Confidence Intervals based on Absolute Deviation for Population Mean of a Positively Skewed Distribution

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1 Iteratioal Joural of Computatioal ad Theoretical Statistics ISSN (220-59) It. J. Comp. Theo. Stat. 5, No. (May-208) Cofidece Itervals based o Absolute Deviatio for Populatio Mea of a Positively Skewed Distributio Moustafa Omar Ahmad Abu-Shawiesh, Shipra Baik 2 ad B.M. Golam Kibria Departmet of Mathematics, Hashemite Uiversity, Al-Zarqa 35, JORDAN 2 Departmet of Physical Scieces, Idepedet Uiversity, Bagladesh, Dhaka 22, BANGLADESH 3 B.M. Golam Kibria, Departmet of Mathematics ad Statistics, Florida Iteratioal Uiversity, Miami, FL 3399, USA Received September 30, 207, Revised December8, 207 Accepted Jauary 23, 208, Published May, 208 Abstract: This paper proposes three alterative cofidece itervals amely, AADM-t, MAAD-t ad MADM-t, which are simple adjustmets to the Studet-t cofidece iterval for estimatig the populatio mea of a positively skewed distributio. The proposed methods are very easy to calculate ad are ot overly computer-itesive. The performace of these cofidece itervals was compared through a simulatio study usig the followig criterio: (a) coverage probability (b) average width ad (c) coefficiet of variatio of width. Simulatio studies idicate that for small sample sizes ad moderate/highly positively skewed distributios, the proposed AADM-t cofidece iterval performs the best ad it is as good as the Studet-t cofidece iterval. Some real-life data are aalyzed which support the fidigs of this paper to some extet. Keywords: Cofidece iterval, Robustess, Absolute deviatio, Coverage probability, Positively skewed distributio, Mote Carlo simulatio.. INTRODUCTION The positively skewed data are frequetly ecoutered i both ecoomics ad health-care fields where experimets with rare diseases or a typical behavior are the orm. The classical Studet-t cofidece iterval is the most widely classical used approach because it is simple to calculate ad robust for both small ad large sample sizes. However whe the populatio distributio is positively skewed, the Studet-t cofidece iterval will oly have a approximate (-α) coverage probability. This coverage probability may be improved by developig differet cofidece iterval methods i order to aalyze the positively skewed data. This paper reviews ad develops some cofidece itervals which hadle both small samples ad positively skewed data. Sice a theoretical compariso amog the iterval is ot possible, a simulatio study has bee coducted to compare the performace of the itervals. The coverage probability (CP), average width (AW) ad coefficiet of variatio of widths (CVW) are cosidered as a performace criterio. They have bee recorded ad compared across cofidece itervals. Smaller width idicates a better cofidece iterval whe coverage probabilities are the same. Higher coverage probability idicates a better cofidece iterval whe widths are the same. This paper is orgaized as follows: The proposed cofidece itervals have bee developed i sectio 2. A Mote Carlo simulatio study has bee coducted i sectio 3. As applicatios, some real life data have bee aalyzed i sectio 4. Some cocludig remarks are give i sectio THE PROPOSED CONFIDENCE INTERVAL ESTIMATORS The mai characteristics for the scale estimators based o the media absolute deviatio for costructig the proposed cofidece itervals will be discussed i this sectio. Let X, X 2,, X be a radom sample which is idepedetly ad idetically distributed ad comes from a positively skewed distributio with ukow μ ad σ. We wat to develop 00(-α)% cofidece iterval for μ. The classical Studet-t cofidece iterval for μ ad the proposed media absolute deviatios cofidece itervals have bee discussed as below: address: mabushawiesh@hu.edu.jo, baik@iub.edu.bd ad kibriag@fiu.edu

