[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-217278-en":3,"doc-seo-217278-105":30,"detail-sidebar-cat-0-en-105":97},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},217278,962085564549,"Aditya","https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","THE LINEAR LEAST SQUARES PREDICTION APPROACH TO POPULATIONS WITH TREND","Finite population estimation is reformulated as a prediction problem for populations with a linear trend and for populations exhibiting autocorrelation. Using established results from linear prediction theory, the optimal predictor for the population total is derived under two distinct systematic sampling strategies. The centered systematic strategy of Madow (1953) is shown to be optimal in some cases with linear trend and nearly optimal under autocorrelation for sufficiently large samples. When model parameters are unknown, an iterative technique estimates population variance and autocorrelation, and model-misspecification effects are examined.","RT-MAE-8511  \nTHE LINEAR LEAST SQUARES PREDICTIONAPPROACH T0 POPULATIONS WITH TREND  \nby  \nHeleno BolfarineandMonica C.Sandoval  \nPalavras Chaves:Finite population;linear trend;(Key words)autocorrelated populations;optimalpredictor;robustness;systematicsampling.  \nClassificacão AMS:62D05(AMS Classification)62M10  \n-Junho de 1985 -  \n# THE LINEAR LEAST SQUAPES PPEDICTION APPROACH TOPOPULATIONS WITH TRED\n\nby  \nHeleno Bolfarine and Monica C.SandovalInstituto de Matematica e EstatisticaUniversidade de São PauloCaixa Postal 2057001498 -SAO PAULO,SP  \n## SUMMARY\n\nFinite populaticn estimation problems areformulated as prediction problems under populations withlinear trend and autocorrelated populations.  Using wel1known results in linear prediction theory,the optimalpredictor for the population total is obtained under twodifferent systematic strategies.  The centered systematicstrategy proposed by Madow(1953)is shown to be optimalin some cases under linear trend and nearly optimal underautocorrelation for relatively large sample sizes.  Underautocorrelation an iterative technique is proposed toestimate the population variance and autocorrelationwhenever they are unknown.  Some effects of failure ofthe assumed models are studied.  \n## 1.INTRODUCTION\n\nConsider a population of size N=nk,the units ofwhich are identified by the labels 1,2,.,N,and orderedin increasing size of the label.Associated with unit i  \nthere is an unknown quantity of interest,yi,i=1,…N.The usua¹random start systematic sampling design dividethe N units into k≥2 clusters S₁,S₂,…,Sk,where S;consists of the units i,i+k,..,i+(n-1)k.  Then itselects one S;at random,estimating the population total,where y₁is the mean of the yvalues for the units in S;and y₂is the mean of the non-sample units,i.e.,the values not in Si·On the otherhand,Madow's(1953)central systematic sampling consistsof the selection of clusterif k is odd,and ifk is even eitherwith probability 1/2.  \nThe population total T is then predicted by,as before.  \nIn the prediction approach,the value of y;istreated as the realization of the random variable Yi,andis the joint distribution of Y₁,Y₂,…,YNwhich isused in the definition of bias,variance and standarderror.So,after the sample has been selected,y₁isknown and estimating T is equivalent to predicting thevalue of,where S is the selected cluster.  \nIn the sequel,we do not distinguish between y;and Y;,since both are unknown.  The theorem that followscharacterizes the optimai predictor under the generallinear model,and is obtained from wel1 known results inlinear prediction theory(Goldberger(1962)).  So itsproof is not presented here.  \n,where EIe]=0 and var[e]=Y,andTheorem1.If  \nwhere label 1 corresponds to the n sample units and label 2to the N-n nons ample units,then among al1 linear estimatorsf satisfying EIT-T]=0,the error variance,var[个-T],isminimized by taking  \nandare respectivelyandwherethe unit vectors of dimensions n and N-n.  The error varianceof this estimator is  \nIn Section 2,the optimal predictor is obtained under thelinear trend population model.  Two optimal strategies areconsidered.  Section 3 presents the optimal predictor underautocorrelated populations.In both cases,effects offlaws in the adopted models are studied  \n2.POPULATIONS WITH LINEAR TREND  \n   \n   \n   \n   \nIt is assumed that associated with unit t there are  \ntwo numbers(t,yt),in such a way that  \n(1)  \ny+=a+bt+e  \nwhere the et are independent,E(et)=0 and Var[etJ=a²,t=1,.,N.  Here E[·]and var[·]denote the expectationand variance operator.  As pointed out before,we areinterested in predicting the sum of the y values,given asample S of size n,selected from the population.  \n2.1.-The Optimal Predictor  \n   \n   \n   \nLet t₁,…,tn the corresponding values of t ins,the selected sample,and y₁,…,yn the correspondingvalues of y.Note that the y values y₁,…,yn werereordered in such a way that,…·,yn)correspondsto the sample values and,…,yn)to the non-sample values.  \nUsing Theorem 1,it follows t","cbCaicNyGod6I3Kl","https://ap.wps.com/l/cbCaicNyGod6I3Kl","pdf",3619153,1,20,"English","en",105,"# SUMMARY\n# 1. INTRODUCTION\n# 2. POPULATIONS WITH LINEAR TREND\n## 2.1 - THE OPTIMAL PREDICTOR\n# 3. EFFECTS OF FLAWS ON THE ASSUMED MODEL","[{\"question\":\"What is the main idea behind using the prediction approach?\",\"answer\":\"The method treats the unknown population quantities as realizations of random variables and uses their joint distribution to define bias, variance, and standard error after sampling. Estimating the population total is then equivalent to predicting the corresponding total under the chosen model.\"},{\"question\":\"How is the optimal predictor obtained for populations with a linear trend?\",\"answer\":\"Under the linear model y_t = a + bt + e with independent errors, the optimal predictor for the population total is derived from results in linear prediction theory. The sampling scheme that minimizes the prediction error variance is identified as a first-order balanced selection.\"},{\"question\":\"What happens when the assumed model is flawed?\",\"answer\":\"If the true model differs from the assumed linear trend model, the paper studies how the bias and optimality change. It introduces conditions for unbiasedness under the alternative model and proposes the resulting form of the optimal predictor, using least-squares estimators of the new model parameters.\"}]","THE LINEAR LEAST SQUARES PREDICTION APPROACH TO POPULATIONS WITH TREND | PDF",1788831877,50,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":92,"head_meta":94,"extra_data":96,"updated_unix":28},"the-linear-least-squares-prediction-approach-to-populations-with-trend","",{"@graph":36,"@context":91},[37,54,74],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/the-linear-least-squares-prediction-approach-to-populations-with-trend/217278/",4,{"url":52,"name":13,"@type":55,"image":56,"author":61,"headline":13,"publisher":63,"fileFormat":66,"inLanguage":23,"description":14,"dateModified":67,"datePublished":68,"encodingFormat":66,"isAccessibleForFree":69,"interactionStatistic":70},"DigitalDocument",{"url":57,"@type":58,"width":59,"height":60},"https://docshare.wps.com/thumbnails/the-linear-least-squares-prediction-approach-to-populations-with-trend/217278.png","ImageObject",300,407,{"name":9,"@type":62},"Person",{"url":41,"name":64,"@type":65},"DocShare","Organization","application/pdf","2026-09-11","2026-09-08",true,{"@type":71,"interactionType":72,"userInteractionCount":20},"InteractionCounter",{"@type":73},"ViewAction",{"@type":75,"mainEntity":76},"FAQPage",[77,83,87],{"name":78,"@type":79,"acceptedAnswer":80},"What is the main idea behind using the prediction approach?","Question",{"text":81,"@type":82},"The method treats the unknown population quantities as realizations of random variables and uses their joint distribution to define bias, variance, and standard error after sampling. Estimating the population total is then equivalent to predicting the corresponding total under the chosen model.","Answer",{"name":84,"@type":79,"acceptedAnswer":85},"How is the optimal predictor obtained for populations with a linear trend?",{"text":86,"@type":82},"Under the linear model y_t = a + bt + e with independent errors, the optimal predictor for the population total is derived from results in linear prediction theory. The sampling scheme that minimizes the prediction error variance is identified as a first-order balanced selection.",{"name":88,"@type":79,"acceptedAnswer":89},"What happens when the assumed model is flawed?",{"text":90,"@type":82},"If the true model differs from the assumed linear trend model, the paper studies how the bias and optimality change. It introduces conditions for unbiasedness under the alternative model and proposes the resulting form of the optimal predictor, using least-squares estimators of the new model parameters.","https://schema.org",{"og:url":52,"og:type":93,"og:title":13,"og:site_name":64,"og:description":14},"article",{"robots":95,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":98},[99,103,107,111,116,120,125,128,132,135,139],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":100,"show_sort_weight":101,"slug":102},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},"Exam",70,"exam",{"id":112,"doc_module":4,"doc_module_name":46,"category_name":113,"show_sort_weight":114,"slug":115},5,"Comic",60,"comic",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":118,"show_sort_weight":29,"slug":119},6,"Technology","technology",{"id":121,"doc_module":4,"doc_module_name":46,"category_name":122,"show_sort_weight":123,"slug":124},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":126,"slug":127},30,"research-report",{"id":129,"doc_module":4,"doc_module_name":46,"category_name":130,"show_sort_weight":21,"slug":131},9,"Religion & Spirituality","religion-spirituality",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":21,"slug":134},"World Cup","world-cup",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":136,"slug":138},10,"Lifestyle","lifestyle",{"id":140,"doc_module":4,"doc_module_name":46,"category_name":141,"show_sort_weight":112,"slug":142},19,"General","general"]