(y | S^,Sp] + E[Var(J | sJ.S^)] 3 2 2 2 2 n S a n S 0 = Var(ii) + E[( ) 1^+ ( ^ 3 2 ^2 2 = [ -r E(i-w) + E(ir)] (1.1.7) ^ 2 whereh BW = "Z'i Thus, from (1.1.6) Var(ii) < a^/n^ = Var(X^) (1.1.8) if and only if Gg 2 20^ ( — + — ) E(w) - —^ E(W) < 0 (1.1.9) Bi <=> (1+p) E(W^) - 2p E(W) <0 But, . °2 ^ • % <syM'4^ (1.1.10) n 1+pF 9 1 9 9 9 9 6;/a^0/(S^/a^) "1 02 a? " where p = (-^)/(—), F = {shoh Kshoh ~ F ^ . =2 n2 11 2 z 2 4 Hence, (1+p) E(W^) - 2p E(W) <0 <=> 2 2 (1+p) E{ —2-J— } _ 2p^ E{ T^- } < 0 <=> (1+pF)^ E{[(1+P)F^ - 2F(l+pF)]/(l+pF)^} < 0 <=> E{[F^-2F-pF^]/(l+pF)^} < 0 . (1.1.11) Now, E{£z2E=Ê^}<E{^!=2r } (1+pF) (1+pF) ^ E(F^-2F) (1.1.12) (l+2p)^ 1 Now, note that F is distributed as F . Hence, — ^ F From (1.1.12) it follows that Var(y) < Var(X^) if E(F^-2F) 0 . ^ 12 Similarly, Var(li) < Var(X-) if E( —:r - — ) <0 . But for 2 p2 F m_ > 5 and > 5, 1 — 2 — 2 m, m (m +2) E(F-2R=^[;l^-2]<0 <=> -m^mg4-2mg+8m^ < 0.
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