2 2 M. Abu-Shawiesh, et. al.: Cofidece Itervals based o Absolute Deviatio for Populatio A. The classical Studet-t cofidece iterval This iterval relies o the ormality assumptio ad is developed by [] as a more robust way for testig hypotheses specifically for small sample sizes ad/or σ is ukow. The ( α)00% cofidece iterval for μ ca be costructed as follows: whe σ is kow. X Z, () 2 For small sample sizes ad ukow σ, the ( α)00% cofidece iterval for μ which is kow as the Studet-t cofidece iterval ca be costructed as follows: t X t, 2 S, (2) where ( 2, ) is the upper α/2 percetage poit of the studet t distributio with (-) degrees of freedom. The classical Studet-t approach is ot very robust uder extreme deviatios from ormality [2]. Additioally, sice the classical Studet-t depeds o the ormality assumptio, it may ot be the best cofidece iterval for asymmetric distributios. I this paper, we assume that X follows a positively skewed distributio. Previous researchers have foud that the Studet-t performs well for small samples sizes ad asymmetric distributios i terms of the coverage probability comig close to the omial cofidece coefficiet although its average widths ad variability were ot as small as other cofidece itervals ([2]-[5]). B. The Proposed Media Absolute Deviatios Cofidece Itervals For a positively skewed distributio, it is kow that the media describes the ceter of a distributio better tha the mea. Thus for a positive skewed data ad because of the robustess of the media, i this sectio, we will cosider three methods based o media absolute deviatios statistics to costruct the cofidece iterval for μ. The proposed cofidece itervals are computatioally simple ad therefore aalytically a more desirable methods. C. The AADM-t Cofidece Iterval The first method we propose i this paper is called the AADM-t cofidece iterval, which is a modificatio of the classical Studet-t cofidece iterval. The ( α)00% AADM-t cofidece iterval for μ is give by: where AADM 2 i X t, 2 AADM, (3) X i MD, MD is the sample media. As stated i [6], if X, X 2,, X ~ N(μ, 2 ), the AADM is a cosistet estimate of σ ad is asymptotically ormally distributed, which is: lim E( AADM ) ( AADM ) N(0, ) 2 Moreover, usig the strog law large umbers, it ca be show that AADM coverges to σ almost surely. D. The MAAD-t Cofidece Iterval The secod method we propose i this paper is called the MAAD-t cofidece iterval, which is aother modificatio of the classical Studet-t cofidece iterval. The ( α)00% MAAD-t cofidece iterval for μ is give by:

3 It. J. Comp. Theo. Stat. 5, No., -3 (May-208) 3 X t, 2 MAAD, (4) where MAAD is defied as X X, i,2,... MAAD media i, (5) This estimator was give by [7] ad they showed that it is more robust tha S. E.The MADM-t Cofidece Iterval The third method we propose i this paper is called the MADM-t cofidece iterval, which is aother modificatio of the classical Studet-t cofidece iterval. This method is based o MADM. The ( α)00% MADM-t cofidece iterval for μ is give by: MADM X t, (6), 2 where MADM was first itroduced by [8] ad is defied as X MD, i,2,... MADM media i, (7) The MADM has importat robustess properties as follows: (i) It has a maximum breakdow poit which is 50% which is twice as much as iterquartile rage (IQR) (ii) It has the smallest gross error sesitivity value which is.67. (iii) It has the sharpest boud of ifluece fuctio. (iv) The efficiecy of it is 37% for the case of ormal distributio. (v) If the MADM is multiplied by.4826, it becomes a ubiased estimator of σ. 3. SIMULATION STUDY Sice a theoretical compariso amog the itervals is difficult, followig [3], a simulatio study has bee coducted to compare the performace of the cofidece itervals. Based o the results of the simulatio studies, the best cofidece iterval will be chose based o coverage probability (CP), average width (AW), coefficiet of variatio of the widths (CVW), sample size () ad skewess level. The program for the simulatio study has bee coducted usig MATLAB(205) programmig laguages. Sice our mai objective is to compare the performace of the classical Studet-t ad the proposed cofidece itervals for positive skewed distributios, the to geerate data, we choose the gamma distributio with various skewess levels for compariso purposes. The probability desity fuctio of the gamma distributio is defied as x f ( x /, ) x exp( ) ( ) where α is a shape parameter ad β is a scale parameter. The mea of this distributio is ad variace is 2 2. We wat to fid some good cofidece itervals which will be useful for a sample comig from a positively skewed distributio. A. The Simulatio Techique The program flowchart for the simulatio study is as follows: (i) Select the sample size (), umber of simulatio rus (M) ad the cofidece sigificace level (α). (ii) Geerate a radom sample of size (), X, X 2,, X, which is a idepedetly ad idetically distributed ad comes from a gamma distributio with two parameters α ad β with the chose populatio skewess usig the MATLAB (205) program. (iii) Costruct cofidece itervals at a (-α)00% cofidece level usig the formulas defied i sectio 2. (iv) For each cofidece iterval costructed, determie if the cofidece iterval icludes µ ad for those cofidece itervals that cotai the mea calculate the width of the cofidece iterval.

