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Shape parameter
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<h2 id="estimation">Estimation</h2> <p>Many <a href="/facts/Estimators/CbkjcKvN">estimators</a> measure location or scale; however, estimators for shape parameters also exist. Most simply, they can be estimated in terms of the higher <a href="/facts/Moment_(mathematics)/GL7vJ1nb">moments</a>, using the <a href="/facts/Method_of_moments_(statistics)/Qfso3fMe">method of moments</a>, as in the <i><a href="/facts/Skewness/rdS8luFa">skewness</a></i> (3rd moment) or <i><a href="/facts/Kurtosis/yqGgry89">kurtosis</a></i> (4th moment), if the higher moments are defined and finite. Estimators of shape often involve <a href="/facts/Higher-order_statistics/pdPLtQBA">higher-order statistics</a> (non-linear functions of the data), as in the higher moments, but linear estimators also exist, such as the <a href="/facts/L-moment/xJuzASyy">L-moments</a>. <a href="/facts/Maximum_likelihood/0Yq2dpQD">Maximum likelihood</a> estimation can also be used. </p> <h2 id="examples">Examples</h2> <p>The following continuous probability distributions have a shape parameter: </p> <ul><li><a href="/facts/Beta_distribution/DbJk4eeV">Beta distribution</a></li> <li><a href="/facts/Burr_distribution/75pvKu6P">Burr distribution</a></li> <li><a href="/facts/Dagum_distribution/rmgoUXvM">Dagum distribution</a></li> <li><a href="/facts/Erlang_distribution/dqFqlxJg">Erlang distribution</a></li> <li><a href="/facts/ExGaussian_distribution/mJU9306M">ExGaussian distribution</a></li> <li><a href="/facts/Exponential_power_distribution/DTLkhMhQ">Exponential power distribution</a></li> <li><a href="/facts/Fr%25C3%25A9chet_distribution/B3p3hQB2">Fréchet distribution</a></li> <li><a href="/facts/Gamma_distribution/lczcdJmw">Gamma distribution</a></li> <li><a href="/facts/Generalized_extreme_value_distribution/hDG41s9y">Generalized extreme value distribution</a></li> <li><a href="/facts/Log-logistic_distribution/ze7Pasb3">Log-logistic distribution</a></li> <li><a href="/facts/Log-t_distribution/0JSwnTYP">Log-t distribution</a></li> <li><a href="/facts/Inverse-gamma_distribution/zpyktNyN">Inverse-gamma distribution</a></li> <li><a href="/facts/Inverse_Gaussian_distribution/mUPsu4EJ">Inverse Gaussian distribution</a></li> <li><a href="/facts/Pareto_distribution/PgMDbLrR">Pareto distribution</a></li> <li><a href="/facts/Pearson_distribution/L0oULaWK">Pearson distribution</a></li> <li><a href="/facts/Skew_normal_distribution/TcAzDMQW">Skew normal distribution</a></li> <li><a href="/facts/Lognormal_distribution/V2RXmmlE">Lognormal distribution</a></li> <li><a href="/facts/Student_t-distribution/DeT1SDqH">Student's t-distribution</a></li> <li><a href="/facts/Tukey_lambda_distribution/mFjgYang">Tukey lambda distribution</a></li> <li><a href="/facts/Weibull_distribution/1fEWxbKl">Weibull distribution</a></li></ul> <p>By contrast, the following continuous distributions do <i>not</i> have a shape parameter, so their shape is fixed and only their location or their scale or both can change. It follows that (where they exist) the <a href="/facts/Skewness/rdS8luFa">skewness</a> and <a href="/facts/Kurtosis/yqGgry89">kurtosis</a> of these distribution are constants, as skewness and kurtosis are independent of location and scale parameters. </p> <ul><li><a href="/facts/Exponential_distribution/yY3EdV0u">Exponential distribution</a></li> <li><a href="/facts/Cauchy_distribution/3HS6D3Xk">Cauchy distribution</a></li> <li><a href="/facts/Logistic_distribution/QlQNt0uC">Logistic distribution</a></li> <li><a href="/facts/Normal_distribution/UapjjPyQ">Normal distribution</a></li> <li><a href="/facts/Raised_cosine_distribution/uceLfxdC">Raised cosine distribution</a></li> <li><a href="/facts/Continuous_uniform_distribution/XbnlVljT">Uniform distribution</a></li> <li><a href="/facts/Wigner_semicircle_distribution/d10fufTU">Wigner semicircle distribution</a></li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Skewness/rdS8luFa">Skewness</a></li> <li><a href="/facts/Kurtosis/yqGgry89">Kurtosis</a></li> <li><a href="/facts/Location_parameter/UQTyS3JU">Location parameter</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Ekawati, Dian; Warsono; Kurniasari, Dian (December 2014). "On the Moments, Cumulants, and Characteristic Function of the Log-Logistic Distribution" (PDF). The Journal for Technology and Science. 25. <a href="http://repository.lppm.unila.ac.id/120/1/23%20On%20the%20Moments,%20Cumulants,%20and%20Characteristic%20Function%20of%20the%20Log-Logistic%20Distribution.pdf" target="_blank">http://repository.lppm.unila.ac.id/120/1/23%20On%20the%20Moments,%20Cumulants,%20and%20Characteristic%20Function%20of%20the%20Log-Logistic%20Distribution.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Everitt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. ISBN 0-521-81099-X <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Birnbaum, Z. W. (1948). "On Random Variables with Comparable Peakedness". The Annals of Mathematical Statistics. 19 (1). Institute of Mathematical Statistics: 76–81. doi:10.1214/aoms/1177730293. ISSN 0003-4851. <a href="https://doi.org/10.1214%2Faoms%2F1177730293" target="_blank">https://doi.org/10.1214%2Faoms%2F1177730293</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>