# Investigating the Significance of a Correlation Coefficient using Jackknife Estimates

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1 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) ISSN (Prit & Olie) Ivestigatig the Sigificace of a Correlatio Coefficiet usig Jackkife Estimates Athoy Akpata a, Idika Okorie b a,b Departmet of Statistics, Abia State Uiversity Uturu,Nigeria a b Abstract Ofte i Applied statistics, populatio parameters are ot kow ad could be iferred usig the available sample data ad this is the uderpiig of statistical iferece. Resamplig techique such as jackkife offers effective estimates of parameters ad its asymptotic distributio. I this paper, we preset the jackkife estimate of the parameters of a simple liear regressio model with particular iterest o the correlatio coefficiet. This procedure provides a effective alterative test statistic for testig the ull hypothesis of o associatio betwee the explaatory variables ad a respose variable. Keywords: Jackkife; simple liear regressio; correlatio coefficiet; ols estimates; bias. 1. Itroductio After estimatio of parameters i applied statistics it is always crucial to assess the accuracy of the estimator by its stadard error ad costructio of cofidece itervals for the parameter [1]. Queouille i 1956 developed a cross validatio procedure kow as jackkife (leave-oe-out procedure) for estimatig the bias of a estimator [2]. Two years later this method was further exteded by Joh Tukey to estimate the variace of a estimator ad the ame Jackkife was coied for this cross validatio method [3] * Correspodig author. address: 441

4 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR)(2015) Volume 22, No 2, pp ,2,3,, ad evaluatig θ ols (J) ad ρ x,y (J) the least squares estimates based o the remaiig observatios [10]. The estimates of θ J ad ρ J, bias ad variace usig the pseudo values θ Ji ad ρ x,y(ji) are θ J = θ Ji (5) With bias bias = θ ols θ Ji (6) Or more succictly bias = θ ols θ J (7) Ad the variace var θ J = θ Ji θ J 2 ( 1) (8) Also, ρ x,y(j) = ρ x,y(ji) (9) With bias bias = ρ x,y ρ x,y(ji) (10) Or bias = ρ x,y ρ x,y(j) (11) Ad variace var ρ x,y(j) = ρ x,y(ji) ρ x,y(j) 2 ( 1) (12) 2.1 Algorithm for Jackkifig Simple Liear Regressio Model Steps: 444

6 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR)(2015) Volume 22, No 2, pp Table 2: ols Estimates S/N θ (0)Ji θ (1)Ji ρ x,y(ji) θ (0)J θ (1)J ρ x,y(j) SE θ (0)J SE θ (1)J SE ρ x,y(j) Table 3: Compariso betwee ols ad Jackkife ols Estimates Estimates ols Jackkife Bias θ (0) SE θ (0) θ (1) SE θ (1) ρ x,y SE ρ x,y Testig the sigificace of the correlatio coefficiet We shall proceed to test the sigificace of the correlatio coefficiet at 5% level of sigificace as follows: H 0 : ρ x,y = 0 H 1 : ρ x,y 0 446

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