4 4 M. Abu-Shawiesh, et. al.: Cofidece Itervals based o Absolute Deviatio for Populatio (v) Repeat (i)-(iv) M times, the compute CP (the proportio of itervals that cotai the true mea out of M itervals), AW ad CVW(ratio of coverage to width). Followig [3], the parameters for the gamma distributio have bee chose ad the radom sample of size, X, X 2,..., X was take from the followig gamma distributios with a commo mea of 0: (a) G(6,0.625) with skewess 0.5; (b) G(4,2.5) with skewess ; (c) G(,0) with skewess 2; (d) G(0.25,40) with skewess 4; (e) G(0.,40) with skewess 6; (f) G(0.063,40) with skewess 8. To check whether our selected four methods are sesitive with or ot, we choose from 5 to 00. The cofidece level for the simulatio study is 0.95 which is the commo cofidece iterval. The umber of M was chose to be More o simulatio techiques, we refer [9] -[0] amog others. B. The Simulatio Results CP, AW ad CVW for selected ad for skewess 0.5,, 2, 4, 6, ad 8 are calculated ad give i Tables I-VI respectively ad i figures -6 respectively. TABLE I. ESTIMATED COVERAGE PROBABILITIES USING GAMMA (6,0.625) WITH SKEWNESS = 0.5 Studet-t AADM-t MAAD-t MADM-t CP AW CVW CP AW CVW CP AW CVW CP AW CVW

5 It. J. Comp. Theo. Stat. 5, No., -3 (May-208) 5 Studet-t AADM-t MAAD-t MADM-t Figure. Estimated Coverage Probabilities usig Gamma (6, 0.625) with Skewess = 0.5 From Table I ad Fig., we observe that the classical Studet-t cofidece iterval has coverage probability close to the omial level, followed by AADM-t ad MAAD-t cofidece itervals. It is also observable that, MAAD ad MADM have the smallest widths as compare to other two selected cofidece itervals. TABLE II. ESTIMATED COVERAGE PROBABILITIES USING THE GAMMA (4, 2.5) WITH SKEWNESS =.0 Studet-t AADM-t MAAD-t MADM-t CP AW CVW CP AW CVW CP AW CVW CP AW CVW

6 6 M. Abu-Shawiesh, et. al.: Cofidece Itervals based o Absolute Deviatio for Populatio Studet-t AADM-t MAAD-t MADM-t Figure 2. Estimated Coverage Probabilities usig the Gamma (4, 2.5) with Skewess =.0 From Table II ad Fig.2, we observe that whe skewess icreases from 0.5 to.0, our proposed AADM-t cofidece iterval followed by MAAD-t cofidece iterval coverage probabilities are close to the omial level with the classical Studet-t cofidece iterval. Smallest widths are observed from our two proposed MAAD ad MADM itervals. TABLE III: ESTIMATED COVERAGE PROBABILITIES USING THE GAMMA (,0) WITH SKEWNESS = 2.0 Studet-t AADM-t MAAD-t MADM-t CP AW CVW CP AW CVW CP AW CVW CP AW CVW

7 It. J. Comp. Theo. Stat. 5, No., -3 (May-208) 7 Studet-t AADM-t MAAD-t MADM-t Figure 3. Estimated Coverage Probabilities usig the Gamma (,.0) with Skewess = 2.0 From Table III ad Fig.3, it is oticeable that our proposed AADM-t ad Studet-t cofidece iterval have similar coverage probability. Itervals with respect to width are performig best as compare to the studet s t iterval. TABLE IV. ESTIMATED COVERAGE PROBABILITIES USING THE GAMMA (0.25,40) WITH SKEWNESS = 4.0 Studet-t AADM-t MAAD-t MADM-t CP AW CVW CP AW CVW CP AW CVW CP AW CVW

8 8 M. Abu-Shawiesh, et. al.: Cofidece Itervals based o Absolute Deviatio for Populatio Studet-t AADM-t MAAD-t MADM-t Figure 4. Estimated Coverage Probabilities usig the Gamma (,.0) with Skewess = 4.0 From Table IV ad Fig.4, we observe that i case of a moderate to highly skewed distributio, the AADM-t cofidece iterval coverage probability is very close to omial level 0.95 as compare to others. Here also our proposed itervals performig very well. TABLE V. ESTIMATED COVERAGE PROBABILITIES USING THE GAMMA (0.,40) WITH SKEWNESS = 6 Studet-t AADM-t MAAD-t MADM-t CP AW CVW CP AW CVW CP AW CVW CP AW CVW

9 It. J. Comp. Theo. Stat. 5, No., -3 (May-208) 9 Studet-t AADM-t MAAD-t MADM-t Figure 5. Estimated Coverage Probabilities usig the Gamma (,.0) with Skewess = 6.0 From Tables V-VI ad Fig.5 ad Fig. 6 we observe that i case of a very highly skewed distributio, the AADM-t cofidece iterval coverage probability is stable ad very close to the omial level 0.95 as compare to others. Here also our proposed itervals performig very well i terms of widths whe the sample sizes are small. TABLE VI. ESTIMATED COVERAGE PROBABILITIES USING THE GAMMA (0.063,40) WITH SKEWNESS =8 Studet-t AADM-t MAAD-t MADM-t CP AW CVW CP AW CVW CP AW CVW CP AW CVW

10 Frequecy 0 M. Abu-Shawiesh, et. al.: Cofidece Itervals based o Absolute Deviatio for Populatio Studet-t AADM-t MAAD-t MADM-t Figure 6. Estimated Coverage Probabilities usig the Gamma (,.0) with Skewess = APPLICATIONS As a applicatio, three examples have bee cosidered to illustrate the performace of the cofidece itervals which have bee cosidered i this paper. These examples have various sample sizes ad level of skewess. MATLAB(205) programmig laguage codes are used to produce ecessary tables ad figures respectively. A. Example- To study the average use of psychotropic drugs from o-atipsychotic drug users, the umber of users of psychotropic drugs was reported for 20 differet categories of drugs []. The followig data represet the umber of users: 43.4, 24,.8, 0, 0., 70., 0.4, 50.0, 3.5, 5.2, 35.7, 27.3, 5, 64.3, 70, 94, 6.9, 9., 38.8, 4.8. We wat to fid the average umber of users of psychotropic drugs for o-atipsychotic drug users. The umber of user is positively skewed with skewess =.57 ad mea = A histogram of the data values showig its positive skewess is give i Fig.7. The proposed cofidece itervals ad their correspodig widths have bee give i Table VII. Histogram of Psychotropic Drug Exposure Data Psychotropic Drug Figure 7. Histogram of Psychotropic Drug Exposure Data

11 Frequecy It. J. Comp. Theo. Stat. 5, No., -3 (May-208) TABLE VII. THE 95% CONFIDENCE INTERVALS FOR PSYCHOTROPIC DRUG EXPOSURE DATA Method Cofidece Iterval Width Studet-t (9.748, ) AADM-t (22.692, 62.08) MAAD-t (26.49, 56.65) MADM-t (30.232, ) We observe that the MADM-t cofidece iterval has the smallest width followed by MAAM-t ad AADM-t. The classical Studet-t cofidece iterval has the highest width. Both the proposed MAAD-t ad MADM-t has the shorter widths compared to the correspodig AADM-t. All the cofidece itervals have approximately short width. Note that the sample size is small ad data are highly skewed. Thus the MADM-t cofidece iterval performs the best i the sese of havig smaller width tha the other two proposed cofidece itervals. B. Example-2 To study the Mosquito survival rates i a wet climate, 8 survival times were reported [2], the followig data represets the time of death: 0.539, 0.292, 0.090, 0.044, 0.00, 0.00, 0.00, We wat to fid the average survival time. Survival rate is positively skewed with skewess =.83 ad mea is 0.3. A histogram of the data values showig its positive skewess is give i Fig.8. The proposed cofidece itervals ad their correspodig widths have bee give i Table VIII. We foud that from the Table VIII, the MADM-t cofidece iterval has the smallest width followed by MAAM-t ad AADM-t. The classical Studet-t cofidece iterval has the highest width. Both the proposed MAAD-t ad MADM-t 5 Histogram of Mosquito Survival Rates Mosquito Survival Rates Figure 8. Histogram of mosquito survival rates data TABLE VIII. THE 95% CONFIDENCE INTERVALS FOR MOSQUITO SURVIVAL RATES DATA Method Cofidece Iterval Width Studet-t (- 0.03, 0.288) 0.39 AADM-t (0.00, 0.247) MAAD-t (0.029, 0.227) 0.98 MADM-t (0.05, 0.5) has the shorter widths compared to the correspodig AADM-t. Here is small ad data is highly skewed. So, the MADM-t cofidece iterval performs the best i the sese of havig smaller width tha the other two proposed cofidece itervals.

12 Frequecy 2 M. Abu-Shawiesh, et. al.: Cofidece Itervals based o Absolute Deviatio for Populatio C. Example-3 The percetage of adults livig with HIV- for 5 regios of the world were reported [3], the followig data represet the HIV- prevalece rate for each regio: 0.6, 2.3, 0.6, 0.3, 0.7, 0.9, 0.3, 0., 0.2, 0.3, 4.5, 5.7, 4.4, 4.8, 7. We wat to fid the average percetage of disorders for a regio. The percetage of adults livig with HIV- is positively skewed with skewess = 2.67 ad mea is A histogram of the data values showig its positive skewess is give i Fig.9. The proposed cofidece itervals ad their correspodig widths have bee give i Table IX. From the Table IX, we observe that the MADM-t cofidece iterval has the smallest width followed by MAAM-t ad AADM-t. The classical Studet-t cofidece iterval has the highest width. Both the proposed MAAD-t ad MADM-t has the shorter widths compared to the correspodig AADM-t. All the cofidece itervals have approximately short width. Also the Studet-t ad the AADM-t cofidece itervals have approximately equal widths. Thus the MADM-t cofidece iterval performs the best i the sese of havig smaller width tha the other two proposed cofidece itervals. 5. SUMMARY AND CONCLUDING REMARKS This paper proposes a umber of cofidece itervals amely, the AADM-t, the MAAD-t ad the MADM-t, which are simple adjustmets to the classical Studet s-t cofidece iterval ad based o the absolute deviatio for estimatig μ of a positively skewed distributio. The proposed methods are very easy to calculate ad are ot overly computer-itesive. The simulatio study shows that the best cofidece iterval based o coverage probability for Histogram of HIV HIV- 2 6 Figure 9: Histogram of HIV- prevalece data TABLE IX. THE 95% CONFIDENCE INTERVALS FOR HIV- PREVALENCE DATA Method Cofidece Iterval Width Studet-t (0.49, 5.28) AADM-t (.29, 4.57) MAAD-t (.604, 4.096) MADM-t (2.573, 3.27) moderately to highly skewed data is the AADM-t followed by MAAD-t ad MADM-t. The best cofidece iterval based o width for moderately to highly skewed data is the MADM-t followed by MAAD-t ad AADM-t. Therefore, the practitioers should decide whether coverage probability or width is importat to their study to choose a cofidece iterval because it is hard to fid a cofidece iterval that will have high coverage probability ad a small width. It is also evidet from the simulatio study that the large sample sizes, the classical Studet-t are iadequate for highly skewed data. Three real life umerical examples are aalyzed which supported our results to some extet. I geeral, the proposed cofidece itervals performed well i the sese that they improved their respective cofidece itervals i terms of either coverage probability or width. Fially, the proposed iterval estimatio methods performed well compared to the classical Studet-t cofidece iterval.

13 It. J. Comp. Theo. Stat. 5, No., -3 (May-208) 3 ACKNOWLEDGEMENTS Authors are thakful to the referee ad the Editor for their variable commets ad suggestios, which improved the presetatio of the paper. This article was partially completed while author, B. M. Golam Kibria was o sabbatical leave (Fall 207). He is grateful to Florida Iteratioal Uiversity for awardig the sabbatical leave which gave him excellet research facilities. REFERENCES [] Studet, The probable error of a mea, Biometrika, pp.-25, 908. [2] D. Boos, ad J. Hughes-Oliver, How large does have to be for z ad t itervals, America Statisticia, pp.2 28, [3] W. Shi, ad B.M.G. Kibria, O some cofidece itervals for estimatig the mea of a skewed populatio, Iteratioal Joural of Mathematical Educatio ad Techology, pp.42 42, [4] F.K. Wag, Cofidece iterval for a mea of o-ormal data. Quality ad Reliability Egieerig Iteratioal, pp , 200. [5] X.H. Zhou, ad P. Dih, Noparametric cofidece itervals for the oe- ad two-sampleproblems, Biostatistics, pp , [6] J. L. Gastwirth, Statistical properties of a measure of tax assessmet uiformity, Joural of Statistical Plaig Iferece, pp. -2, 982. [7] C. Wu, Y, Zaho, Z. Wag, The media absolute deviatios ad their applicatios to Shewhart X cotrol charts, Commuicatios i Statistics-Simulatio ad Computatio, pp , [8] F.R. Hampel, The ifluece curve ad its role i robust estimatio, Joural of the America Statistical Associatio, pp , 974. [9] B.M.G. Kibria, ad S. Baik, Parametric ad oparametric cofidece itervals for estimatig the differece of meas of two skewed populatios, Joural of Applied Statistics. pp , 203. [0] S. Baik, ad B.M.G. Kibria, Estimatig the populatio stadard deviatio with cofidece iterval: A simulatio study uder skewed ad symmetric coditios, Iteratioal Joural of Statistics i Medical Research, pp , 204. [] R.E. Johso, ad B.H. McFarlad, Atipsychotic drug exposure i a health maiteace orgaizatio, Medical Care, pp , 993. [2] J.D. Charlwood, M.H. Birley, H. Dagoro, R. Paru, ad P.R. Holmes, Assessig survival rates of aopheles farauti (Diptera: Culicidae) from Papua New Guiea, Joural of Aimal Ecology, pp , 985. [3] J. Hemelaar, E. Gouws, P.D. Ghys, ad S. Osmaov, Global ad regioal distributio of HIV- geetics subtypes ad recombiats i 2004, AIDS, pp.w3 W23, 2006.

Chapter 8. Confidence Interval Estimation. Copyright 2015, 2012, 2009 Pearson Education, Inc. Chapter 8, Slide 1

Chapter 8. Confidence Interval Estimation. Copyright 2015, 2012, 2009 Pearson Education, Inc. Chapter 8, Slide 1 Chapter 8 Cofidece Iterval Estimatio Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 1 Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for